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or considering the implementation of, ... June 27, 2011 TRB Transportation Research E-Circular E-C154: Development of Warranty Programs for Hot-Mix Asphalt is a synopsis of current information for the development of warranty programs for hot-mix asphalt pavements. The opinions expressed in this e-circular are those of the document's author and do not necessarily reflect the views of the Transportation Research Board, the National Academies. July 06, 2011 TRB Transportation Research E-Circular E-C153: Dynamic Traffic Assignment: A Primer is designed to help explain the basic concepts and definitions of dynamic traffic assignment (DTA) models and addresses the application, selection, planning, and execution of a DTA model. The report also describes the general DTA modeling procedure and modeling issues that may concern a model user. June 22, 2011 TRB Transportation Research E-Circular E-C152: Adapting Transportation to the Impacts of Climate Change: State of the Practice 2011 focuses on transportation adaptation practices that can be implemented to yield potential benefits now and in the longer term. The document highlights what climate change adaptation means for the transportation industry. E-C152 was produced under the auspices of ... July 22, 2011 TRB’s Transportation Research E-Circular E-C151: Modeling Operating Speed is a synthesis of existing operating speed models developed in different regions of the world. The models are grouped according to roadway type. Limitations and deficiencies in existing operating speed models and suggestions for future work are also identified. Practitioner perspectives on the potential use of speed predicti... June 16, 2011 TRB Transportation Research E-Circular E-C 150: Advancing Regional Transportation Operations: A National Workshop is a summary of the December 2008 workshop held in Washington, D.C. The purpose of the workshop was to advance the state of the practice in regional approaches to managing and operating the transportation system in a multiagency, multimodal, and cross-functional manner
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with the goal of... July 11, 2011 TRB Transportation Research Electronic Circular E-C149: 75 Years of the Fundamental Diagram for Traffic Flow Theory: Greenshields Symposium includes the papers that were presented at a July 2008 conference that explored traffic flow theory’s history, recent developments, technological impacts, current status, and potential research needs. March 31, 2011 TRB Transportation Research E-Circular E-C148: Critical Issues in Aviation and the Environment explores the environmental media affected by aviation activities and processes that link aviation and the environment. The circular consists of nine individually authored sections representing the authors' expert opinions on these issues. E-C148 updates and expands upon previous circulars while maintaini... January 08, 2011 TRB’s Transportation Research Circular E-C147: Development in Asphalt Binder Specifications includes papers that explore test methods and specifications for high-temperature permanent deformation, fatigue cracking, and low-temperature cracking. December 22, 2010 TRB Electronic Circular E-C146: Trucking 101: An Industry Primer is designed to provide a basic picture of the structure of the U.S. trucking industry for public officials, policy makers, engineers, administrators, planners, academic researchers, journalists, and anyone who needs to think about issues affecting, or affected by, trucking. The report emphasizes that many of the distinct parts of the... August 18, 2010 TRB’s Transportation Research Circular E-C145: Joint International Light Rail Conference--Growth and Renewal includes 24 peer-reviewed research papers that were presented as part of the April 2009 Joint International Light Rail Conference. Collectively, the papers provide up-to-date information on planning, design, construction, maintenance, and operations involved in running a light rail sys... April 16, 2010 TRB’s Transportation Research Circular E-C144: Research Needs Statements for Climate Change and Transportation includes a series of specific research needs statements on climate change and transportation designed as a resource to be used by universities, students, research organizations, government agencies, and other interested
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parties as they contemplate and fund research in this area. April 12, 2010 TRB Transportation Research Circular E-C143: Modal Primer on Greenhouse Gas and Energy Issues for the Transportation Industry is primer designed to provide transportation decision makers with an inclusive, educated, and objective overview of the current state of the transportation industry from a greenhouse gas and energy standpoint. May 10, 2010 TRB’s Transportation Research Circular E-C142: Methodology for the Development and Inclusion of Crash Modification Factors in the First Edition of the Highway Safety Manual explores the literature review procedure and inclusion process related to development of Part D—Crash Modification Factors of the Highway Safety Manual (HSM), which is expected to be released later this year. The development of... November 12, 2009 TRB Transportation Research Circular E-C141: Colorado's Full-Scale Field Testing of Rockfall Attenuator Systems explores the durability and performance of several currently available mesh and cable net products that can be used to construct rockfall attenuator systems, for which no design guidelines exist. October 12, 2009 TRB Transportation Research Circular E-C140: A Review of the Fundamentals of Asphalt Oxidation: Chemical, Physicochemical, Physical Property, and Durability Relationships explores the current physicochemical understanding of the chemistry, kinetics, and mechanisms of asphalt oxidation and its influence on asphalt durability. October 09, 2009 TRB’s Transportation Research Circular E-C139: Use of Accelerated Pavement Testing to Evaluate Maintenance and Pavement Preservation Treatments explores experiences where accelerated pavement testing (APT) has been successfully used for the assessment of pavement preservation and maintenance strategies. The goal of the report is to motivate future APT research for evaluation of maintenance and pav... September 17, 2009 TRB’s Transportation Research Circular E-C138: Critical Issues in Aviation and the Environment 2009 explores the major environmental media affected by aviation activities and the key processes that link aviation and the
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environment, including climate change, alternative fuels, and sustainability. The circular focuses on the state of science, rather than on policy, and the authors of various portio... May 19, 2009 TRB’s Transportation Research Circular E-C137: Glossary of Highway Quality Assurance Terms contains terms of common usage and accepted practice for terminology used in the highway quality assurance field. The report is designed as a reference document. June 18, 2009 TRB Transportation Research Circular E-C136: Implementation Status for Geotechnical LRFD in State DOTs explores various states’ experiences in the practical implementation of Load and Resistance Factor Design (LRFD) in geotechnical and substructure design. July 11, 2009 TRB’s Transportation Research Circular E-C135 Maintenance Management includes papers presented at the 12th American Association of State Highway and Transportation Officials–TRB Maintenance Management Conference held in Annapolis, Maryland, July 19-23, 2009. The objective of this series of conferences is to provide a forum every 3 to 4 years for the exchange of new ideas and developments in the ma... September 11, 2009 TRB’s Transportation Research Circular E-C134: Influence of Roadway Surface Discontinuities on Safety is designed to help highway engineers evaluate roadway maintenance guidelines and priorities. The report addresses safety issues related to roadway roughness, holes, and bumps; the positive effects of road surface discontinuities; pavement edges; friction variations; water accumulations; surface c... April 23, 2009 TRB’s Transportation Research Circular E-C133: Glossary of Regional Transportation Systems Management and Operations Terms is designed to provide clear definitions of terms as they are typically used in the context of regional transportation systems management and operations. June 28, 2009 TRB’s Transportation Research Circular E-C132: Young Impaired Drivers provides overview of the information presented and discussions among the participants at a June 3-4, 2008, workshop that explored the risks posed by young impaired drivers and how these risks might be
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ameliorated. Background papers prepared for the workshop are also included. The workshop included perspectives on the issue... December 03, 2008 Transportation Research Circular E-C131, Transportation Asset Management: Strategic Workshop for Department of Transportation Executives summarizes a workshop held in Washington, D.C., in December 2006 that explored advancements in asset management that have the potential to lead to growth in asset management applications in the United States. October 16, 2008 TRB’s Transportation Research Circular E-C130: Geophysical Methods Commonly Employed for Geotechnical Site Characterization explores how geotechnical, geophysical tools are used to image or characterize the shallow subsurface of the Earth, typically to depths of less than several hundred feet. October 08, 2008 TRB’s Transportation Research Circular E-C129: Use of Inclinometers for Geotechnical Instrumentation on Transportation Projects explores the state of the practice and representative applications on the use of inclinometer systems for measuring ground deformation and performance of geotechnical design elements on transportation projects. The report examines inclinometer components and installation ... October 27, 2008 TRB’s Transportation Research Circular E-C128 includes papers that were presented at the 10th International Conference on Bridge and Structure Management held on October 20-22, 2008, in Buffalo, New York. The conference brought together practitioners, administrators, and researchers from around the world to exchange information on the development, implementation, and utilization of effective... February 19, 2008 TRB Transportation Research Circular E-C127: Implementation of an Airport Pavement Management System explores the role and data requirements for an Airport Pavement Management System (APMS). The report also examines the benefits, costs, and common challenges associated with the implementation of an APMS. June 20, 2008 TRB Transportation Research Circular E-C126: Surface Transportation Weather and Snow Removal and Ice Control Technology includes papers that were presented at the 4th National Conference on Surface Transportation Weather (June 16–17, 2008), and the 7th International
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Symposium on Snow and Ice Control (June 17–19, 2008), which were both held in Indianapolis, Indiana. The E-Circular includes pa... March 01, 2008 TRB’s Transportation Research Circular E-C125: Evolving Role of Statewide Transportation Planning in an Era of Regional Funding and Governance reports on a July 7-8, 2006 peer exchange in La Jolla, California that explored this topic. December 07, 2007 TRB’s Transportation Research Circular E-C124: Practical Approaches to Hot-Mix Asphalt Mix Design and Production Quality Control Testing explores the state-of-the-practice for Superpave mix design and specification, including a summary of changes made to the Superpave mix design procedure as a result of National Cooperative Highway Research Program research findings, American Association of State ... November 26, 2007 TRB’s Transportation Research Circular E-C123, Traffic Safety and Alcohol Regulation: A Symposium provides an overview of the information presented at a June 5-6, 2006, symposium that examined the role of alcohol regulation in traffic safety. The report explores discussions among symposium participants and includes the background papers prepared for the workshop. November 07, 2007 Transportation Research Circular E-C122, Asphalt Emulsion Technology: Review of Asphalt Emulsion Residue Procedures, explores methods used worldwide for the recovery of asphalt emulsion residue and examines new avenues for research and practice on the issue. In January 2014, TRB published Transportation Research Circular E-C182 : Progress Toward Performance-Graded Emulsified Asphalt Specifications... August 31, 2007 Transportation Research Circular E-C121, Information Assets to Support Transportation Decision Making: Report of a Peer Exchange of State Transportation Organizations, summarizes discussions at an April 17-18, 2007, peer exchange of state department of transportation officials and other professionals that focused on data and information uses, management strategies, needs, and gaps in their organiz... August 24, 2007 Transportation Research Circular E-C120, Traffic Monitoring Data: Successful Strategies in Collection and Analysis summarizes sessions that took place
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as part of a May 2007 workshop in Washington, D.C., designed as a forum to exchange knowledge about successful strategies in the collection and the analysis of traffic data. August 19, 2007 TRB’s Transportation Research Circular E-C119: North American Freight Transportation Data Workshop summarizes sessions that took place as part of a May 2007 workshop in Washington, D.C., that allowed data users and providers to discuss freight transportation data from a North American perspective. The workshop highlighted changes in government-supplied data sources and explored data users’ n... July 11, 2007 TRB’s Transportation Research Circular E-C118: Lessons Learned from the AASHO Road Test and Performance of the Interstate Highway Systemincludes six papers that reflect on the performance of Interstate Highway System pavements and lessons learned from the American Association of State Highway Officials (AASHO) road test from 1956 to 1961 in Ottawa, Illinois. May 23, 2007 TRB’s Transportation Research Circular E-C117: The Domain of Truck and Bus Safety Research explores information and perspectives on truck and bus safety research. The report is designed to help establish a knowledge base and help identify potential future activities. April 26, 2007 TRB’s Transportation Research Circular E-C116, Geotechnical Challenges of the Interstate Highway System: The First 50 Years and a Look Ahead, explores what geotechnical engineers have learned since the construction of the Interstate System and what they see as future challenges for the discipline. The circular is based on presentations made during a session at the 2006 TRB 85th Annual Meetin... April 18, 2007 TRB’s Transportation Research Circular E-C115: Challenges of Data for Performance Measures: A Workshop is the proceedings of a July 8, 2006, workshop in San Diego, California. The workshop explored ongoing issues and research directions in data management processes for performance measures. April 09, 2007 TRB’s Transportation Research E-Circular 114: Research
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Opportunities in Radio Frequency Identification Transportation Applications summarizes a conference by the same title that was held October 17 to 18, 2006, in Washington, D.C. This conference focused on current and future research on radio frequency identification (RFID) technologies in transportation applications, which include the tagg... January 29, 2007 TRB has released Transportation Research Circular E-C113, Artificial Intelligence in Transportation: Information for Application includes six articles that explore five general artificial intelligence (AI) areas including knowledge-based systems, neural networks, fuzzy sets, genetic algorithms, and agent-based models. The circular is designed to serve as an informational resource for transpo... March 01, 2007 TRB’s Transportation Research Circular E-C112, Joint International Light Rail Conference: A World of Applications and Opportunities is the proceedings of the Joint International Light Rail (LRT) Conference that was held in St. Louis, Missouri, on April 9-11, 2006. The conference focused on planning and urban integration, vehicle design and innovation, infrastructure use, security and fare en... January 10, 2007 TRB’s Transportation Research Circular E-C111, Integrating Roadway, Traffic, and Crash Data: A Peer Exchange is the proceedings of a November 1-2, 2006, meeting held in Washington, D.C. The peer exchange was designed to bring both data and highway safety professionals together to share experiences and identify key issues relating to integrating roadway, traffic, and crash data sources used t... January 17, 2007 TRB's Transportation Research Circular E-C110: Geometric Design Strategic Research documents efforts leading up to and resulting from the Strategic Geometric Design Research Needs Workshop held in Williamsburg, Virginia, in July 2004. The report also contains research problem statements for possible use as a long-range geometric design research program by organizations such as the American Associa... December 04, 2006 TRB’s Transportation Research Circular E-C109: Transportation Information Assets and Impacts: An Assessment of Needs explores the relationship between data, as
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the raw material, and information as the processed, useful product supporting decisions. The report also examines the value of information in specific real decisions and outlines an ongoing process designed to help ensure that transpo... October 19, 2006 TRB’s Transportation Research Circular E-C108: Geospatial Information Technologies for Asset Management is the proceedings of an October 30-31, 2005, peer exchange held in Kansas City, Missouri. The peer exchange focused on moving spatial technology applications to the next level by managing change, data integration, and communication. Participants at the exchange identified research t... October 17, 2006 TRB’s Transportation Research Circular E-C107: Control of Cracking in Concrete: State of the Art explores causes of cracking, testing, and ways of minimizing strains and stresses that can cause cracking in transportation structures such as bridge structures, pavements, and footings. November 15, 2006 TRB Transportation Research Circular E-C106: Environmental Geospatial Information for Transportation summarizes a May 3-4, 2006, peer exchange that took place in Washington, D.C. The peer exchange highlighted several examples of successful collaboration, data sharing, and the effective use of environmental geospatial information systems data in transportation planning. The report focus... September 25, 2006 TRB Transportation Research Circular E-C105: Factors Affecting Compaction of Asphalt Pavements includes invited papers that were presented at a January 2005 workshop that explored asphalt practitioners’ concerns related to specifying and achieving density during hot-mix asphalt (HMA) pavement construction. Issues covered by the papers include optimizing HMA construction temperatures, advance... October 06, 2006 TRB’s Transportation Research Circular E-C104, Fifty Years of Interstate Structures: Past, Present, and Future documents some of the key legislation, specifications development, and advances in research, design, and construction for bridges and other highway structures since the signing of the Federal-Aid Highway Act of 1956. September 18, 2006 TRB’s Transportation Research Circular E-C103, Concrete: The Sustainable Infrastructure
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Material for the 21st Century examines some of the research innovations in the 20th century that led to advancements in concrete. The circular also explores what the future may hold as a result of the continuing advancements in high-performance durable concrete. August 21, 2006 TRB Transportation Research Circular E-C102: Asphalt Emulsion Technology includes four papers designed to serve as an overview of the chemistry, production, quality assurance testing, and application of bituminous emulsions. The papers were produced following a technical session at the 2005 TRB 84th Annual Meeting on bituminous emulsions. In January 2014, TRB published Transportation Researc... August 16, 2006 TRB’s Transportation Research Circular E-C101, Driver Education: The Path Ahead includes a series of papers that were presented at a workshop held on September 12–13, 2005, in Washington, D.C. Issues covered by papers in the circular include novice drivers, content of driver education, instructional methods for young drivers, student competency measures, novice driver training effectiv... July 12, 2006 TRB's 100th E-Circular: Linking Transportation and Land Use explores the results from the Land Use Peer Exchange during the summer meeting of the TRB Ports, Waterways, Freight, and International Trade Conference and the Joint Meeting of the TRB Planning, Data, Finance, Administration, Freight, and Management Committees in Boston, Massachusetts, on July 12–13, 2005. The purpose of the peer exchange... June 16, 2006 TRB’s Transportation Research Circular E-C099, Statewide Transportation Planning: Making Connections includes the presentations, resource papers, and summaries of views expressed by conference speakers, panelists, and participants at a May 18–20, 2003, conference held in Duck Key, Florida. The presentations, breakout sessions, and informal discussions held during the conference were oriented... July 24, 2006 TRB's Transportation Research Circular E-C098, Maintenance Management 2006: Presentations from the 11th AASHTO–TRB Maintenance Management Conference, contains papers presented at the 11th AASHTO–TRB Maintenance Management Conference
