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What happens in the reliable broadcast algorithm if the accuracy property of the failure detector is violated? | The following are correctness arguments to satisfy the algorithm of changing a failure detector W to a failure detector S. The failure detector W is weak in completeness, and the failure detector S is strong in completeness. They are both weak in accuracy. It transforms weak completeness into strong completeness. It preserves the perpetual accuracy. It preserves the eventual accuracy.If all arguments above are satisfied, the reduction of a weak failure detector W to a strong failure detector S will agree with the algorithm within the distributed computing system. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
What happens in the reliable broadcast algorithm if the accuracy property of the failure detector is violated? | Designing an algorithm for atomic broadcasts is relatively easy if it can be assumed that computers will not fail. For example, if there are no failures, atomic broadcast can be achieved simply by having all participants communicate with one "leader" which determines the order of the messages, with the other participants following the leader. However, real computers are faulty; they fail and recover from failure at unpredictable, possibly inopportune, times. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a country with n ≥ 2 cities. For every pair of different cities x, y, there exists a direct route (single direction) either from x to y or from y to x. Show that there exists a city that we can reach from every other city either directly or through exactly one intermediate city. | It now must be shown that for every single tour covering all cities, there are values for the dummy variables u i {\displaystyle u_{i}} that satisfy the constraints. Without loss of generality, define the tour as originating (and ending) at city 1. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a country with n ≥ 2 cities. For every pair of different cities x, y, there exists a direct route (single direction) either from x to y or from y to x. Show that there exists a city that we can reach from every other city either directly or through exactly one intermediate city. | To prove this, it is shown below (1) that every feasible solution contains only one closed sequence of cities, and (2) that for every single tour covering all cities, there are values for the dummy variables u i {\displaystyle u_{i}} that satisfy the constraints. To prove that every feasible solution contains only one closed sequence of cities, it suffices to show that every subtour in a feasible solution passes through city 1 (noting that the equalities ensure there can only be one such tour). For if we sum all the inequalities corresponding to x i j = 1 {\displaystyle x_{ij}=1} for any subtour of k steps not passing through city 1, we obtain: ( n − 1 ) k ≤ ( n − 2 ) k , {\displaystyle (n-1)k\leq (n-2)k,} which is a contradiction. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prove that if a^2 is even, a is even. | In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to prove that the sum of two even integers is always even: Consider two even integers x and y. Since they are even, they can be written as x = 2a and y = 2b, respectively, for some integers a and b. Then the sum is x + y = 2a + 2b = 2(a+b). Therefore x+y has 2 as a factor and, by definition, is even. Hence, the sum of any two even integers is even.This proof uses the definition of even integers, the integer properties of closure under addition and multiplication, and the distributive property. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prove that if a^2 is even, a is even. | Here is a proof which should be appreciable with limited mathematical background: Statement: The product of any two even natural numbers is also even. Proof: Any even natural number is of the form 2n, where n is a natural number. Therefore, let us assume that we have two even numbers which we will denote by 2k and 2l. Their product is (2k)(2l) = 4(kl) = 2(2kl). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
If process i fails, then eventually all processes j≠i fail
Is the following true? If no process j≠i fails, nothing can be said about process i | The algorithm assumes that: the system is synchronous. processes may fail at any time, including during execution of the algorithm. a process fails by stopping and returns from failure by restarting. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
If process i fails, then eventually all processes j≠i fail
Is the following true? If no process j≠i fails, nothing can be said about process i | there is a failure detector which detects failed processes. message delivery between processes is reliable. each process knows its own process id and address, and that of every other process. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
What happens in our "Consensus-Based Total-Order Broadcast" algorithm, if the set of messages delivered in a round is not sorted deterministically after deciding in the consensus abstraction, but before it is proposed to consensus? | Out-of-order delivery occurs when sequenced packets arrive out of order. This may happen due to different paths taken by the packets or from packets being dropped and resent. HOL blocking can significantly increase packet reordering.Reliably broadcasting messages across a lossy network among a large number of peers is a difficult problem. While atomic broadcast algorithms solve the single point of failure problem of centralized servers, those algorithms introduce a head-of-line blocking problem. The Bimodal Multicast algorithm, a randomized algorithm that uses a gossip protocol, avoids head-of-line blocking by allowing some messages to be received out-of-order. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
What happens in our "Consensus-Based Total-Order Broadcast" algorithm, if the set of messages delivered in a round is not sorted deterministically after deciding in the consensus abstraction, but before it is proposed to consensus? | Saks' research in computational complexity theory, combinatorics, and graph theory has contributed to the study of lower bounds in order theory, randomized computation, and space–time tradeoff. In 1984, Saks and Jeff Kahn showed that there exist a tight information-theoretical lower bound for sorting under partially ordered information up to a multiplicative constant.In the first super-linear lower bound for the noisy broadcast problem was proved. In a noisy broadcast model, n + 1 {\displaystyle n+1} processors P 0 , P 1 , … , P n {\displaystyle P_{0},P_{1},\ldots ,P_{n}} are assigned a local input bit x i {\displaystyle x_{i}} . Each processor may perform a noisy broadcast to all other processors where the received bits may be independently flipped with a fixed probability. The problem is for processor P 0 {\displaystyle P_{0}} to determine f ( x 1 , x 2 , … , x n ) {\displaystyle f(x_{1},x_{2},\ldots ,x_{n})} for some function f {\displaystyle f} . Saks et al. showed that an existing protocol by Gallager was indeed optimal by a reduction from a generalized noisy decision tree and produced a Ω ( n log ( n ) ) {\displaystyle \Omega (n\log(n))} lower bound on the depth of the tree that learns the input.In 2003, P. Beame, Saks, X. Sun, and E. Vee published the first time–space lower bound trade-off for randomized computation of decision problems was proved. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Would it make sense to add the total-order property to the best-effort broadcast? | Out-of-order delivery occurs when sequenced packets arrive out of order. This may happen due to different paths taken by the packets or from packets being dropped and resent. HOL blocking can significantly increase packet reordering.Reliably broadcasting messages across a lossy network among a large number of peers is a difficult problem. While atomic broadcast algorithms solve the single point of failure problem of centralized servers, those algorithms introduce a head-of-line blocking problem. The Bimodal Multicast algorithm, a randomized algorithm that uses a gossip protocol, avoids head-of-line blocking by allowing some messages to be received out-of-order. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Would it make sense to add the total-order property to the best-effort broadcast? | It would be more efficient to wait with the transmission until the channel conditions are improved, or at least to give scheduling priority to less expensive users. Round-robin scheduling does not utilize this. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A sequential object is a tuple T = (Q, q0, O, R, ∆), where:
● Q is a set of states.
● q0 ∈ Q is an initial state.