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held in Charleston, South Carolina, July 16-20, 2006. The report includes papers on outsourcing, pavements, roadside, winter operations, bridges, maintenance managemen... July 12, 2006 TRB’s Transportation Research Circular E-C097, International Perspectives on Urban Street Design: Proceedings of the Context-Sensitive Design Workshop includes four papers that were presented during the 2005 TRB 84th Annual Meeting. The papers explore street design in small European towns; context-sensitive design aspects of arterial streets in Berlin, Germany; and issues relative to histori... June 02, 2006 TRB’s Transportation Research Circular E-C096, Drugs and Traffic: A Symposium provides an overview of the information presented and the discussions among the participants, as well as the background papers prepared for a June 20-21, 2005, symposium held in Woods Hole, Massachusetts. The goal of the symposium was to synthesize and summarize available information on the role of drugs in traffic... April 18, 2006 TRB’s Transportation Research Circular E-C095, Operations Data for Planning Applications: Identifying Needs, Opportunities, and Best Practices summarizes a May 4, 2005, Washington, DC, peer exchange that focused on opportunities to improve the linkages between transportation planning and operations. The report includes a summary of questions addressed by participants about the relationship b... April 13, 2006 TRB’s Transportation Research Circular E-C094, Safety Data Analysis and Evaluation: Research Problem Statements contains research problem statements produced by members and friends of TRB’s Safety Data, Research, and Analysis Committee (ANB20). The committee is concerned with methods of gathering, storing, and, in particular, using the transportation safety data for informed decision making.... April 03, 2006 TRB’s Transportation Research Circular E-C093: 6th National Conference on Transportation Asset Management summarizes the content of the sessions and presentations from the November 1–3, 2005, conference in Kansas City, Missouri. The circular includes a summary of each conference session as well as summaries of the individual topics
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included in the session. The circular is intended as a... February 10, 2006 TRB’s Transportation Research Circular E-C092, Maintenance and Operations of Transportation Facilities: 2005 Strategic Vision explores the major trends that affect maintenance; cites current and emerging innovations in management systems, technology, and intelligent transportation systems; and examines some of the key maintenance challenges of this century. February 24, 2006 TRB Transportation Research Circular E-C091, Innovations in Statewide Planning: A Peer Exchange summarizes a July 2005 peer exchange on innovations in statewide planning. January 19, 2006 TRB Transportation Research Circular E-C090, Design–Build: A Quality Process summarizes an all-day workshop that took place at the 2005 TRB 84th Annual Meeting. The workshop focused on achieving quality on design-build projects from the preparation of the request for proposals, to the actual design, to the use of incentives to achieve desired levels of quality, and to final construction. January 17, 2006 TRB’s Transportation Research Circular E-C089: Critical Issues in Aviation and the Environment 2005 updates a 2004 summary of critical issues in aviation and the environment. The 2005 report identifies priority research that may yield potential benefits to environmental media affected by aviation activities (noise, air quality, and water quality) and processes that link aviation and the envi... January 19, 2006 TRB Conference Proceedings E-C088: Commodity Flow Survey Conference is the proceeding from a conference held in on July 8-9, 2005, in Boston, Massachusetts. The conference examined the commodity flow survey (CFS), reviewed data sources being used to supplement information from the CFS, and explored ideas to improve future iterations of the CFS. January 11, 2006 TRB Electronic Circular E-C087: Rubblization of Portland Cement Concrete Pavements includes non-peer reviewed papers that were written based on presentations made during sessions on rubblization of portland cement concrete pavements at the TRB 84 Annual Meeting
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in January 2005. Hot-mix asphalt (HMA) overlays can also be considered for application to an existing, deteriorated portland cement ... December 06, 2005 TRB Electronic Circular E-C086: Evaluation of Chemical Stabilizers: State-of-the-Practice Report is designed to provide information on practices that agencies have found to be successful and tp provide a reasonable degree of uniformity and standardization in the evaluation of chemical stabilizers used in soil stabilization. The circular is intended to provide the potential users of any chemical st... January 19, 2006 TRB Transportation Research Circular E-C085, Railroad Operational Safety: Status and Research Needs summarizes the proceedings of the Midyear Meeting of the Transportation Research Board’s (TRB) Railroad Operational Safety Subcommittee held September 10–12, 2002, in Irvine, California, which examined human factors-related research issues facing the railroad enterprise. The report includes research... January 06, 2006 TRB Research Circular E-C084 presents the top 16 pedestrian research problem statements, prioritized from a list of approximately 80 research problem statements by TRB’s Technical Activities Division Committee on Pedestrians. January 14, 2006 TRB Transportation Research Circular E-C083: National Roundabout Conference: 2005 Proceedings includes presentations made during the May 22-25, 2005, conference in Vail, Colorado. Issues examined during the conference included the range of settings where roundabouts may be used, design elements and criteria, and methods of estimating safety and operations impacts. The conference also e... January 18, 2006 TRB Transportation Research Circular E-C082, Statewide Multimodal Transportation Planning Proceedings: 2004 Peer Exchange focuses on cost estimating for transportation planning and incorporating safety into the transportation planning process. The report provides an overview of the presentation and roundtable discussions that took place during the peer exchange on July 27-28, 2004, in Park C... January 13, 2006 TRB Research Circular E-C081: A Research Program for Improvement of the Highway Capacity Manual describes a research program designed
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to improve the Highway Capacity Manual (HCM). The proposed program was developed by the members of the TRB Highway Capacity and Quality of Service Committee with the assistance of feedback from users of the HCM. The program includes 38 research studies that wo... December 02, 2005 TRB Electronic Circular E-C080: Freight Data for State Transportation Agencies documents a peer exchange that was held July 11, 2005, in Boston, Massachusetts. At the peer exchange, participants identified major challenges and issues relating to freight data. Some of those challenges and issues include the determination of data and information needs, logistics and supply chain analysis, frei... September 19, 2005 TRB Electronic Circular E-C079: Calibration to Determine Load and Resistance Factors for Geotechnical and Structural Design explores how load and resistance factors are developed. The circular includes information on how to estimate load and resistance factors where adjustment of these factors is justified based on local experience and data. The circular also includes criteria for docu... October 31, 2005 TRB Electronic Circular E-C078 Roadway Pavement Preservation 2005 contains papers written for the TRB First National Conference on Roadway Preservation held in Kansas City, Missouri, on October 31-November 1, 2005. The conference addressed all aspects of successfully implemented roadway pavement preservation activities, including management, engineering, economics, the establishment of strat... August 04, 2005 TRB’s Electronic Circular 77--Enhancing the Value of Data Programs: A Peer Exchange contains the proceedings of a peer exchange that took place on July 23-24, 2001, in Vail, Colorado. The purpose of the exchange was to share ideas on raising the awareness of data programs and ensuring that data add value to the operation of state departments of transportation. The forum examined what s... June 14, 2005 TRB Electronic Circular E-C076: Asset Management in Planning and Operations--A Peer Exchange summarizes a September
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7–8, 2004, peer exchange that focused on the expanding role of asset management in planning and operations as a comprehensive approach to managing agency resources and transportation systems. The peer exchange was designed to gather additional information about the state of the... September 01, 2005 TRB Electronic Circular E-C075: Statewide Travel Demand Modeling—A Peer Exchange includes presentations made during the September 23–24, 2004, peer exchange and includes comparison of travel demand modeling efforts taking place in select states. May 19, 2005 TRB Electronic Circular E-C074: Glossary of Highway Quality Assurance Terms—Third Update contains terms of common usage and accepted practice in the area of highway quality assurance (QA). The report is designed as a reference document that contains common usage of highway QA terminology. May 17, 2005 TRB Transportation Research Circular E-C073: Performance Measures to Improve Transportation Planning Practice: A Peer Exchange summarizes the results of a peer review on the use of performance measures to improve transportation planning and its relationship to project programming. The one-day peer review focused on how state departments of transportation are using performance measures to imp... January 06, 2005 TRB’s Electronic Circular 72: Implementing Impaired Driving Countermeasures: Putting Research into Action—A Symposium summarizes a symposium that was held on August 21-22, 2003, in Irvine, California. The circular provides an overview of the information presented and discussions among the participants, as well as the background papers prepared for the symposium. January 06, 2005 TRB’s Electronic Circular 71: Data for Understanding Our Nation’s Travel: National Household Travel Survey Conference summarizes a conference held on November 1-2, 2004, in Washington, D.C. The conference was designed to bring together a diverse set of data users who understand the data’s usefulness in order to provide feedback to inform the design of future national travel behavior surveys.... December 08, 2004
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TRB's E-Circular 70 - Optimizing the Dissemination and Implementation of Research Results: A Summary of Workshop and Midyear Meeting Activities examines the issues and challenges associated with optimizing the dissemination and implementation of research results. This TRB E-Circular also documents the problem exploration process and the potential priority actions from these efforts. August 12, 2004 TRB’s Electronic Circular E-C069: Critical Issues in Aviation and the Environment identifies priority research focused on existing and future environmental impacts of aviation on the environment and the opportunities for mitigating or avoiding those impacts. This is first in what is planned to be an annual summary of critical issues in aviation and the environment to be developed by TRB’s En... September 17, 2004 TRB Circular E-C068: New Simple Performance Test for Asphalt Mixes includes a series of papers that examine practical and reliable laboratory tests that could be considered for ranking the rutting potential of hot-mix asphalt paving mixtures. The circular also includes a literature review on performance tests. August 04, 2004 TRB’s Transportation Research Circular E-C067: Context-Sensitive Design Around the Country: Some Examples describes Context-Sensitive Design (CSD) projects presented during the 2003 TRB 82nd Annual Meeting. CSD is defined as a project development process, including geometric design, that attempts to address safety and efficiency while being responsive to or consistent with the road’s natural... June 22, 2004 TRB Transportation Research E-Circular 66 is the proceedings of the National Metropolitan Planning Organization Peer Exchange that took place on May 20-21, 2003, in Duck Key, Florida. The proceedings include information on near-term planning research topics and needs, short-term needs for transportation planning capacity building, and long-term research topics for the TRB Committee on Metrop... June 15, 2004 TRB’s Electronic Circular E-C065: Transportation Security Education and Training summarizes the transportation security education and training presentations at
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the TRB’s 83rd Annual Meeting, January 11–15, 2004, and emphasizes the important security duties transportation professional’ play in protecting against and reacting to security events. May 07, 2004 TRB’s Transportation Research Circular E-C064 -- Data Requirements in Transportation Reauthorization Legislation: What Is Included and Impacts on the Data Community highlights a TRB conference held on November, 19, 2003, in Washington, DC, that examined data issues associated with the programmatic proposals being considered for surface transportation reauthorization legislation, including new and ... June 30, 2004 TRB Electronic Circular 63: Sixth International Symposium on Snow Removal and Ice Control Technology is the compendium of papers for the June 7-9, 2004, Spokane, Washington symposium. Issues addressed at the symposium included winter weather (information, models, and data quality); winter maintenance (policy, management, and performance); customers’ perspectives on winter operations; environ... March 19, 2004 TRB Transportation Research Circular E-C062: Addressing Fiscal Constraint and Congestion Issues in State Transportation Planning is the proceedings from a peer exchange held on July 14–16, 2002, in Wood’s Hole, Massachusetts. March 18, 2004 TRB Transportation Research Circular E-C061: Data Partnerships: Making Connections for Effective Transportation Planning is the proceedings of a peer exchange held on May 21, 2003, in Duck Key, Florida. The meeting addressed some of the benefits and challenges facing the development of transportation data partnerships. December 16, 2003 TRB Transportation Research Circular EC060: Using Simulation to Evaluate Impacts of Airport Security -- 2003 Simulation Workshop documents the proceedings of a January 12, 2003, workshop. The primary objective of this workshop was to present, demonstrate, and discuss the passenger and baggage screening simulation techniques used to comply with the Transportation Security Administration’s (TSA) con... January 07, 2004 TRB Transportation Research Circular E-C059: Accelerated Highway Construction, Workshop Series Summary summarizes three workshops held in Washington, D.C., Indianapolis, Indiana, and
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Pittsburgh, Pennsylvania, in 2000-2002. The objective of this workshop series was to provide a forum for the exchange of new ideas and developments in the field of accelerated construction. January 13, 2004 TRB Transportation Research Circular E-C058 includes the proceedings of the 9th National Light Rail Transit Conference held November 16-18, 2003, in Portland, Oregon. The ninth national conference focuses on the planning, design, construction, operation, maintenance, and administration of LRT systems. TABLE OF CONTENTS FRONT MATTER Cover 2003 LRT Conference Planning Committee Forward Conten... November 07, 2003 TRB Transportation Research Circular EC057 - The Roadway INFOstructure: What? Why? How? summarizes an August 21-23, 2002, workshop in Irvine, California, which examined the Federal Highway Administration’s vision for a Roadway INFOstructure. The Roadway INFOstructure was designed as the road and highway component of the Integrated Network of Transportation Information. November 05, 2003 TRB Transportation Research Circular E-C056: The Future of MEMS: Microelectromechanical Systems in Transportation Engineering introduces microelectromechanical systems (MEMS) into transportation engineering by describing the general concept and the issues related to their development. MEMS are a relatively new innovation in the integrated circuit field. Although there are a few developed applicati... September 18, 2003 TRB Transportation Research Circular E-C055: A National Forum on Assessing Historic Significance for Transportation Programs presents the proceedings of a forum held in Washington, D.C., on May 23–25, 1999. August 12, 2003 TRB Transportation Research Circular E-C054: Third National Community Impact Assessment Conference includes the proceedings of the conference, which was held in Madison, Wisconsin, on August 19-21, 2002. Community Impact Assessment is an iterative process of understanding potential impacts of proposed transportation activities on affected communities and their sub-populations throughout transporta... July 14, 2003 TRB Transportation Research Circular EC053: 9th International Bridge Management Conference Supplement includes two papers that cover Dynamic Load Tests in
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Bridge Management and Hybrid Knowledge Representation in Bridge Management Systems. This circular is a supplement to Circular E-C049 ( . Papers in both Circulars were presented at the 9th Internati... July 18, 2003 TRB Transportation Research Circular EC052: Maintenance Management 2003 contains papers presented at the Tenth AASHTO-TRB Maintenance Management Conference held in Duluth, Minnesota, July 13-17, 2003. The objective of this series of conferences is to provide a forum every three to four years for the exchange of new ideas and developments in the maintenance and operations management of transportati... May 15, 2003 TRB Transportation Research Circular E-C051: Future Aviation Activities - 12th International workshop summarizes a September 18-20, 2002 Transportation Research Board and Federal Aviation Administration workshop on forecasting long-term trends and developments in commercial, business, and personal air transport. May 22, 2003 TRB Transportation Research Circular EC050: Transportation and Economic Development 2002 summarizes presentations from a May 5-7, 2002, conference that offered transportation, economic development, and planning professionals a broader understanding of the most timely and important issues linking transportation and economic development. April 01, 2003 TRB Transportation Research Circular E-C049 contains papers on bridge management concepts, strategies and health indices, asset management, joints, coatings and concrete repair, life - cycle costs, load testing, utilizing the Internet and performance measures, deterioration and reliability, future directions and challenges in bridge management systems, management system implementation, safety and ... January 08, 2003 TRB Transportation Research Circular E-C048: Freight Transportation Research Needs Statements identifies critical issues in freight transportation as a resource for researchers in the freight area and serves as a seedbed for further discussion and analysis from a wider cross-section of freight practitioners. December 05, 2002 TRB Transportation Research Circular E-C047: Financial Aspects of Equipment Information includes assistance on analyzing various financial acquisition choices and their impacts on
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budget, total costs, and fleet upgrade factors. November 14, 2002 TRB Transportation Research Circular E-C046: Using Spatial Data, Tools, and Technologies to Improve Program Delivery chronicles a peer exchange that focused on optimum use of spatial data, tools, and information to facilitate decision making and to deliver multimodal transportation programs. October 18, 2002 TRB Transportation Research Circular E-C045: 2002 California HOV Summit Proceedings The proceedings from TRB's May 3, 2002, California HOV Summit, which examined the latest events and research findings surrounding high-occupancy vehicle lanes, are now available in . October 31, 2002 TRB Transportation Research Circular Number E-C044: Bailey Method for Gradation Selection in Hot-Mix Asphalt Mixture Design includes a synopsis of current information on a systematic approach to selecting and adjusting aggregate gradation in hot-mix asphalt design. September 01, 2002 TRB Transportation Research Circular E-C043: Significance of Restricted Zone in Superpave Aggregate Gradation Specification presents the historical basis of the restricted zone in hot-mix asphalt (HMA) mixes and summarizes the published research conducted on the restricted zone to determine its significance within the Superpave gradation specification. A characteristic hump in a mix gradation with... August 29, 2002 TRB Transportation Research E-Circular Number E-C042: Airport - Airspace Simulations: A New Outlook documents proceedings from a workshop produced by TRB's Committee on Airspace and Airfield Capacity and Delay. July 31, 2002 TRB Transportation Research E-Circular 41: Supporting the Establishment of Safety Transportation Networks includes an introduction to safety-conscious planning and a toolkit to organize and conduct a safety-conscious planning forum. August 08, 2002 TRB Transportation Research E-Circular E-C040: Aviation Demand Forecasting -- A Survey of Methodologies includes examples of the diversity of techniques used in forecasting aviation system demand and market analyses. September 01, 2002 TRB Transportation Research Circular E-C039: Conference on Transportation Improvements: Experiences Among Tribal, Local, State, and
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Federal Governments is the proceedings from a conference held on October 18–21, 2001 in Albuquerque, New Mexico that focused on the complexity of broad transportation issues of importance for Native American nations. June 10, 2002 TRB Transportation Research Circular E-C038: Standards for Testing, Evaluating, and Locating Roadside Safety Features includes presentations by international experts on standards for testing, evaluating, and locating roadside safety features--from a meeting of the TRB Committee on Roadside Safety Features (A2A04) Subcommittee on International Research Activities (A2A04(2)). May 20, 2002 TRB Transportation Research Circular E-C037: Glossary of Highway Quality Assurance Terms includes standards for usage of highway quality assurance terminology.
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Title: URL Source: Markdown Content: # Lecture notes: Studying distributed systems – Reliable Broadcast M2 MOSIG: Large-Scale Data Management and Distributed Systems Thomas Ropars 2024 This lecture studies the implementation of reliable broadcast primitives 1. Considering a crash-stop failure model, it studies broadcast algorithms that differ according to the reliability and ordering guarantees they offer. # 1 Motivations In a distributed system, it is often useful to be able to send a message to a group of processes. This is the service a broadcast primitives implements. Since faults might occur, we are interested in reliable broadcast primitives. Defining the prop-erties of a reliable broadcast primitive is not as simple as it seems. A simple property would be: all processes deliver the same set of messages. But what should we do if a sender crash after it has sent its message to some processes but not to all? # 2 Safety and liveness When defining the expected properties for a distributed algorithms, we need to define properties that cover two categories: safety properties and liveness properties. Roughly speaking, a safety property stipulates that “ nothing bad ” will ever happen; a liveness property stipulates that “ something good ”will eventually happen. More precisely, a safety property is a property whose violation can be observed by looking only at the prefix of an execution (i.e., by looking only at an execution up to a certain time). This is not the case for a liveness property. A liveness property is a property whose violation cannot be observed on a prefix of an execution: if the property does not hold on a prefix of an execution, it might still hold later. # 3 Best-effort broadcast For this algorithm and each of the following ones, we adopt the same approach. We start be defining
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the required safety and liveness properties of the algorithm. Then we propose an implementation that satisfies these properties. > 1Acknowledgments: This lecture is strongly inspired from Chapter 3 of the book of Cachin, Guerraoui, and Rodrigues 1For all algorithms, we consider a crash-stop failure model for processes and we assume quasi-reliable channels. 3.1 Specification The best-effort broadcast abstraction has the following properties: • Integrity: Each process delivers message m at most once, and only if was broadcasted by some process. • Validity: If a correct process broadcasts a message m, then every correct process eventually delivers m. Integrity is a safety property while validity is a liveness property. 3.2 Implementation Figure 1 presents an implementation of the best-effort broadcast primitive. The group of processes into which message m is broadcasted is noted Π. > 1Implements: > 2BestEffortBroadcast, instance beb. > 4Uses: > 5QuasiReliablePointToPointLinks, instance qrl. > 7Upon beb.broadcast(m): > 8for all q ∈Π: > 9qrl.send(q, m) > 11 Upon qrl.deliver(p, m) > 12 Trigger beb.deliver(m) Figure 1: Implementation of Best-Effort Broadcast. The implementation is trivial as it simply relies on the properties of quasi-reliable links. Performance: To analyze the performance of the algorithm, we consider 2 metrics: 1) The num-ber of communication steps required to terminate one operation; 2) The total number of messages exchanged during one operation. We want to compute these metrics as of function of N , the number of processes in Π.The algorithm presented in Figure 1 requires 1 communication step and exchanges O(N ) mes-sages. # 4 Regular Reliable Broadcast With the best-effort broadcast primitive, if a sender crashes, some processes might deliver a message m while the others don’t. The processes do not agree on the set of messages to deliver. In practice, 2it can be problematic if not all correct processes deliver
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the same set of messages. The goal of the (regular) Reliable Broadcast primitive 2 is to ensure this property. 4.1 Specification The reliable broadcast abstraction has the following properties: • Integrity: Each process delivers message m at most once, and only if was broadcasted by some process. • Validity: If a correct process broadcasts a message m, then every correct process eventually delivers m. • Agreement: If a message m is delivered by some correct process, then m is eventually delivered by every correct process. Compared to best-effort broadcast, we introduce an additional Agreement property, which is a liveness property 3. 4.2 Implementation Figure 2 presents an implementation of the reliable broadcast primitive. For this implementation, we rely on the best-effort broadcast primitive. We also rely on a perfect failure detector. The algorithm handles two main cases to ensure agreement : • If a process delivers a message m for which the source has already crashed, it broadcasts (using best-effort broadcast) m again (line 31). • When a process becomes aware that a process q has crashed, it best-effort broadcasts all the messages from q it has already delivered (line 36). Performance : If no process crash (best case), it takes 1 communication step and O(N ) messages to rb-deliver a message to all processes. The worst case is if all processes crash one by one and if the message is delivered only by one process before the sender crashes, and the process who delivered the message in the next one to crash. In this case, N communication step and and O(N 2) messages are required. # 5 Uniform reliable broadcast With the reliable broadcast primitive described above, we can have a scenario where a faulty process delivers a message m before it crashes, and where the correct processes never
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deliver this message: The agreement property only applies to correct processes. > 2From this point on, when we simply say “ reliable broadcast ”, we are referring to the regular reliable broadcast abstraction. > 3The fact that this agreement property is a liveness property can be counter-intuitive since if message mis not delivered by any process, the property is ensured. However, considering the definition of liveness given above, it fits in this category: We can’t conclude that the property is not ensured by considering a prefix of the execution. 313 Implements: > 14 ReliableBroadcast, instance rb. > 16 Uses: > 17 BestEffortBroadcast, instance beb > 18 PerfectFailureDetector, instance P > 20 Variables: > 21 correct = Π # The set of processes considered correct > 22 from[N] = { ∅, ∅, ..., ∅} # N is the total number of processes > 24 Upon rb.broadcast(m): > 25 beb.broadcast([self, m]) # self is the id of the process > 27 Upon beb.deliver(p, [s, m]) # s is the id of the original source > 28 if m /∈ from[s]: > 29 Trigger rb.deliver(m) > 30 from[s] = from[s] ∪ m > 31 if s /∈ correct: > 32 beb.broadcast([s, m]) > 34 Upon event crash(q) raised by P: > 35 correct = correct \ {q} # remove q from the set > 36 for all m ∈ from[q]: > 37 beb.broadcast([q, m]) Figure 2: Implementation of Reliable Broadcast. In some scenarii, such behavior is not acceptable. In any scenario where processes interact with the outside world , we do not want this kind of behavior. Imagine that the messages are displayed on a screen 4, or that the messages are orders for transferring money. In such a case, we cannot assume anymore that it is not a problem that a process
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delivered m because that process crashed afterwards. We want to implement a broadcast primitive with a stronger agreement property, which is called Uniform Agreement . We find the need for such a uniform property in many other abstractions. 5.1 Specification The uniform reliable broadcast abstraction has the following properties: • Integrity: Each process delivers message m at most once, and only if was broadcasted by some process. • Validity: If a correct process broadcasts a message m, then every correct process eventually delivers m. • Uniform Agreement: If a message m is delivered by some process (whether correct or not), then m is eventually delivered by every correct process. > 4 The user might have read the message before the process crashes and it might not be acceptable to consider that the message did not exist. 45.2 Implementation Figure 3 presents an implementation of the uniform reliable broadcast primitive. For this imple-mentation, we still rely on the best-effort broadcast abstraction. We also rely on a perfect failure detector. > 38 Implements: > 39 UniformReliableBroadcast, instance urb. > 41 Uses: > 42 BestEffortBroadcast, instance beb > 43 PerfectFailureDetector, instance P > 45 Variables: > 46 correct = Π # The set of processes considered correct > 47 delivered = ∅ > 48 pending = ∅ > 49 for all m: > 50 ack[m] = ∅ > 52 Upon urb.broadcast(m): > 53 pending = pending ∪ [self,m] > 54 beb.broadcast([self, m]) > 56 Upon beb.deliver(p, [s, m]) # s is the id of the source > 57 ack[m] = ack[m] ∪ p > 58 if [s, m] /∈ pending: > 59 pending = pending ∪ [, m] > 60 beb.broadcast([s, m]) > 61 tryDeliver([s,m]) > 63 Upon event crash(q) raised by P: > 64 correct = correct \ {q} > 65 for
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all [s, m] ∈ pending: > 66 tryDeliver([s,m]) > 68 Function canDeliver(m): > 69 return (correct ⊆ ack[m]) # we receive m from all correct processes > 71 Function tryDeliver([s, m]): > 72 if canDeliver(m) and m /∈ delivered: > 73 delivered = delivered ∪ m > 74 Trigger urb.deliver([s, m]) Figure 3: Implementation of Uniform Reliable Broadcast. The proposed solution is called All-ack Uniform Reliable Broadcast. The basic idea is that a process can deliver a message only when it has received a copy of that message from all correct processes (condition tested by the function canDeliver() ), which implies that they have all received the message. The ack array includes one entry per message, that stores the list of processes from which a given message has already been received (line 57). 5Performance : If no process crash (best case), it takes 2 communication steps to rb-deliver amessage to all processes. In the first step, the source rb-broadcast to all ( N messages sent). In the second step, all processes except the source rb-broadcast to all ( (N − 1) × N messages sent). The total number of messages sent is N 2. In the worst case, when processes crash in sequence, N + 1 steps are required to terminate. # 6 FIFO reliable broadcast In addition to agreement guarantees, some applications have also requirements on the order in which messages are delivered. For instance, if we implement a distributed game, and we broadcast the successive positions of each player in the game. We consider now the implementation of a FIFO (reliable) broadcast, that extends the reliable broadcast abstraction to guarantee if a process broadcasts two messages, they are delivered in the order they were sent. 6.1 Specification The FIFO broadcast abstraction has the following properties: • Integrity: Each
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process delivers message m at most once, and only if was broadcasted by some process. • Validity: If a correct process broadcasts a message m, then every correct process eventually delivers m. • Agreement: If a message m is delivered by some correct process, then m is eventually delivered by every correct process. • FIFO delivery: If some process broadcasts message m1 before it broadcasts message m2, then no correct process delivers m2 unless it has already delivered m1. 6.2 Implementation Figure 3 presents an implementation of the FIFO reliable broadcast primitive. A per-process se-quence number piggybacked on each message allows deciding when a message can be delivered. # References C. Cachin, R. Guerraoui, and L. Rodrigues. Introduction to Reliable and Secure Distributed Programming . Springer Publishing Company, Incorporated, 2nd edition, 2011. 676 Implements: > 77 FIFOReliableBroadcast, instance frb. > 79 Uses: > 80 ReliableBroadcast, instance rb. > 82 Variables: > 83 lsn = 0 # local sequence number > 84 pending = ∅ > 85 next[N] = {1, 1, ..., 1} # seq. number of the next message to deliver > 87 Upon frb.broadcast(m): > 88 lsn = lsn + 1 > 89 rb.broadcast([m, lsn]) > 91 Upon rb.deliver(p, [m, lsn]): > 92 pending = pending ∪ {[p, m ,lsn]} > 93 while exists (p, m′, lsn ′) ∈ pending such that lsn ′ = next[p]: > 94 next[p] = next[p] + 1 > 95 pending = pending \ { m′} > 96 Trigger frb.deliver( m′) Figure 4: Implementation of FIFO Broadcast. 7
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Title: URL Source: Markdown Content: # 3. Reliable Broadcast He said: “I could have been someone”; She replied: “So could anyone.” (The Pogues) This chapter covers broadcast communication abstractions. These are used to dis-seminate information among a set of processes and differ according to the reliability of the dissemination. For instance, best-effort broadcast guarantees that all correct processes deliver the same set of messages if the senders are correct. Stronger forms of reliable broadcast guarantee this property even if the senders crash while broad-casting their messages. Even stronger broadcast abstractions are appropriate for the arbitrary-fault model and ensure consistency with Byzantine process abstractions. We will consider several related abstractions for processes subject to crash faults: best-effort broadcast, (regular) reliable broadcast, uniform reliable broadcast, stub-born broadcast, probabilistic broadcast , and causal broadcast . For processes in the crash-recovery model, we describe stubborn broadcast , logged best-effort broad-cast , and logged uniform reliable broadcast . Finally, for Byzantine processes, we introduce Byzantine consistent broadcast and Byzantine reliable broadcast . For each of these abstractions, we will provide one or more algorithms implementing it, and these will cover the different models addressed in this book. # 3.1 Motivation 3.1.1 Client–Server Computing In traditional distributed applications, interactions are often established between two processes. Probably the most representative of this sort of interaction is the now classic client–server scheme. According to this model, a server process exports an interface to several clients . Clients use the interface by sending a request to the server and by later collecting a reply. Such interaction is supported by point-to-point communication protocols. It is extremely useful for the application if such a protocol C. Cachin et al., Introduction to Reliable and Secure Distributed Programming ,DOI: 10.1007/978-3-642-15260-3 3, c© Springer-Verlag Berlin Heidelberg 2011 73 74 3 Reliable Broadcast is
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reliable . Reliability in this context usually means that, under some assumptions (which are, by the way, often not completely understood by most system design-ers), messages exchanged between the two processes are not lost or duplicated, and are delivered in the order in which they were sent. Typical implementations of this abstraction are reliable transport protocols such as TCP on the Internet. By using a reliable point-to-point communication protocol, the application is free from dealing explicitly with issues such as acknowledgments, timeouts, message retransmissions, flow control, and a number of other issues that are encapsulated by the protocol interface. 3.1.2 Multiparticipant Systems As distributed applications become bigger and more complex, interactions are no longer limited to bilateral relationships. There are many cases where more than two processes need to operate in a coordinated manner. Consider, for instance, a multiuser virtual environment where several users interact in a virtual space. These users may be located at different physical places, and they can either directly interact by exchanging multimedia information, or indirectly by modifying the environment. It is convenient to rely here on broadcast abstractions. These allow a process to send a message within a group of processes, and make sure that the processes agree on the messages they deliver. A naive transposition of the reliability requirement from point-to-point protocols would require that no message sent to the group be lost or duplicated, i.e., the processes agree to deliver every message broadcast to them. However, the definition of agreement for a broadcast primitive is not a simple task. The existence of multiple senders and multiple recipients in a group introduces degrees of freedom that do not exist in point-to-point communication. Consider, for instance, the case where the sender of a message fails by crashing. It may happen that some recipients deliver the
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last message sent while others do not. This may lead to an inconsistent view of the system state by different group members. When the sender of a message exhibits arbitrary-faulty behavior, assuring that the recipients deliver one and the same message is an even bigger challenge. The broadcast abstractions in this book provide a multitude of reliability guar-antees. For crash-stop processes they range, roughly speaking, from best-effort ,which only ensures delivery among all correct processes if the sender does not fail, through reliable , which, in addition, ensures all-or-nothing delivery semantics, even if the sender fails, to totally ordered , which furthermore ensures that the delivery of messages follow the same global order, and terminating , which ensures that the pro-cesses either deliver a message or are eventually aware that they should never deliver the message. For arbitrary-faulty processes, a similar range of broadcast abstractions exists. The simplest one among them guarantees a form of consistency , which is not even an issue for crash-stop processes, namely, to ensure that two correct processes, if they deliver a messages at all, deliver the same message. The reliable broadcast abstractions and total-order broadcast abstractions among arbitrary-faulty processes 3.2 Best-Effort Broadcast 75 additionally provide all-or-nothing delivery semantics and totally ordered delivery, respectively. In this chapter, we will focus on best-effort and reliable broadcast abstractions. Stronger forms of broadcast will be considered in later chapters. The next three sections present broadcast abstractions with crash-stop process abstractions. More general process failures are considered afterward. # 3.2 Best-Effort Broadcast A broadcast abstraction enables a process to send a message, in a one-shot operation, to all processes in a system, including itself. We give here the specification and an algorithm for a broadcast communication primitive with a weak form of reliability, called best-effort broadcast . 3.2.1 Specification
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With best-effort broadcast, the burden of ensuring reliability is only on the sender. Therefore, the remaining processes do not have to be concerned with enforcing the reliability of received messages. On the other hand, no delivery guarantees are offered in case the sender fails. Best-effort broadcast is characterized by the follow-ing three properties depicted in Module 3.1: validity is a liveness property, whereas the no duplication property and the no creation property are safety properties. They descend directly from the corresponding properties of perfect point-to-point links. Note that broadcast messages are implicitly addressed to all processes. Remember also that messages are unique, that is, no process ever broadcasts the same message twice and furthermore, no two processes ever broadcast the same message. > Module 3.1: Interface and properties of best-effort broadcast > Module: Name: BestEffortBroadcast, instance beb . > Events: Request: 〈beb ,Broadcast |m〉: Broadcasts a message mto all processes. > Indication: 〈beb ,Deliver |p,m〉: Delivers a message mbroadcast by process p. > Properties: BEB1: Validity: If a correct process broadcasts a message m, then every correct process eventually delivers m. > BEB2: No duplication: No message is delivered more than once. > BEB3: No creation: If a process delivers a message mwith sender s, then mwas previously broadcast by process s.76 3 Reliable Broadcast > Algorithm 3.1: Basic Broadcast > Implements: > BestEffortBroadcast, instance beb . > Uses: > PerfectPointToPointLinks, instance pl . > upon event 〈beb ,Broadcast |m〉do forall q∈Πdo trigger 〈pl ,Send |q, m 〉; > upon event 〈pl ,Deliver |p,m〉do trigger 〈beb ,Deliver |p,m〉; 3.2.2 Fail-Silent Algorithm: Basic Broadcast We provide here algorithm “Basic Broadcast” (Algorithm 3.1) that implements best-effort broadcast using perfect links. This algorithm does not make any assumption on failure detection: it is a fail-silent algorithm. The algorithm is straightforward. Broadcasting a message
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simply consists of sending the message to every process in the system using perfect point-to-point links, as illustrated by Fig. 3.1 (in the figure, white arrowheads represent request/indication events at the module interface and black arrowheads represent message exchanges). The algorithm works because the properties of perfect links ensure that all correct processes eventually deliver the message, as long as the sender of a message does not crash. Correctness. The properties of best-effort broadcast are trivially derived from the properties of the underlying perfect point-to-point links. The no creation property follows directly from the corresponding property of perfect links. The same applies to no duplication , which relies in addition on the assumption that messages broad-cast by different processes are unique. Validity is derived from the reliable delivery property and the fact that the sender sends the message to every other process in the system. > pqrsbeb−deliver beb−deliver beb−deliver beb−deliver beb−broadcast > Figure 3.1: Sample execution of basic broadcast 3.3 Regular Reliable Broadcast 77 Performance. For every message that is broadcast, the algorithm requires a single communication step and exchanges O(N ) messages. # 3.3 Regular Reliable Broadcast Best-effort broadcast ensures the delivery of messages as long as the sender does not fail. If the sender fails, some processes might deliver the message and others might not deliver it. In other words, they do not agree on the delivery of the message. Actually, even if the process sends a message to all processes before crashing, the delivery is not ensured because perfect links do not enforce the delivery when the sender fails. Ensuring agreement even when the sender fails is an important property for many practical applications that rely on broadcast. The abstraction of (regular) reliable broadcast provides exactly this stronger notion of reliability. 3.3.1 Specification Intuitively, the semantics of
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a reliable broadcast algorithm ensure that the correct processes agree on the set of messages they deliver, even when the senders of these messages crash during the transmission. It should be noted that a sender may crash before being able to transmit the message, in which case no process will del-iver it. The specification of reliable broadcast in Module 3.2 extends the properties of the best-effort broadcast abstraction (Module 3.1) with a new liveness property called agreement . The other properties remain unchanged (but are repeated here for completeness). The very fact that agreement is a liveness property might seem > Module 3.2: Interface and properties of (regular) reliable broadcast > Module: Name: ReliableBroadcast, instance rb . > Events: Request: 〈rb ,Broadcast |m〉: Broadcasts a message mto all processes. > Indication: 〈rb ,Deliver |p,m〉: Delivers a message mbroadcast by process p. > Properties: RB1: Validity: If a correct process pbroadcasts a message m, then peventually delivers m. > RB2: No duplication: No message is delivered more than once. > RB3: No creation: If a process delivers a message mwith sender s, then mwas previously broadcast by process s. > RB4: Agreement: If a message mis delivered by some correct process, then mis eventually delivered by every correct process. 78 3 Reliable Broadcast counterintuitive, as the property can be achieved by not having any process ever deliver any message. Strictly speaking, it is, however, a liveness property as it can always be ensured in extensions of finite executions. We will see other forms of agreement that are safety properties later in the book. 3.3.2 Fail-Stop Algorithm: Lazy Reliable Broadcast We now show how to implement regular reliable broadcast in a fail-stop model. In our algorithm, depicted in Algorithm 3.2, which we have called “Lazy Reliable Broadcast,” we make use of the
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best-effort broadcast abstraction described in the previous section, as well as the perfect failure detector abstraction P introduced earlier. Algorithm 3.2: Lazy Reliable Broadcast > Implements: > ReliableBroadcast, instance rb . > Uses: > BestEffortBroadcast, instance beb ;PerfectFailureDetector, instance P. > upon event 〈rb ,Init 〉do > correct := Π; > from [p]:= [∅]N; > upon event 〈rb ,Broadcast |m〉do trigger 〈beb ,Broadcast |[DATA ,self , m ]〉; > upon event 〈beb ,Deliver |p,[DATA ,s, m ]〉do if m∈from [s]then trigger 〈rb ,Deliver |s,m〉; > from [s]:= from [s]∪ { m}; > if s∈correct then trigger 〈beb ,Broadcast |[DATA ,s, m ]〉; > upon event 〈 P ,Crash |p〉do > correct := correct \ { p}; > forall m∈from [p]do trigger 〈beb ,Broadcast |[DATA ,p, m ]〉; To rb -broadcast a message, a process uses the best-effort broadcast primitive to disseminate the message to all. The algorithm adds some implementation-specific parameters to the exchanged messages. In particular, its adds a message descriptor (D ATA ) and the original source of the message (process s) in the message header .The result is denoted by [D ATA , s, m] in the algorithm. A process that receives the message (when it beb -delivers the message) strips off the message header and rb -delivers it immediately. If the sender does not crash, then the message will be 3.3 Regular Reliable Broadcast 79 rb -delivered by all correct processes. The problem is that the sender might crash. In this case, the process that delivers the message from some other process detects that crash and relays the message to all others. We note that this is a language abuse: in fact, the process relays a copy of the message (and not the message itself). At the same time, the process also maintains a variable correct
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, denoting the set of processes that have not been detected to crash by P. Our algorithm is said to be lazy in the sense that it retransmits a message only if the original sender has been detected to have crashed. The variable from is an array of sets, indexed by the processes in Π, in which every entry s contains the messages from sender s that have been rb -delivered. It is important to notice that, strictly speaking, two kinds of events can force a process to retransmit a message. First, when the process detects the crash of the source, and, second, when the process beb -delivers a message and realizes that the source has already been detected to have crashed (i.e., the source is not anymore in correct ). This might lead to duplicate retransmissions when a process beb -delivers a message from a source that fails, as we explain later. It is easy to see that a pro-cess that detects the crash of a source needs to retransmit the messages that have already been beb -delivered from that source. On the other hand, a process might beb -deliver a message from a source after it detected the crash of that source: it is, thus, necessary to check for the retransmission even when no new crash is detected. Correctness. The no creation (respectively validity ) property of our reliable broad-cast algorithm follows from the no creation (respectively validity ) property of the underlying best-effort broadcast primitive. The no duplication property of reliable broadcast follows from our use of a variable from that keeps track of the messages that have been rb -delivered at every process and from the assumption of unique messages across all senders. Agreement follows here from the validity property of the underlying best-effort broadcast primitive,
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from the fact that every process relays every message that it rb -delivers when it detects the sender, and from the use of a perfect failure detector. Performance. If the initial sender does not crash then the algorithm requires a single communication step and O(N ) messages to rb -deliver a message to all processes. Otherwise, it may take O(N ) steps and O(N 2) messages in the worst case (if the processes crash in sequence). 3.3.3 Fail-Silent Algorithm: Eager Reliable Broadcast In the “Lazy Reliable Broadcast” algorithm (Algorithm 3.2), when the accuracy property of the failure detector is not satisfied, the processes might relay messages unnecessarily. This wastes resources but does not impact correctness. On the other hand, we rely on the completeness property of the failure detector to ensure the broadcast agreement . If the failure detector does not ensure completeness then the processes might omit to relay messages that they should be relaying (e.g., messages broadcast by processes that crashed), and hence might violate agreement .80 3 Reliable Broadcast Algorithm 3.3: Eager Reliable Broadcast > Implements: > ReliableBroadcast, instance rb . > Uses: > BestEffortBroadcast, instance beb . > upon event 〈rb ,Init 〉do > delivered := ∅; > upon event 〈rb ,Broadcast |m〉do trigger 〈beb ,Broadcast |[DATA ,self , m ]〉; > upon event 〈beb ,Deliver |p,[DATA ,s, m ]〉do if m∈delivered then > delivered := delivered ∪ { m}; > trigger 〈rb ,Deliver |s,m〉; > trigger 〈beb ,Broadcast |[DATA ,s, m ]〉; In fact, we can circumvent the need for a failure detector (i.e., the need for its completeness property) by adopting an eager scheme: every process that gets a mes-sage relays it immediately. That is, we consider the worst case, where the sender process might have crashed, and we relay every message. This relaying
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phase is exactly what guarantees the agreement property of reliable broadcast. The resulting algorithm (Algorithm 3.3) is called “Eager Reliable Broadcast.” The algorithm assumes a fail-silent model and does not use any failure detec-tor: it relies only on the best-effort broadcast primitive described in Sect. 3.2. In Fig. 3.2, we illustrate how the algorithm ensures agreement even if the sender crashes: process p crashes and its message is not beb -delivered by processes r and by s. However, as process q retransmits the message, i.e., beb -broadcasts it, the remaining processes also beb -deliver it and subsequently rb -deliver it. In our “Lazy > rb−deliver rb−deliver rb−broadcast pqrsrb−deliver rb−deliver Figure 3.2: Sample execution of reliable broadcast with faulty sender 3.4 Uniform Reliable Broadcast 81 Reliable Broadcast” algorithm, process q will be relaying the message only after it has detected the crash of p. Correctness. All properties, except agreement , are ensured as in the “Lazy Reliable Broadcast.” The agreement property follows from the validity property of the underlying best-effort broadcast primitive and from the fact that every correct process immediately relays every message it rb -delivers. Performance. In the best case, the algorithm requires a single communication step and O(N 2) messages to rb -deliver a message to all processes. In the worst case, should the processes crash in sequence, the algorithm may incur O(N ) steps and O(N 2) messages. # 3.4 Uniform Reliable Broadcast With regular reliable broadcast, the semantics just require the correct processes to deliver the same set of messages, regardless of what messages have been deliv-ered by faulty processes. In particular, a process that rb -broadcasts a message might rb -deliver it and then crash, before the best-effort broadcast abstraction can even beb -deliver the message to any other process. Indeed, this scenario may occur
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in both reliable broadcast algorithms that we presented (eager and lazy). It is thus pos-sible that no other process, including correct ones, ever rb -delivers that message. There are cases where such behavior causes problems because even a process that rb -delivers a message and later crashes may bring the application into a inconsistent state. We now introduce a stronger definition of reliable broadcast, called uniform reliable broadcast . This definition is stronger in the sense that it guarantees that the set of messages delivered by faulty processes is always a subset of the messages delivered by correct processes. Many other abstractions also have such uniform variants. 3.4.1 Specification Uniform reliable broadcast differs from reliable broadcast by the formulation of its agreement property. The specification is given in Module 3.3. Uniformity is typically important if the processes interact with the external world, e.g., print something on a screen, authorize the delivery of money through a bank machine, or trigger the launch of a rocket. In this case, the fact that a process has delivered a message is important, even if the process has crashed afterward. This is because the process, before crashing, could have communicated with the external world after having delivered the message. The processes that did not crash should also be aware of that message having been delivered, and of the possible external action having been performed. Figure 3.3 depicts an execution of a reliable broadcast algorithm that is not uni-form. Both processes p and q rb -deliver the message as soon as they beb -deliver 82 3 Reliable Broadcast > Module 3.3: Interface and properties of uniform reliable broadcast > Module: Name: UniformReliableBroadcast, instance urb . > Events: Request: 〈urb , Broadcast | m 〉: Broadcasts a message m to all processes. > Indication: 〈urb , Deliver
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| p, m 〉: Delivers a message m broadcast by process p. > Properties: URB1–URB3: Same as properties RB1–RB3 in (regular) reliable broadcast (Mod-ule 3.2). > URB4: Uniform agreement: If a message mis delivered by some process (whether correct or faulty), then mis eventually delivered by every correct process. > rb−deliver rb−broadcast pqrsrb−deliver > Figure 3.3: Nonuniform reliable broadcast it, but crash before they are able to relay the message to the remaining processes. Still, processes r and s are consistent among themselves (neither has rb -delivered the message). 3.4.2 Fail-Stop Algorithm: All-Ack Uniform Reliable Broadcast Basically, our “Lazy Reliable Broadcast” and “Eager Reliable Broadcast” algo-rithms do not ensure uniform agreement because a process may rb -deliver a message and then crash. Even if this process has relayed its message to all processes (through a best-effort broadcast primitive), the message might not reach any of the remaining processes. Note that even if we considered the same algorithms and replaced the best-effort broadcast abstraction with a reliable broadcast one, we would still not implement a uniform broadcast abstraction. This is because a process may deliver a message before relaying it to all processes. Algorithm 3.4, named “All-Ack Uniform Reliable Broadcast,” implements uni-form reliable broadcast in the fail-stop model. Basically, in this algorithm, a process 3.4 Uniform Reliable Broadcast 83 Algorithm 3.4: All-Ack Uniform Reliable Broadcast > Implements: > UniformReliableBroadcast, instance urb . > Uses: > BestEffortBroadcast, instance beb .PerfectFailureDetector, instance P. > upon event 〈urb , Init 〉 do > delivered := ∅; > pending := ∅; > correct := Π; > forall mdo ack [m]:= ∅; > upon event 〈urb , Broadcast | m 〉 do > pending := pending ∪ { (self , m )}; > trigger 〈beb , Broadcast | [D ATA , self , m ] 〉;
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> upon event 〈beb , Deliver | p, [D ATA , s, m ] 〉 do > ack [m]:= ack [m]∪ { p}; > if (s, m )∈pending then > pending := pending ∪ { (s, m )}; > trigger 〈beb , Broadcast | [D ATA , s, m ] 〉; > upon event 〈 P , Crash | p 〉 do > correct := correct \ { p}; > function candeliver( m) returns Boolean is return (correct ⊆ ack [m]) ; > upon exists (s, m )∈pending such that candeliver (m) ∧ m ∈ delivered do > delivered := delivered ∪ { m}; > trigger 〈urb , Deliver | s, m 〉;delivers a message only when it knows that the message has been beb -delivered and thereby seen by all correct processes. All processes relay the message once, after they have seen it. Each process keeps a record of processes from which it has already received a message (either because the process originally sent the message or because the process relayed it). When all correct processes have retransmitted the message, all correct processes are guaranteed to deliver the message, as illustrated in Fig. 3.4. The algorithm uses a variable delivered for filtering out duplicate messages and a variable pending , used to collect the messages that have been beb -delivered and seen, but that still need to be urb -delivered. The algorithm also uses an array ack with sets of processes, indexed by all possi-ble messages. The entry ack [m] gathers the set of processes that the process knows have seen m. Of course, the array can be implemented with a finite amount of 84 3 Reliable Broadcast > urb−broadcast pqrsurb−deliver urb−deliver urb−deliver > Figure 3.4: Sample execution of all-ack uniform reliable broadcast memory by using a sparse
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representation. Note that the last upon statement of the algorithm is triggered by an internal event defined on the state of the algorithm. Correctness. The validity property follows from the completeness property of the failure detector and from the validity property of the underlying best-effort broad-cast. The no duplication property relies on the delivered variable to filter out duplicates. No creation is derived from the no creation property of the underlying best-effort broadcast. Uniform agreement is ensured by having each process wait to urb -deliver a message until all correct processes have seen and relayed the message. This mechanism relies on the accuracy property of the perfect failure detector. Performance. When considering the number of communication steps, in the best case, the algorithm requires two communication steps to urb -deliver a message to all processes. In such scenario, in the first step it sends N messages and in the second step N (N −1) messages, for a total of N 2 messages. In the worst case, if the processes crash in sequence, N + 1 steps are required. Therefore, uniform reliable broadcast requires one step more to deliver a message than its regular counterpart. 3.4.3 Fail-Silent Algorithm: Majority-Ack Uniform Reliable Broadcast The “All-Ack Uniform Reliable Broadcast” algorithm of Sect. 3.4.2 (Algorithm 3.4) is not correct if the failure detector is not perfect. Uniform agreement would be violated if accuracy is not satisfied and validity would be violated if completeness is not satisfied. We now give a uniform reliable broadcast algorithm that does not rely on a per-fect failure detector but assumes a majority of correct processes, i.e., N > 2f if we assume that up to f processes may crash. We leave it as an exercise to show why the majority assumption is needed in the fail-silent model, without any
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failure detector. Algorithm 3.5, called “Majority-Ack Uniform Reliable Broadcast,” is similar to Algorithm 3.4 (“All-Ack Uniform Reliable Broadcast”) in the fail-silent model, except that processes do not wait until all correct processes have seen a message, but only until a majority quorum has seen and retransmitted the message. Hence, the algorithm can be obtained by a small modification from the previous one, affecting only the condition under which a message is delivered. 3.5 Stubborn Broadcast 85 > Algorithm 3.5: Majority-Ack Uniform Reliable Broadcast > Implements: > UniformReliableBroadcast, instance urb . > Uses: > BestEffortBroadcast, instance beb .// Except for the function candeliver (·)below and for the absence of 〈 > Crash 〉events // triggered by the perfect failure detector, it is the same as Algorithm 3.4. > function > candeliver( m)returns Boolean is return #( ack [m]) > N/ 2; Correctness. The algorithm provides uniform reliable broadcast if N > 2f . The no duplication property follows directly from the use of the variable delivered . The no creation property follows from the no creation property of best-effort broadcast. To argue for the uniform agreement and validity properties, we first observe that if a correct process p beb -delivers some message m then p eventually urb -delivers m.Indeed, if p is correct, and given that p beb -broadcasts m according to the algorithm, then every correct process beb -delivers and hence beb -broadcasts m. As we assume a majority of the processes to be correct, p eventually beb -delivers m from more than N/ 2 processes and urb -delivers it. Consider now the validity property. If a correct process p urb -broadcasts a mes-sage m then p beb -broadcasts m, and hence p beb -delivers m eventually; according to the above observation, p eventually also urb -delivers m. Consider now
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uniform agreement , and let q be any process that urb -delivers m. To do so, q must have beb -delivered m from a majority of the processes. Because of the assumption of a correct majority, at least one correct process must have beb -broadcast m. Hence, all correct processes eventually beb -deliver m by the validity property of best-effort broadcast, which implies that all correct processes also urb -deliver m eventually according to the observation made earlier. Performance. The performance of the algorithm is similar to the performance of the “All-Ack Uniform Reliable Broadcast” algorithm. # 3.5 Stubborn Broadcast This section presents a stubborn broadcast abstraction that works with crash-stop process abstractions in the fail-silent system model, as well as with crash-recovery process abstractions in the fail-recovery model. 3.5.1 Specification The stubborn broadcast abstraction hides a retransmission mechanism and delivers every message that is broadcast by a correct process an infinite number of times, 86 3 Reliable Broadcast > Module 3.4: Interface and properties of stubborn best-effort broadcast > Module: Name: StubbornBestEffortBroadcast, instance sbeb . > Events: Request: 〈sbeb ,Broadcast |m〉: Broadcasts a message mto all processes. > Indication: 〈sbeb ,Deliver |p,m〉: Delivers a message mbroadcast by process p. > Properties: SBEB1: Best-effort validity: If a process that never crashes broadcasts a message m,then every correct process delivers man infinite number of times. > SBEB2: No creation: If a process delivers a message mwith sender s, then mwas previously broadcast by process s. similar to its point-to-point communication counterpart. The specification of best-effort stubborn broadcast is given in Module 3.4. The key difference to the best-effort broadcast abstraction (Module 3.1) defined for fail-no-recovery settings lies in the stubborn and perpetual delivery of every message broadcast by a process that does not crash. As a direct consequence, the no duplication
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property of best-effort broadcast is not ensured. Stubborn broadcast is the first broadcast abstraction in the fail-recovery model considered in this chapter (more will be introduced in the next two sections). As the discussion of logged perfect links in Chap. 2 has shown, communication abstrac-tions in the fail-recovery model usually rely on logging their output to variables in stable storage. For stubborn broadcast, however, logging is not necessary because every delivered message is delivered infinitely often; no process that crashes and recovers finitely many times can, therefore, miss such a message. The very fact that processes now have to deal with multiple deliveries is the price to pay for saving expensive logging operations. We discuss a logged best-effort broadcast in the next section, which eliminates multiple deliveries, but adds at the cost of logging the messages. The stubborn best-effort broadcast abstraction also serves as an example for stronger stubborn broadcast abstractions, implementing reliable and uniform reli-able stubborn broadcast variants, for instance. These could be defined and imple-mented accordingly. 3.5.2 Fail-Recovery Algorithm: Basic Stubborn Broadcast Algorithm 3.6 implements stubborn best-effort broadcast using underlying stubborn communication links. Correctness. The properties of stubborn broadcast are derived directly from the properties of the stubborn links abstraction used by the algorithm. In particular, 3.6 Logged Best-Effort Broadcast 87 > Algorithm 3.6: Basic Stubborn Broadcast > Implements: > StubbornBestEffortBroadcast, instance sbeb . > Uses: > StubbornPointToPointLinks, instance sl . > upon event 〈sbeb ,Recovery 〉do > // do nothing > upon event 〈sbeb ,Broadcast |m〉do forall q∈Πdo trigger 〈sl ,Send |q, m 〉; > upon event 〈sl ,Deliver |p,m〉do trigger 〈sbeb ,Deliver |p,m〉; validity follows from the fact that the sender sends the message to every process in the system. Performance. The algorithm requires a single communication step for a process to deliver a message, and
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exchanges at least N messages. Of course, the stubborn links may retransmit the same message several times and, in practice, an optimization mechanism is needed to acknowledge the messages and stop the retransmission. # 3.6 Logged Best-Effort Broadcast This section and the next one consider broadcast abstractions in the fail-recovery model that rely on logging. We first discuss how fail-recovery broadcast algorithms use stable storage for logging and then present a best-effort broadcast abstraction and its implementation. 3.6.1 Overview Most broadcast specifications we have considered for the fail-stop and fail-silent models are not adequate for the fail-recovery model. As explained next, even the strongest one of our specifications, uniform reliable broadcast, does not provide use-ful semantics in a setting where processes that crash can later recover and participate in the computation. For instance, suppose a message m is broadcast by some process p. Consider another process q, which should eventually deliver m. But q crashes at some instant, recovers, and never crashes again; in the fail-recovery model, q is a correct process. For a broadcast abstraction, however, it might happen that process q delivers m and crashes immediately afterward, without having processed m, that is, before the application had time to react to the delivery of m. When the process recovers later, it 88 3 Reliable Broadcast > Module 3.5: Interface and properties of logged best-effort broadcast > Module: Name: LoggedBestEffortBroadcast, instance lbeb . > Events: Request: 〈lbeb ,Broadcast |m〉: Broadcasts a message mto all processes. > Indication: 〈lbeb ,Deliver |delivered 〉: Notifies the upper layer of potential up-dates to variable delivered in stable storage (which log-delivers messages according to the text). > Properties: LBEB1: Validity: If a process that never crashes broadcasts a message m, then every correct process eventually log-delivers m. > LBEB2: No duplication: No message is
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log-delivered more than once. > LBEB3: No creation: If a process log-delivers a message mwith sender s, then m > was previously broadcast by process s. has no memory of m, because the delivery of m occurred asynchronously and could not be anticipated. There should be some way for process q to find out about m upon recovery, and for the application to react to the delivery of m. We have already encountered this problem with the definition of logged perfect links in Sect. 2.4.5. We adopt the same solution as for logged perfect links: the module maintains a variable delivered in stable storage, stores every delivered messages in the variable, and the higher-level modules retrieve the variable from stable storage to determine the delivered messages. To notify the layer above about the delivery, the broadcast abstraction triggers an event 〈 Deliver | delivered 〉. We say that a message m is log-delivered from sender s whenever an event 〈 Deliver | delivered 〉 occurs such that delivered contains a pair (s, m ) for the first time. With this implementation, a process that log-delivers a message m, subsequently crashes, and recovers again will still be able to retrieve m from stable storage and to react to m. 3.6.2 Specification The abstraction we consider here is called logged best-effort broadcast to emphasize that it log-delivers messages by “logging” them to local stable storage. Its specifi-cation is given in Module 3.5. The logged best-effort broadcast abstraction has the same interface and properties as best-effort broadcast with crash-stop faults (Mod-ule 3.1), except that messages are log-delivered instead of delivered. As we discuss later, stronger logged broadcast abstractions (regular and uniform) can be designed and implemented on top of logged best-effort broadcast. 3.6 Logged Best-Effort Broadcast 89 Algorithm 3.7: Logged Basic Broadcast >
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Implements: > LoggedBestEffortBroadcast, instance lbeb . > Uses: > StubbornPointToPointLinks, instance sl . > upon event 〈lbeb ,Init 〉do > delivered := ∅;store (delivered ); > upon event 〈lbeb ,Recovery 〉do retrieve (delivered ); > trigger 〈lbeb ,Deliver |delivered 〉; > upon event 〈lbeb ,Broadcast |m〉do forall q∈Πdo trigger 〈sl ,Send |q, m 〉; > upon event 〈sl ,Deliver |p,m〉do if (p, m )∈delivered then > delivered := delivered ∪ { (p, m )};store( delivered ); > trigger 〈lbeb ,Deliver |delivered 〉; 3.6.3 Fail-Recovery Algorithm: Logged Basic Broadcast Algorithm 3.7, called “Logged Basic Broadcast,” implements logged best-effort broadcast. Its structure is similar to Algorithm 3.1 (“Basic Broadcast”). The main differences are the following: 1. The “Logged Basic Broadcast” algorithm uses stubborn best-effort links between every pair of processes for communication. They ensure that every message that is sent by a process that does not crash to a correct recipient will be delivered by its recipient an infinite number of times. 2. The “Logged Basic Broadcast” algorithm maintains a log of all delivered mes-sages. When a new message is received for the first time, it is added to the log, and the upper layer is notified that the log has changed. If the process crashes and later recovers, the upper layer is also notified (as it may have missed a notification triggered just before the crash). Correctness. The no creation property is derived from that of the underlying stub-born links, whereas no duplication is derived from the fact that the delivery log is checked before delivering new messages. The validity property follows from the fact that the sender sends the message to every other process in the system. Performance. The algorithm requires a single communication step for a process to deliver a message, and exchanges at least N messages. Of course,
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stubborn links 90 3 Reliable Broadcast may retransmit the same message several times and, in practice, an optimization mechanism is needed to acknowledge the messages and stop the retransmission. Additionally, the algorithm requires a log operation for each delivered message. # 3.7 Logged Uniform Reliable Broadcast In a manner similar to the crash-no-recovery case, it is possible to define both reliable and uniform variants of best-effort broadcast for the fail-recovery setting. 3.7.1 Specification Module 3.6 defines a logged uniform reliable broadcast abstraction, which is appropriate for the fail-recovery model. In this variant, if a process (either cor-rect or not) log-delivers a message (that is, stores the variable delivered containing the message in stable storage), all correct processes should eventually log-deliver that message. The interface is similar to that of logged best-effort broadcast and its properties directly correspond to those of uniform reliable broadcast with crash-stop processes (Module 3.3). > Module 3.6: Interface and properties of logged uniform reliable broadcast > Module: Name: LoggedUniformReliableBroadcast, instance lurb . > Events: Request: 〈lurb , Broadcast | m 〉: Broadcasts a message m to all processes. > Indication: 〈lurb , Deliver | delivered 〉: Notifies the upper layer of potential up-dates to variable delivered in stable storage (which log-delivers messages according to the text). > Properties: LURB1–LURB3: Same as properties LBEB1–LBEB3 in logged best-effort broad-cast (Module 3.5). > LURB4: Uniform agreement: If a message mis log-delivered by some process (whether correct or faulty), then mis eventually log-delivered by every correct process. 3.7.2 Fail-Recovery Algorithm: Logged Majority-Ack Uniform Reliable Broadcast Algorithm 3.8, called “Logged Majority-Ack Uniform Reliable Broadcast,” implements logged uniform broadcast, assuming that a majority of the processes 3.7 Logged Uniform Reliable Broadcast 91 Algorithm 3.8: Logged Majority-Ack Uniform Reliable Broadcast Implements: LoggedUniformReliableBroadcast, instance lurb . Uses: StubbornBestEffortBroadcast, instance sbeb . upon event 〈
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lurb ,Init 〉 do delivered := ∅; pending := ∅; forall m do ack [m] := ∅;store( pending , delivered ); upon event 〈 lurb ,Recovery 〉 do retrieve( pending , delivered ); trigger 〈 lurb ,Deliver | delivered 〉; forall (s, m ) ∈ pending do trigger 〈 sbeb ,Broadcast | [D ATA , s, m ] 〉; upon event 〈 lurb ,Broadcast | m 〉 do pending := pending ∪ { (self , m )};store( pending ); trigger 〈 sbeb ,Broadcast | [D ATA , self , m ] 〉; upon event 〈 sbeb ,Deliver | p, [D ATA , s, m ] 〉 do if (s, m ) ∈ pending then pending := pending ∪ { (s, m )};store( pending ); trigger 〈 sbeb ,Broadcast | [D ATA , s, m ] 〉; if p ∈ ack [m] then ack [m] := ack [m] ∪ { p}; if #( ack [m]) > N/ 2 ∧ (s, m ) ∈ delivered then delivered := delivered ∪ { (s, m )};store( delivered ); trigger 〈 lurb ,Deliver | delivered 〉;is correct. It log-delivers a message m from sender s by adding (s, m ) to the deliv-ered variable in stable storage. Apart from delivered , the algorithm uses two other variables, a set pending and an array ack , with the same functions as in “All-Ack Uniform Reliable Broadcast” (Algorithm 3.4). Variable pending denotes the mes-sages that the process has seen but not yet lurb -delivered, and is logged. Variable ack is not logged because it will be reconstructed upon recovery. When a message has been retransmitted by a majority of the processes, it is log-delivered. Together with the assumption of a correct majority, this ensures that at least one correct pro-cess has logged the message,
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and this will ensure the retransmission to all correct processes. Correctness. Consider the agreement property and assume that some correct pro-cess p log-delivers a message m. To do so, a majority of the processes must have 92 3 Reliable Broadcast retransmitted the message. As we assume a majority of the processes is correct, at least one correct process must have logged the message (in its variable pending ). This process will ensure that the message is eventually sbeb -broadcast to all correct processes; all correct processes will hence sbeb -deliver the message and acknow-ledge it. Hence, every correct process will log-deliver m. To establish the validity property, assume some process p lurb -broadcasts a message m and does not crash. Eventually, the message will be seen by all correct processes. As a majority of pro-cesses is correct, these processes will retransmit the message and p will eventually lurb -deliver m. The no duplication property is trivially ensured by the definition of log-delivery (the check that (s, m ) ∈ delivered before adding (s, m ) to delivered only serves to avoid unnecessary work). The no creation property is ensured by the underlying links. Performance. Suppose that some process lurb -broadcasts a message m. All correct processes log-deliver m after two communication steps and two causally related logging operations (the variable pending can be logged in parallel to broadcasting the D ATA message). # 3.8 Probabilistic Broadcast This section considers randomized broadcast algorithms, whose behavior is partially determined by a controlled random experiment. These algorithms do not provide deterministic broadcast guarantees but, instead, only make probabilistic claims about such guarantees. Of course, this approach can only be used for applications that do not require full reliability. On the other hand, full reliability often induces a cost that is too high, especially
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for large-scale systems or systems exposed to attacks. As we will see, it is often possible to build scalable probabilistic algorithms that exploit randomization and provide good reliability guarantees. Moreover, the abstractions considered in this book can almost never be mapped to physical systems in real deployments that match the model completely; some uncertainty often remains. A system designer must also take into account a small probability that the deployment fails due to such a mismatch. Even if the probabilis-tic guarantees of an abstraction leave room for error, the designer might accept this error because other sources of failure are more significant. 3.8.1 The Scalability of Reliable Broadcast As we have seen throughout this chapter, in order to ensure the reliability of broad-cast in the presence of faulty processes (and/or links with omission failures), a process needs to send messages to all other processes and needs to collect some form of acknowledgment . However, given limited bandwidth, memory, and proces-sor resources, there will always be a limit to the number of messages that each process can send and to the acknowledgments it is able to collect in due time. If the 3.8 Probabilistic Broadcast 93 > (a) (b) > Figure 3.5: Direct vs. hierarchical communication for sending messages and receiv-ing acknowledgments group of processes becomes very large (say, thousands or even millions of mem-bers in the group), a process sending out messages and collecting acknowledgments becomes overwhelmed by that task (see Fig. 3.5a). Such algorithms inherently do not scale . Sometimes an efficient hardware-supported broadcast mechanism is avail-able, and then the problem of collecting acknowledgments, also known as the ack implosion problem, is the worse problem of the two. There are several ways to make algorithms more scalable. One way is to use some form of hierarchical scheme to send
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messages and to collect acknowledgments, for instance, by arranging the processes in a binary tree, as illustrated in Fig. 3.5b. Hierarchies can reduce the load of each process but increase the latency of the com-munication protocol. Additionally, hierarchies need to be reconfigured when faults occur (which may not be a trivial task), and even with this sort of hierarchies, the obligation to send and receive information, directly or indirectly, to and from every other process remains a fundamental scalability problem of reliable broadcast. In the next section we discuss how randomized approaches can circumvent this limitation. 3.8.2 Epidemic Dissemination Nature gives us several examples of how a randomized approach can implement a fast and efficient broadcast primitive. Consider how an epidemic spreads through a population. Initially, a single individual is infected; every infected individual will in turn infect some other individuals; after some period, the whole population is infected. Rumor spreading or gossiping uses exactly the same mechanism and has proved to be a very effective way to disseminate information. A number of broadcast algorithms have been designed based on this principle and, not surprisingly, these are often called epidemic , rumor mongering , gossip , or probabilistic broadcast algorithms. Before giving more details on these algorithms, we first define the abstraction that they implement, which we call probabilistic broadcast . To illustrate how algorithms can implement the abstraction, we assume a model where processes can only fail by crashing. 94 3 Reliable Broadcast > Module 3.7: Interface and properties of probabilistic broadcast > Module: Name: ProbabilisticBroadcast, instance pb . > Events: Request: 〈pb ,Broadcast |m〉: Broadcasts a message mto all processes. > Indication: 〈pb ,Deliver |p,m〉: Delivers a message mbroadcast by process p. > Properties: PB1: Probabilistic validity: There is a positive value εsuch that when a correct pro-cess
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broadcasts a message m, the probability that every correct process eventually delivers mis at least 1−ε. > PB2: No duplication: No message is delivered more than once. > PB3: No creation: If a process delivers a message mwith sender s, then mwas previously broadcast by process s. 3.8.3 Specification The probabilistic broadcast abstraction is depicted in Module 3.7. Its interface is the same as for best-effort broadcast (Module 3.1), and also two of its three properties, no duplication and no creation , are the same. Only the probabilistic validity property is weaker than the ordinary validity property and accounts for a failure probability ε,which is typically small. As for previous communication abstractions introduced in this chapter, we assume that messages are implicitly addressed to all processes in the system, i.e., the goal of the sender is to have its message delivered to all processes of a given group, constituting what we call the system. 3.8.4 Randomized Algorithm: Eager Probabilistic Broadcast Algorithm 3.9, called “Eager Probabilistic Broadcast,” implements probabilistic broadcast. The sender selects k processes at random and sends them the message. In turn, each of these processes selects another k processes at random and forwards the message to those processes, and so on. The parameter k is called the fanout of a gossip algorithm. The algorithm may cause a process to send the message back to the same process from which it originally received the message, or to send it to another process that has already received the message. Each step consisting of receiving a message and resending it is called a round of gossiping . The algorithm performs up to R rounds of gossiping for each message. The description of the algorithm uses a function picktargets (k), which takes the fanout k as input and outputs a set of
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processes. It returns k random samples chosen from Π \ { self } according to the uniform distribution without replacement. The 3.8 Probabilistic Broadcast 95 Algorithm 3.9: Eager Probabilistic Broadcast > Implements: > ProbabilisticBroadcast, instance pb . > Uses: > FairLossPointToPointLinks, instance fll . > upon event 〈pb , Init 〉 do > delivered := ∅; > procedure gossip (msg ) is forall t ∈ picktargets (k) do trigger 〈 fll ,Send | t, msg 〉; > upon event 〈pb , Broadcast | m 〉 do > delivered := delivered ∪ { m}; > trigger 〈pb , Deliver | self , m 〉;gossip( [G OSSIP , self , m, R ]); > upon event 〈fll , Deliver | p, [G OSSIP , s, m, r ] 〉 do if m ∈ delivered then > delivered := delivered ∪ { m}; > trigger 〈pb , Deliver | s, m 〉; > if r > 1then gossip( [G OSSIP , s, m, r − 1] ); function random (S) implements the random choice of an element from a set S for this purpose. The pseudo code looks like this: > function picktargets (k) returns set of processes is > targets := ∅; > while #( targets ) candidate := random (Π \ { self }); > if candidate ∈targets then > targets := targets ∪ { candidate }; > return targets ; The fanout is a fundamental parameter of the algorithm. Its choice directly im-pacts the performance of the algorithm and the probability of successful reliable delivery (in the probabilistic validity property of probabilistic broadcast). A higher fanout not only increases the probability of having the entire population infected but also decreases the number of rounds required to achieve this. Note also that the al-gorithm induces a significant
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amount of redundancy in the message exchanges: any given process may receive the same message many times. A three-round execution of the algorithm with fanout three is illustrated in Fig. 3.6 for a system consisting of nine processes. However, increasing the fanout is costly. The higher the fanout, the higher the load imposed on each process and the amount of redundant information exchanged 96 3 Reliable Broadcast > (a) round 1 (b) round 2 (c) round 3 > Figure 3.6: Epidemic dissemination or gossip (with fanout 3)over the network. Therefore, to select the appropriate fanout value is of particular importance. Note that there are runs of the algorithm where a transmitted message may not be delivered to all correct processes. For instance, all processes that receive the message directly from the sender may select exactly the same set of k target pro-cesses and forward the message only to them, and the algorithm may stop there. In such a case, if k is much smaller than N , not all processes will deliver the message. As another example, there might be one process that is simply never selected by any process and never receives the message. This translates into the fact that reliable delivery is not guaranteed, that is, the probability that some process never delivers the message is nonzero. But by choosing large enough values of k and R in relation to N , this probability can be made arbitrarily small. Correctness. The no creation and no duplication properties are immediate from the underlying point-to-point links and from the use of the variable delivered .For the probabilistic validity property, the probability that for a particular broad-cast message, all correct processes become infected and deliver the message depends on the fanout k and on the maximum number of rounds R.We now
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derive a simple estimate of the probability that a particular correct pro-cess delivers a message. Suppose that the underlying fair-loss links deliver every message sent by the first infected correct process (i.e., the original sender) but no further message; in other words, only the sender disseminates the broadcast mes-sage. In every round, a fraction of γ = k/N processes become infected like this (some may have been infected before). The probability that a given correct pro-cess remains uninfected is at most 1 − γ. Hence, the probability that this process is infected after R rounds is at least about E1 = 1 − (1 − γ)R.Toward a second, more accurate estimate, we eliminate the simplification that only one process infects others in a round. Suppose a fraction of d = ( N − f )/N processes are correct; assume further that in every round, the number of actually infected processes is equal to their expected number. Denote the expected number of infected and correct processes after round r by Ir . Initially, only the sender is infected and I0 = 1 . After round r for r > 0, we observe that Ir−1 correct processes stay infected. Among the remaining N − Ir−1 processes, we expect that a fraction of d is correct and a fraction of γ of them becomes infected: 3.8 Probabilistic Broadcast 97 > 00.2 0.4 0.6 0.8 10510 15 20 Probability of delivery Number of rounds (R) E1 E2 > Figure 3.7: Illustration of gossip delivery probability to one correct process using the “Eager Probabilistic Broadcast” algorithm with R = 1 , . . . , 20 rounds, in terms of estimates E1 and E2 from the text Ir = Ir−1 + dγ (N − Ir−1). As all Ir processes infect others in round r +
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1 , the infections in round r + 1 spread about as fast as if one process would have infected the others during additional Ir rounds. Summing this up over all R rounds, we obtain our second estimate: the probability of some correct process being infected after R rounds is about E2 = 1 − (1 − γ)∑ R−1 > r=0 Ir . The two estimates E1 and E2 of the delivery probability for one process are plot-ted in Fig. 3.7 for a system of N = 100 processes, assuming that f = 25 faulty processes crash initially, and fanout k = 10 . Performance. The number of rounds needed for a message to be delivered by all correct processes also depends on the fanout. Every round involves one communi-cation step. The algorithm may send O(N ) messages in every round and O(N R ) messages in total, after running for R rounds; generally, the number of messages sent by the algorithm is dominated by the messages of the last round. 3.8.5 Randomized Algorithm: Lazy Probabilistic Broadcast The “Eager Probabilistic Broadcast” algorithm described earlier uses only gossiping to disseminate messages, where infected processes push messages to other pro-cesses. A major disadvantage of this approach is that it consumes considerable resources and causes many redundant transmissions, in order to achieve reliable delivery with high probability. A way to overcome this limitation is to rely on epi-demic push-style broadcast only in a first phase, until many processes are infected, 98 3 Reliable Broadcast Algorithm 3.10: Lazy Probabilistic Broadcast (part 1, data dissemination) > Implements: > ProbabilisticBroadcast, instance pb . > Uses: > FairLossPointToPointLinks, instance fll ;ProbabilisticBroadcast, instance upb .// an unreliable implementation > upon event 〈pb , Init 〉 do > next := N; > lsn := 0; > pending :=
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∅;stored := ∅; > procedure gossip (msg ) is forall t ∈ picktargets (k) do trigger 〈 fll ,Send | t, msg 〉; > upon event 〈pb , Broadcast | m 〉 do > lsn := lsn + 1 ; > trigger 〈upb , Broadcast | [D ATA , self , m, lsn ] 〉; > upon event 〈upb , Deliver | p, [D ATA , s, m, sn ] 〉 do if random ([0 , 1]) > α then > stored := stored ∪ { [DATA ,s, m, sn ]}; > if sn =next [s]then > next [s]:= next [s] + 1 ; > trigger 〈pb , Deliver | s, m 〉; > else if sn > next [s]then > pending := pending ∪ { [DATA ,s, m, sn ]}; > forall missing ∈[next [s], . . . , sn −1] do if no m′exists such that [DATA ,s, m ′,missing ]∈pending then > gossip ([ REQUEST ,self , s, missing , R −1]) ; starttimer (Δ, s, sn );and to switch to a pulling mechanism in a second phase afterward. Gossiping until, say, half of the processes are infected is efficient. The pulling phase serves a backup to inform the processes that missed the message in the first phase. The second phase uses again gossip, but only to disseminate messages about which processes have missed a message in the first phase. This idea works for scenarios where every sender broadcasts multiple messages in sequence. For describing an implementation of this idea in a compact way, we assume here that the first phase is realized by an unreliable probabilistic broadcast abstraction, as defined by Module 3.7, with a large probability ε that reliable delivery fails, in its probabilistic validity property. Concretely, we expect that a constant fraction of the
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processes, say, half of them, obtains the message after the first phase. The primitive could typically be implemented on top of fair-loss links (as the “Eager Probabilistic Broadcast” algorithm) and should work efficiently, that is, not cause an excessive amount of redundant message transmissions. Algorithm 3.10–3.11, called “Lazy Probabilistic Broadcast,” realizes probabilis-tic broadcast in two phases, with push-style gossiping followed by pulling. The 3.8 Probabilistic Broadcast 99 Algorithm 3.11: Lazy Probabilistic Broadcast (part 2, recovery) > upon event 〈fll ,Deliver |p,[REQUEST ,q, s, sn, r ]〉do if exists msuch that [DATA ,s, m, sn ]∈stored then trigger 〈fll ,Send |q,[DATA ,s, m, sn ]〉; > else if r > 0then > gossip ([ REQUEST ,q, s, sn, r −1]) ; > upon event 〈fll ,Deliver |p,[DATA ,s, m, sn ]〉do > pending := pending ∪ { [DATA ,s, m, sn ]}; > upon exists [DATA ,s, x, sn ]∈pending such that sn =next [s]do > next [s]:= next [s] + 1 ; > pending := pending \ { [DATA ,s, x, sn ]}; > trigger 〈pb ,Deliver |s,x〉; > upon event 〈 > Timeout |s, sn 〉do if sn > next [s]then > next [s]:= sn + 1 ; algorithm assumes that each sender is transmitting a stream of numbered messages. Message omissions are detected based on gaps in the sequence numbers of received messages. Each message is disseminated using an instance upb of unreliable prob-abilistic broadcast. Each message that is retained by a randomly selected set of receivers for future retransmission. More precisely, every process that upb -delivers a message stores a copy of the message with probability α during some maximum amount of time. The purpose of this approach is to distribute the load of storing messages for future retransmission among all processes. Omissions can be detected using
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sequence numbers associated with messages. The array variable next contains an entry for every process p with the sequence num-ber of the next message to be pb -delivered from sender p. The process detects that it has missed one or more messages from p when the process receives a message from p with a larger sequence number than what it expects according to next [p]. When a process detects an omission, it uses the gossip algorithm to disseminate a retrans-mission request. If the request is received by one of the processes that has stored a copy of the message then this process retransmits the message. Note that, in this case, the gossip algorithm does not have to ensure that the retransmission request reaches all processes with high probability: it is enough that the request reaches, with high probability, one of the processes that has stored a copy of the missing message. With small probability, recovery will fail. In this case, after a timeout with delay Δ has expired, a process simply jumps ahead and skips the missed messages, such that subsequent messages from the same sender can be delivered. The pseudo code of Algorithm 3.10–3.11 uses again the function picktargets (k) from the previous section. The function random ([0 , 1]) used by the algorithm returns a random real number from the interval [0 , 1] . The algorithm may invoke multiple timers, where operation starttimer (Δ, parameters ) starts a timer instance identified by parameters with delay Δ.100 3 Reliable Broadcast Garbage collection of the stored message copies is omitted in the pseudo code for simplicity. Note also that when a timeout for some sender s and sequence number sn occurs, the pseudo code may skip some messages with sender s in pending that have arrived meanwhile (be it
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through retransmissions or delayed messages from s)and that should be processed; a more complete implementation would deliver these messages and remove them from pending . Correctness. The no creation and no duplication properties follow from the under-lying point-to-point links and the use of sequence numbers. The probability of delivering a message to all correct processes depends here on the fanout (as in the “Eager Probabilistic Broadcast” algorithm) and on the reliabil-ity of the underlying dissemination primitive. For instance, if half of the processes upb -deliver a particular message and all of them were to store it (by setting α = 0 )then the first retransmission request to reach one of these processes will be suc-cessful, and the message will be retransmitted. This means that the probability of successful retransmission behaves like the probability of successful delivery in the “Eager Probabilistic Broadcast” algorithm. Performance. Assuming an efficient underlying dissemination primitive, the broad-casting of a message is clearly much more efficient than in the “Eager Probabilistic Broadcast” algorithm. It is expected that, in most cases, the retransmission request message is much smaller that the original data message. Therefore, this algorithm is also much more resource-effective than the “Eager Probabilistic Broadcast” algorithm. Practical algorithms based on this principle make a significant effort to optimize the number of processes that store copies of each broadcast message. Not surpris-ingly, the best results can be obtained if the physical network topology is taken into account: for instance, in a wide-area system with processes in multiple LANs, an omission in a link connecting a LAN with the rest of the system affects all pro-cesses in that LAN. Thus, it is desirable to have a copy of the message in each LAN (to recover from local omissions) and a copy outside the LAN (to recover from the omission
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in the link to the LAN). Similarly, the retransmission procedure, instead of being completely random, may search first for a copy in the local LAN and only afterward at more distant processes. # 3.9 FIFO and Causal Broadcast So far, we have not considered any ordering guarantee among messages delivered by different processes. In particular, when we consider a reliable broadcast abstraction, messages can be delivered in any order by distinct processes. In this section, we introduce reliable broadcast abstractions that deliver messages according to first-in first-out (FIFO) order and according to causal order . FIFO order ensures that messages broadcast by the same sender process are delivered in the order in which they were sent. Causal order is a generalization of FIFO order that additionally preserves the potential causality among messages from multiple senders. These orderings are orthogonal to the reliability guarantees.
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Title: URL Source: Markdown Content: # Some problems I have proposed # Warut Suksompong 1. Asian Pacific Mathematics Olympiad 2022, Problem 3 Find all positive integers k r .3. Crux Mathematicorum, September 2021, Problem 4669 For a given positive integer n, a 4 n × 4n table is partitioned into 16 n2 unit squares, each of which is coloured in one of 4 given colours. A set of four cells is called colourful if the centers of the cells form a rectangle with sides parallel to the sides of the table, and the cells are coloured in all four different colours. Determine the maximum number of colourful sets. 4. International Mathematical Olympiad 2021 Shortlist, Problem C4 The kingdom of Anisotropy consists of n cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from X to Y is a sequence of roads such that one can move from X to Y along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection. Let A and B be two distinct cities in Anisotropy. Let NAB denote the maximal number of
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paths in a diverse collection of paths from A to B. Similarly, let NBA denote the maximal number of paths in a diverse collection of paths from B to A.Prove that the equality NAB = NBA holds if and only if the number of roads going out from A is the same as the number of roads going out from B.5. Asian Pacific Mathematics Olympiad 2021, Problem 2 For a polynomial P and a positive integer n, define Pn as the number of positive integer pairs ( a, b ) such that a < b ≤ n and |P (a)| − | P (b)| is divisible by n.1Determine all polynomials P with integer coefficients such that Pn ≤ 2021 for all positive integers n.6. Asian Pacific Mathematics Olympiad 2021, Problem 3 Let ABCD be a cyclic convex quadrilateral and Γ be its circumcircle. Let E be the intersection of the diagonals AC and BD , let L be the center of the circle tangent to sides AB , BC , and CD , and let M be the midpoint of the arc BC of Γ not containing A and D. Prove that the excenter of triangle BCE opposite E lies on the line LM .7. International Mathematical Olympiad 2020 Shortlist, Problem C7 Consider any rectangular table having finitely many rows and columns, with a real number a(r, c ) in the cell in row r and column c. A pair ( R, C ), where R is a set of rows and C a set of columns, is called a saddle pair if the following two conditions are satisfied: (i) For each row r′, there is r ∈ R such that a(r, c ) ≥ a(r′, c ) for all c ∈ C;(ii) For each column c′, there is c
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∈ C such that a(r, c ) ≤ a(r, c ′) for all r ∈ R.A saddle pair ( R, C ) is called a minimal pair if for each saddle pair ( R′, C ′) with R′ ⊆ R and C′ ⊆ C, we have R′ = R and C′ = C.Prove that any two minimal pairs contain the same number of rows. 8. Asian Pacific Mathematics Olympiad 2020, Problem 3 Determine all positive integers k for which there exist a positive integer m and a set S of positive integers such that any integer n > m can be written as a sum of distinct elements of S in exactly k ways. 9. Asian Pacific Mathematics Olympiad 2020, Problem 4 Let Z denote the set of all integers. Find all polynomials P (x) with integer coeffi-cients that satisfy the following property: For any infinite sequence a1, a 2, . . . of integers in which each integer in Z appears exactly once, there exist indices i < j and an integer k such that ai+ai+1 +· · · +aj = P (k). 10. Crux Mathematicorum, January 2020, Problem 4504 Find all positive integers ( a, b, c, x, y, z ), a ≤ b ≤ c and x ≤ y ≤ z, for which the following two equations hold: a + b + c = xy + yz + zx, x + y + z = abc. 11. Crux Mathematicorum, November 2019, Problem 4481 2Find all functions f : R → R such that f (x2 + y2) = f (x + y)f (x − y) + 2 f (y)y for all x, y ∈ R.12. Crux Mathematicorum, October 2019, Problem 4477 Given a positive integer n, let a1 ≥ · · · ≥ an ≥ 0 and
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b1 ≥ · · · ≥ bn ≥ 0 be integers such that 1. a1 + · · · + ai ≥ b1 + · · · + bi for all i = 1 , . . . , n − 1; 2. a1 + · · · + an = b1 + · · · + bn.Assume that there are n boxes, with box i containing ai balls. In each move, Alice is allowed to take two boxes with an unequal number of balls, and move one ball from the box with more balls to the other box. Prove that Alice can perform a finite number of moves after which each box i contains bi balls. 13. International Mathematical Olympiad 2019 Shortlist, Problem C2 You are given a set of n blocks, each weighing at least 1; their total weight is 2 n.Prove that for every real number r with 0 ≤ r ≤ 2n − 2 you can choose a subset of the blocks whose total weight is at least r but at most r + 2. 14. International Mathematical Olympiad 2019 Shortlist, Problem C8 Alice has a map of Wonderland, a country consisting of n ≥ 2 towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns. Alice wants to know whether there is at least one town in Wonderland
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with at most one outgoing road. Prove that she can always find out by asking at most 4 n questions. 15. Asian Pacific Mathematics Olympiad 2019, Problem 1 Let Z+ be the set of positive integers. Determine all functions f : Z+ → Z+ such that a2 + f (a)f (b) is divisible by f (a) + b for all positive integers a and b.16. Mathematics Magazine, December 2018, Problem Q1086 Find the largest natural number n with the property that there exist n points on the plane, not all collinear, such that no three non-collinear points are vertices of an obtuse triangle. 317. International Mathematical Olympiad 2018 Shortlist, Problem N7 (with Pakawut Jiradilok) Let n ≥ 2018 be an integer, and let a1, a 2, . . . , a n, b 1, b 2, . . . , b n be pairwise distinct positive integers not exceeding 5 n. Suppose that the sequence a1 b1 , a2 b2 , . . . , an bn forms an arithmetic progression. Prove that the terms of the sequence are equal. 18. International Mathematical Olympiad 2017 Shortlist, Problem C3 Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: (1) Choose any number of the form 2 j , where j is a non-negative integer, and put it into an empty cell. (2) Choose two (not necessarily adjacent) cells with the same number in them; denote that number by 2 j . Replace the number in one of the cells with 2 j+1 and erase the number in the other cell. At the end of the game, one cell contains the number 2 n, where n is a
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given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of n.19. International Mathematical Olympiad 2017 Shortlist, Problem N3 Determine all integers n ≥ 2 with the following property: for any integers a1, a 2, . . . , a n whose sum is not divisible by n, there exists an index 1 ≤ i ≤ n such that none of the numbers ai, a i + ai+1 , . . . , a i + ai+1 + · · · + ai+n−1 is divisible by n. (We let ai = ai−n when i > n .) 20. Asian Pacific Mathematics Olympiad 2017, Problem 1 We call a 5-tuple of integers arrangeable if its elements can be labeled a, b, c, d, e in some order so that a − b + c − d + e = 29. Determine all 2017-tuples of integers n1, n 2, . . . , n 2017 such that if we place them in a circle in clockwise order, then any 5-tuple of numbers in consecutive positions on the circle is arrangeable. 21. Asian Pacific Mathematics Olympiad 2017, Problem 5 (with Pakawut Jiradilok) Let n be a positive integer. A pair of n-tuples ( a1, . . . , a n) and ( b1, . . . , b n) with integer entries is called an exquisite pair if |a1b1 + · · · + anbn| ≤ 1. Determine the maximum number of distinct n-tuples with integer entries such that any two of them form an exquisite pair. 422. International Mathematical Olympiad 2016 Shortlist, Problem N1 For any positive integer k, denote the sum of digits of k in its decimal representation by S(k). Find all polynomials P (x) with integer
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coefficients such that for any positive integer n ≥ 2016, the integer P (n) is positive and S(P (n)) = P (S(n)) . 23. Asian Pacific Mathematics Olympiad 2016, Problem 3 Let AB and AC be two distinct rays not lying on the same line, and let ω be a circle with center O that is tangent to ray AC at E and ray AB at F . Let R be a point on segment EF . The line through O parallel to EF intersects the line AB at P . Let N be the intersection of lines P R and AC , and let M be the intersection of line AB and the line through R parallel to AC . Prove that line M N is tangent to ω.24. Asian Pacific Mathematics Olympiad 2016, Problem 4 The country Dreamland consists of 2016 cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer k such that no matter how Starways establishes its flights, the cities can always be partitioned into k groups so that from any city it is not possible to reach another city in the same group by using at most 28 flights. 25. Asian Pacific Mathematics Olympiad 2015, Problem 1 Let ABC be a triangle, and let D be a point on side BC . A line through D intersects side AB at X and ray AC at Y . The circumcircle of triangle BXD intersects the circumcircle ω of triangle ABC again at point Z 6 = B. The lines ZD and ZY intersect ω again at V and W , respectively. Prove that AB = V W .26. Asian Pacific Mathematics Olympiad
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2015, Problem 4 (with Pakawut Jiradilok) Let n be a positive integer. Consider 2 n distinct lines on the plane, no two of which are parallel. Of the 2 n lines, n are colored blue, the other n are colored red. Let B be the set of all points on the plane that lie on at least one blue line, and R the set of all points on the plane that lie on at least one red line. Prove that there exists a circle that intersects B in exactly 2 n − 1 points, and also intersects R in exactly 2 n − 1 points. 27. Asian Pacific Mathematics Olympiad 2015, Problem 5 (with Pakawut Jiradilok) Determine all sequences a0, a 1, a 2, . . . of positive integers with a0 ≥ 2015 such that for all integers n ≥ 1: (i) an+2 is divisible by an;(ii) |sn+1 − (n + 1) an| = 1, where sn+1 = an+1 − an + an−1 − · · · + ( −1) n+1 a0.528. Asian Pacific Mathematics Olympiad 2014, Problem 2 Let S = {1, 2, . . . , 2014 }. For each non-empty subset T ⊆ S, one of its members is chosen as its representative . Find the number of ways to assign representatives to all non-empty subsets of S so that if a subset D ⊆ S is a disjoint union of non-empty subsets A, B, C ⊆ S, then the representative of D is also the representative of at least one of A, B, C.29. Asian Pacific Mathematics Olympiad 2014, Problem 3 Find all positive integers n such that for any integer k there exists an integer a for which a3 + a − k is divisible by n.30. International Mathematical Olympiad 2013, Problem 4 (with
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Potcharapol Suteparuk) Let ABC be an acute-angled triangle with orthocenter H, and let W be a point on the side BC , lying strictly between B and C. The points M and N are the feet of the altitudes from B and C, respectively. Denote by ω1 the circumcircle of BW N ,and let X be the point on ω1 such that W X is a diameter of ω1. Analogously, denote by ω2 the circumcircle of CW M , and let Y be the point on ω2 such that W Y is a diameter of ω2. Prove that X, Y and H are collinear. 31. United States of America Mathematical Olympiad 2013, Problem 3 Let n be a positive integer. There are n(n+1) 2 marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing n marks. Initially, each mark has the black side up. An operation is to choose a line parallel to one of the sides of the triangle, and flipping all the marks on that line. A configuration is called admissible if it can be obtained from the initial configuration by performing a finite number of operations. For each admissible configuration C, let f (C) denote the smallest number of operations required to obtain C from the initial configuration. Find the maximum value of f (C), where C varies over all admissible configurations. 32. International Mathematical Olympiad 2012 Shortlist, Problem C1 Several positive integers are written in a row. Iteratively, Alice chooses two ad-jacent numbers x and y such that x > y and x is to the left of y, and replaces the pair ( x, y ) by either ( y + 1 , x ) or ( x − 1, x ). Prove that she
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can perform only finitely many such iterations. 33. International Mathematical Olympiad 2012 Shortlist, Problem N1 Call admissible a set A of integers that has the following property: If x, y ∈ A (possibly x = y) then x2 + kxy + y2 ∈ A for every integer k.Determine all pairs m, n of nonzero integers such that the only admissible set containing both m and n is the set of all integers. 634. United States of America Junior Mathematical Olympiad 2012, Problem 5 For distinct positive integers a, b 1 + 2 xn > 2 + · · · + 2011 xn > 2011 = an+1 + 1 . 7
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Title: Proof by Contrapositive: If n^2 is Even then n is Even URL Source: Markdown Content: Proof by Contrapositive: If n^2 is Even then n is Even - YouTube =============== • NaN / NaN Back ![Image 1 SA Skip navigation Search Search with your voice Sign in SA Proof by Contrapositive: If n^2 is Even then n is Even Search Watch later Share Copy link Info Shopping Tap to unmute 2x !Image 3") 0:00 / 0:00 •Watch full video Live • • Scroll for details Sign in to confirm you’re not a bot This helps protect our community. [Learn more]( [Sign in]( Proof by Contrapositive: If n^2 is Even then n is Even ====================================================== [![Image 6: Wrath of Math]( [Wrath of Math]( Wrath of Math Verified 242K subscribers Join Subscribe Subscribed 620 Share Download Download 38K views 3 years ago[#Proofs]( 38,934 views • Jul 9, 2021 • #Proofs Show less Using the contrapositive, we prove that if n^2 is even then n is even. A proof by contrapositive is not necessary here, we'll touch on how it could be done directly, but this is definitely a case where the contrapositive is more obvious. We'll also recap what the contrapositive actually is, and why…...more ...more [![Image 7]( ### Wrath of Math 242K subscribers]( [Videos]( [About]( [Videos]( 8]( Support me on Patreon!]( 9]( Follow on Instagram!]( 10]( Follow on
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TikTok!]( Show less []( Proof by Contrapositive: If n^2 is Even then n is Even ====================================================== 38,934 views 38K views Jul 9, 2021 620 Share Download Download Save 40 Comments ----------- Sort comments Sort by Top comments Newest first !Image 11: Default profile photo]( Can't wait for the calculator documentary. Wait its out im going to watch it. Show less Read more Like 4 Dislike !Image 24 and, therefore, makes n an even number was hard for me to grasp the first time around. Show less Read more Like 3 Dislike ![Image 28]( ❤ by Wrath of Math
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Reply ![Image 29]( · 1 reply ![Image 30]( · 1 reply ![Image 31]( ### [@Zero-ef4sc]( [5 months ago]( Why can't n^2 = 2 which disproves the statement? Since squareroot of 2 is irrational, it isn't even. Though, I'd assume that we're limiting ourselves to integers in which case it makes sense, but that's never stated. Show less Read more Like Dislike Reply 1 reply 1 reply ![Image 32]( ### [@LilJollyJoker]( [8 months ago]( Thank You! Show less Read more Like 1 Dislike ![Image 33]( ❤ by Wrath of Math Reply ![Image 34]( · 1 reply ![Image 35]( · 1 reply ![Image 36]( ### [@krasimirronkov17]( [3 years ago]( Can you make a video on proving the fundamental theorem of arithmetic Show less Read more Like Dislike Reply ![Image 37]( ### [@mutalemwenya10a85]( [7 months ago]( Where did the 2 come from? Show less Read more Like Dislike Reply ![Image 38]( ### [@marriamhaji7440]( [1 year ago]( Thank you so much ![Image 39: ❤]( Show less Read more Like Dislike ![Image 40]( ❤ by Wrath of Math Reply ![Image 41]( · 1 reply ![Image 42]( · 1 reply ![Image 43]( ### [@Blendingpassion]( [2 years ago]( Could you proof this same statement using the direct method? Show less Read more Like Dislike Reply ![Image 44]( ### [@kibromhabtom964]( [2 years ago]( Nice explanation nice proof Thanks . Show less Read more Like Dislike ![Image 45]( ❤ by Wrath of Math Reply ![Image 46]( · 1 reply ![Image 47]( · 1 reply ![Image 48]( ### [@aashsyed1277]( [3 years ago]( *provided n is a integer Show less Read more Like 2 Dislike ![Image 49]( ❤ by Wrath of Math Reply ![Image 50]( · 1 reply ![Image 51]( · 1 reply ![Image 52]( ### [@ZOROZURO-s8k]( [11 months ago]( Beautifully explained ![Image 53: 😊]( Show less Read more Like Dislike
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![Image 54]( ❤ by Wrath of Math Reply ![Image 55]( · 1 reply ![Image 56]( · 1 reply ![Image 57]( ### [@SeeTv.]( [3 years ago]( In this case couldn't you just prove the converse, i.e. "when n is even then n^2 is even" ? Show less Read more Like 1 Dislike ![Image 58]( ❤ by Wrath of Math Reply ![Image 59]( · 4 replies ![Image 60]( · 4 replies ![Image 61]( ### [@edvinshanuka]( [1 year ago]( Can you explain if n² is even, then a is even Show less Read more Like Dislike Reply 1 reply 1 reply ![Image 62]( ### [@tech-guide7894]( [1 year ago]( Nice proof why don't you teach on youtube Show less Read more Like 1 Dislike ![Image 63]( ❤ by Wrath of Math Reply ![Image 64]( · 2 replies ![Image 65]( · 2 replies ![Image 66]( ### [@swx6074]( [1 year ago]( Impressive Show less Read more Like Dislike ![Image 67]( ❤ by Wrath of Math Reply ![Image 68]( · 2 replies ![Image 69]( · 2 replies ![Image 70]( ### [@FortunatePalichani]( [6 months ago]( ![Image 71: 😊]( 72: 😢]( 73: 😅]( 74: 😅]( 75: 😅]( Show less Read more Like Dislike Reply ![Image 76]( Comments 40 ----------- Top comments Newest first ![Image 77]( Description ----------- Proof by Contrapositive: If n^2 is Even then n is Even 620 Likes 38,934 Views 2021 Jul 9 Using the contrapositive, we prove that if n^2 is even then n is even. A proof by contrapositive is not necessary here, we'll touch on how it could be done directly, but this is definitely a case where the contrapositive is more obvious. We'll also recap what the contrapositive actually is, and why we care about it. [#Proofs]( ★DONATE★ ◆ Support Wrath of Math on Patreon for early access to new videos and other
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exclusive benefits: [![Image 78]( ◆ Donate on PayPal: [ Thanks to Robert Rennie, Barbara Sharrock, and Rolf Waefler for their generous support on Patreon! Thanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: [ Follow Wrath of Math on... ● Instagram: [![Image 79]( ● Facebook: [![Image 80]( ● Twitter: [![Image 81]( My Music Channel: [![Image 82]( ...more Show less [![Image 83]( ### Wrath of Math 242K subscribers]( [Videos]( [About]( [Videos]( 84]( Support me on Patreon!]( 85]( Follow on Instagram!]( 86]( Follow on TikTok!]( NaN / NaN [![Image 87]( 11:59 11:59 Now playing]( [### Proof by Contrapositive Brain Gainz Brain Gainz • • 21K views 5 years ago]( [![Image 88]( 6:56 6:56 Now playing]( [### Proof that Square Root 2 is Irrational Center of Math Center of Math • • 201K views 9 years ago]( [![Image 89]( 18:08 18:08 Now playing]( [### Mathematical Induction Practice Problems The Organic Chemistry Tutor The Organic Chemistry Tutor Verified • • 2.1M views 7 years ago]( [![Image 90]( 10:58 10:58 Now playing]( [### Proof by Contraposition Eddie Woo Eddie Woo Verified • • 18K views 4 years ago]( [![Image 91]( 13:39 13:39 Now playing]( [### Proof by contradiction Starfish Maths Starfish Maths • • 19K views 4 years ago]( [![Image 92]( 11:39 11:39 Now playing]( [### The Surprising Math of a Three Person Duel Wrath of Math Wrath of Math Verified • • 65K views 1 month ago]( [![Image 93]( 22:19 22:19 Now playing]( [### Proof by Contradiction | Explanation + 5 Examples Brain Gainz Brain Gainz • • 76K views 5 years ago]( [![Image 94]( 18:25 18:25 Now playing]( [### The SAT Question Everyone Got Wrong Veritasium Veritasium Verified • • 16M views 1 year ago]( [![Image 95](
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7:55 7:55 Now playing]( ### 1.2.1 Proof by Contradiction MIT OpenCourseWare MIT OpenCourseWare • • 116K views 8 years ago Nakia Rimmer (CalcCoach.com) • • 37K views 6 years ago]( ![Image 98 BriTheMathGuy BriTheMathGuy Verified • • 363K views 9 months ago]( [![Image 105]( 6:45 6:45 Now playing]( [### Directly Prove that if x^2 - 6x + 5 is even, then x is odd; math textbook "Book of Proof" R. Hammack My One Fiftieth Of A Dollar My One Fiftieth Of A Dollar • • 1K views 2 years
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ago]( [![Image 106]( 18:46 18:46 Now playing]( [### Proof By Cases | 2 Examples Brain Gainz Brain Gainz • • 12K views 4 years ago]( Show more
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Title: How do I prove the statement "All even integers are divisible by 2." using proof by contradiction? : r/learnmath URL Source: Markdown Content: How do I prove the statement "All even integers are divisible by 2." using proof by contradiction? : r/learnmath =============== Skip to main content. * * * 386K Members Online •5 yr. ago [deleted] How do I prove the statement "All even integers are divisible by 2." using proof by contradiction? ================================================================================================== I got this question as a homework and honestly it doesn't make sense to me. Even numbers are by definition integers that are divisible by 2. So I don't see what I am suppose to prove when the statement is true by definition. Read more Share Share [![Image 4: u/monday_com avatar]( That feeling when you've completed all your tasks! It's exhilarating! Well, with monday.com’s work management platform, get more done in less time with no-code automations, real-time communication, and notifications! Smash that done button! Try now! Sign Up monday.com ![Image 5: Thumbnail image: That feeling when you've completed all your tasks! It's exhilarating! Well, with monday.com’s work management platform, get more done in less time with no-code automations, real-time communication, and notifications! Smash that done button! Try now!]( Sort by: Best Open comment sort options * Best * Top *
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New * Controversial * Old * Q&A ![Image 6: u/theadamabrams avatar definition of "even integer"? Without knowing that, it's impossible to say what kind of proof to write. "An even integer is an integer that is divisible by two" is a perfectly valid definition, and in that case there is literally nothing to prove, but there are other definitions, such as "An even integer is a number that can be represented as two times some integer." Reply reply } Share Share 2 more replies 2 more replies More replies, but often times excercises are not about proving something you didn't know, but rather building _mental agility_ for when you have to apply the same methods to more complicated proofs, which is very useful in my experience. I remember back in the days i didn't really have the experience to tell when it's more convenient wether to use a proof by contradiction or contrapositive, or even a direct one, which is quite embarrassing when you're asked to do them on the spot Reply reply } Share Share []( [impossiblePie287]( •[5y ago]( Edited 5y ago When we try
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to prove anything using the method of contradiction, we start by assuming the opposite of what we are trying to prove. When we continue to progress with our assumption, we suddenly come at a contradiction that proves that our assumption is wrong and the fact that we made our assumption against is correct. So let's suppose 2n is an even integer where n is any integer. Now let's assume that 2n is not divisible by 2 i.e. any even integer doesn't have 2 as it's factor. But our assumption comes to an contradiction as we can see that 2n (an even integer) actually has 2 as a factor. This means that our initial assumption (2 isn't a factor of an even integer 2n) is wrong and hence 2 is a factor of 2n i.e. an even integer is divisible by 2. Reply reply } Share Share []( Mirehi mod 3 = 2 & (x - 1) mod 2 = 1 (x - 2) mod 2 = 0 x mod 3 = 0 & x mod 2 = 0 (x + 1) mod 3 = 1 & (x + 1) mod 2 = 1 (x + 2) mod 2 = 0 There's just no other place for a prime to be, so this means there is a number you can construct: (6 * y) - 1 OR (6 * y) + 1 which is even and prime, if you'd assume there's an even number which isn't divisible by 2 I think it's pretty hilarious to think about Reply reply } Share Share New
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to Reddit? Create your account and connect with a world of communities. المواصلة باستخدام Google المواصلة باستخدام Google Continue with Email Continue With Phone Number By continuing, you agree to ourUser Agreement. * * * 386K Members Online [### Proving that the sum of two even integers gives an even integer]( 1 upvote ·11 comments * * * * [Prove that every integer that is divisible by 10 must be an even number]( 9: r/askmath icon]( r/askmath]( yr. ago ![Image 10: r/askmath icon]( This subreddit is for questions of a mathematical nature. Please read the subreddit rules below before posting. * * * 198K Members Online [### Prove that every integer that is divisible by 10 must be an even number]( 2 comments * * * * [Are Negative Numbers Even and Odd?]( 11: r/maths icon]( r/maths]( mo. ago ![Image 12: A banner for the subreddit]( ![Image 13: r/maths icon]( r/maths is a community dedicated to problem-solving in mathematics across various topics and levels. * * * 59K Members Online [### Are Negative Numbers Even and Odd?]( 10 upvotes ·20 comments * * * * Promoted ![Image 14: sidebar promoted post thumbnail]( do even numbers look like good guys and odd look like bad guys .]( 15: r/RandomThoughts icon]( r/RandomThoughts]( mo. ago ![Image 16: A banner for the subreddit]( ![Image 17: r/RandomThoughts icon]( Unbeckoned, they pop into your head. * * * 875K Members Online [### Why do even numbers look like
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good guys and odd look like bad guys .]( 105 upvotes ·63 comments * * * * How can I prove that no odd number is even?. * * * 386K Members Online ### How can I prove that no odd number is even?. * * * 386K Members Online [### Proof that the sum of consecutive numbers cannot be powers of 2?]( 35 upvotes ·55 comments * * * * [The sum of two even integers is even, the sum of an even and an odd integer is odd, and the sum of two odd integers is even. What is the generalization of this statement to residue classes mod 3?]( 24: r/learnmath icon]( r/learnmath]( yr. ago ![Image 25: r/learnmath icon]( Post all of your math-learning resources here. Questions, no matter how basic, will be answered (to the best ability of the
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online subscribers). * * * 386K Members Online ### The sum of two even integers is even, the sum of an even and an odd integer is odd, and the sum of two odd integers is even. What is the generalization of this statement to residue classes mod 3?. * * * 386K Members Online ### Why is it that a number whose digits add up to a number divisible by 3 is also divisible by 3?]( 31: r/learnmath icon](
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r/learnmath]( yr. ago !Image 32: r/learnmath icon. * * * 386K Members Online ### Factoring: Even numbers are always divisible by 2, why aren't odd numbers always divisible by some fixed number besides 1? (that aren't primes). * * * 386K Members Online ### Can someone explain how subtracting integers work? have two answers?]( 37: r/learnmath icon]( r/learnmath]( mo. ago !Image 38: r/learnmath icon. * * * 386K Members Online ### Does 2^(-1) have two answers?. * * * 386K Members Online [### need help with math riddle]( 7 upvotes ·14 comments
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* * * * Help understanding endless primes proof from book. * * * 386K Members Online ### Help understanding endless primes proof from book = 1 work when proving sqrt(2) is irrational?]( 47: r/learnmath icon]( r/learnmath]( yr. ago !Image 48: r/learnmath icon. * * * 386K Members Online [### Why does the assumption gcd(a, b) = 1 work when proving sqrt(2) is irrational?]( 3 upvotes ·25 comments * * * * [Given an integer N, print all the even numbers from 0 to N in descending order.]( 49: r/learnpython icon]( r/learnpython]( yr. ago ![Image 50: r/learnpython icon]( Subreddit for posting questions and asking for general advice about all topics related to learning python. * * * 932K Members Online [### Given an integer N, print all the even numbers from
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0 to N in descending order.]( 6 comments * * * * If we erased all math, how different do you think it would eventually be?. * * * 386K Members Online ### If we erased all math, how different do you think it would eventually be?. * * * 386K Members Online ### How do I know how many times a number is divisible by 2 till you reach an odd number?. * * * 386K Members Online [### Where can I read that long proof in logic that 1+1=2?]( 65 upvotes ·50 comments * *
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Title: URL Source: Markdown Content: Math 2001, January 14, 2020 # A very first proof Definition. An integer n is called even if it has the form n = 2 k for some integer k. Theorem. If n is an even integer, then n2 is an even integer. Proof. Suppose n is an even integer. Then n has the form n = 2 k for some integer k.Therefore, squaring both sides, n2 = (2 k)2 = 4 k2 = 2(2 k2). Since k is an integer, so is 2 k2.Therefore n2 has the form n2 = 2 `, where ` = 2 k2 is an integer. Therefore, n2 is even. # Some other phrasings Definition 1. Let n be an integer. We say that n is even if it is equal to twice some other integer. Theorem 2. The square of an even integer is even. Proof. Assume that n is an even integer. Then we may write n = 2 k with k being an integer. Squaring both sides, we see that n2 = (2 k)2.We may rewrite this as n2 = 2(2 k2). Now, we have expressed n2 as twice the integer 2 k2.Therefore, n2 is even. Proof. Let n be an even integer. Then, by the definition of an even integer, n = 2 k for some k ∈ Z.Squaring, we obtain n2 = (2 k)2 = 4 k2 = 2(2 k2). Note that 2 k2 is an integer, since it is a product of integers. Therefore n2 = 2 r with r = 2 k2 ∈ Z.Hence n2 is even, by the definition of an even integer. 1The wordy proof Let us prove the theorem together. For the sake of argument, please imagine that you have selected an even integer, and call it n. It
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could be any even integer at all. Now, since it is even, it must be twice another integer. Let’s call the new one k. In other words, n is twice k, which we can write as n = 2 k.Next, the equation n = 2 k can be squared on both sides to yield another correct equation, namely: n2 = (2 k)2.We can rewrite the right-hand side of this somewhat, so that we get n2 =2(2 k2). So far, what we’ve learned is that for whatever even n you picked, there is an integer k so that together they satisfy this equation, namely, n2 = 2(2 k2). But this equation says that n2 is twice 2 k2.And 2 k2 is an integer, since it is a product of integers. So n2 is twice an integer. But that means n2 is even. So, let’s sum up the argument. I have described a method whereby, no matter which even integer n you had in mind, I showed you how to see that n2 is even. More precisely, given that you knew what to double to obtain n, I was able to use that knowledge to show you what to double to obtain n2. I gave you recipe for turning knowledge of n into knowledge of n2: if n = 2 k then n2 = 2(2 k2). # An example run through You pick 6. Six is even since it is twice 3. So we will apply the proof recipe our example, n = 6 and k = 3. In particular, the proof says that n2 should be twice 2 k2. That is, 36 should be twice 18. It worked! 2
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Title: Renaming URL Source: Markdown Content: **Note:** You are looking at a static copy of the former PineWiki site, used for class notes by James Aspnes and output y i, with the requirements: Termination Every nonfaulty process eventually decides. Uniqueness If p i ≠ p j then y i ≠ y j. Anonymity The code executed by any process depends only on its input x i: for any execution of processes p 1..p n with inputs x 1..x n, and any permutation π of [1..n], there is a corresponding execution
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of processes p π(1)..p π(n) with inputs x 1..x n in which p π(i) performs exactly the same operations as p i and obtains the same output y i. The last condition is like non-triviality for consensus: it excludes algorithms where p i just returns i in all executions. Typically we do not have to do much to prove anonymity other than observing that all processes are running the same code. We will be considering renaming in a shared-memory system, where we only have atomic registers to work with. 2. Performance -------------- Conventions on counting processes: * N = number of possible original names. * n = maximum number of processes. * k = number of processes that actually execute the algorithm. Ideally, we'd like any performance measures we get to depend on k alone if possible (giving an **adaptive** algorithm). Next best would be something polynomial in n and k. Anything involving N is bad. 3. Order-preserving renaming ---------------------------- One variant of renaming considered in the Attiya et al. paper is **order-preserving renaming**: here we require that y i< y j whenever x i< x j. Unfortunately, this requires a very large output namespace: with t failures, any asynchronous algorithm for order-preserving renaming requires 2^t(n-t+1)-1 possible output names. This lower bound applies regardless of the model, as long as some processes may start after other processes have already been assigned names. For the wait-free case, we have t = n-1, and the bound becomes just 2 n-1. This is a simpler case than the general t-failure case but the essential idea is the same: if I've only seen a few of the processes, I need to leave room for the others. Claim There is no order-preserving renaming algorithm for n processes using fewer than 2 n-1 names. Proof By
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induction on n. For n=1, we use 2 1-1=1 names; this is the base case. For larger n, suppose we use m names, and consider an execution in which one process p n runs to completion first. This consumes one name y n and leaves k names less than y n and m-k-1 names greater than y n. By setting all the inputs x i for i = 2 n-1-1 and m-k-1 >= 2 n-1-1, so m = k+(m-k-1)+1 >= 2(2 n-1-1)+1 = 2 n-1. 4. Wait-free renaming with 2n-1 names ------------------------------------- Here we use Algorithm 55 from AttiyaWelch * s ← 1 * for ever: * a[i] ← s * view ← snapshot(a) * if view[j] = s for some j, then: * r ← | { j with view[j] ≠ ⊥ and j ≤ i } | * s ← r-th positive integer not in { view[j] | j ≠ i and view[j] = ⊥ } * else: * return s The array a holds proposed names for each process (indexed by the original names), or ⊥ for processes that have not proposed
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a name yet. If a process proposes a name and finds that no other process has proposed the same name, it takes it; otherwise it chooses a new name by first computing its rank r among the active processes and then choosing the r-th unused name. Because the rank is at most n and there are at most n-1 names used by the other processes, this always gives proposed names in the range [1..2n-1]. But it remains to show that the algorithm does in fact satisfy uniqueness and termination. For uniqueness, consider two process with original names i and j. Suppose that i and j both decide on s. Then i sees a view in which a[i] = s and a[j] ≠ s, after which it no longer updates a[i]. Similarly, j sees a view in which a[j] = s and a[i] ≠ s, after which it no longer updates a[j]. If i's view is obtained first, then j can't see a[i] ≠ s, but the same holds if j's view is obtained first. So in either case we get a contradiction, proving uniqueness. Termination is a bit trickier. Here we argue that no process can run forever without picking a name, by showing that if we have a set of processes that are doing this, the one with smallest original name eventually picks a name. More formally, call a process **trying** if it runs for infinitely many steps without choosing a name. Then in any execution with at least one trying process, eventually we reach a configuration where all processes have either finished or are trying. In some subsequent configuration, all the processes have written to the a array at least once; after this point, any process reading the a array will see the same set of original names
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and compute the same rank r for itself. Now look at the trying process i with the smallest original name, and suppose it has rank r. Let F = { z 1< z 2 ... } be the set of "free names" that are not proposed in a by any of the finished processes. Observe that no trying process j != i ever proposes a name in { z 1 ... z r }, because any such process has rank greater than r. This leaves z r open for i to claim, provided the other names in { z 1 ... z r } eventually become free. But this will happen, because only trying processes may have proposed these names (early on in the execution, when the finished processes hadn't finished yet), and the trying processes eventually propose new names that are not in this range. So eventually process i proposes z r, sees no conflict, and finishes, contradicting the assumption that it is trying. Note that we haven't proved any complexity bounds on this algorithm at all, but we know that the snapshot alone takes at least Omega(N) time and space. 5. Lower bounds --------------- Most work on lower bounds has concentrated on bounding the size of the name space. This is tricky, and the current known lower bounds depend on a reduction to problems in topology. See TopologicalMethodsInDistributedComputing. 6. Long-lived renaming ---------------------- In **long-lived renaming** a process can release a name for later use by other processes (or the
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{
"page_id": null,
"source": 7357,
"title": "from dpo"
}
|
same process, if it happens to run choose-name again). Now the bound on the number of names needed is 2k-1, where k is the maximum number of concurrently active processes. The algorithm above can be converted to a long-lived renaming algorithm by adding the following release-name procedure: procedure release-name(i) * a[i] ← ⊥ Here the termination requirement is weakened slightly, to say that some process always makes progress in choose-name (see ObstructionFreedom, Science of Computer Programming 25:1–39, 1995, gives a renaming protocol that is somewhat easier to understand and doesn't require taking snapshots over huge arrays. A downside is that the basic version requires k(k+1)/2 names to handle k active processes (we'll be lazy and only prove k 2). 7.1. Splitters -------------- The Moir-Anderson renaming protocol uses a network of **splitters**.1 at least one process either goes right or gets stop; and (b) at least one process either goes down or gets stop. Another way of saying this is that of all the processes that arrive at a splitter, some process doesn't go down and some process doesn't go right. By arranging splitters in a grid, this property guarantees that every row or column that gets at least one process gets to
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{
"page_id": null,
"source": 7357,
"title": "from dpo"
}
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keep it—which means that with k processes, no process reaches row k+1 or column k+1. Here is the actual code for a splitter. We use two atomic registers: a register X, initialized to ⊥, that can hold original names, and a register Y, initialized to false, that holds a single flag bit (the register names are from the paper). Splitter(id) 1. X ← id 2. if Y = true: go right 3. Y ← true 4. if X = id: stop 5. else: go down Claim 1 If at least one process completes the splitter, at least one process stops or goes right. Proof Let p be the last process to write to X. Then either (a) p sees true in Y and goes right, or (b) p sees false in Y, reads id from X, and stops. Claim 2 If at least one process completes the splitter, at least one process stops or goes down. Proof First observe that if no process ever writes to Y, then no process completes the splitter, because the only way a process can finish the splitter without writing to Y is if it sees true in Y in line 2 (which must have been written by some other process). So if at least one process finishes, at least one process writes Y. Let p be any such process. From the code, having reached line 3, it either stops in line 4 or goes down in line 5. Claim 3 At most one process stops. Proof Let S be the set of processes that reach line 3. The every process in S reads Y before any process in S writes Y. It follows that every process in S writes X before any process in S reads X. If some process p is not the
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{
"page_id": null,
"source": 7357,
"title": "from dpo"
}
|
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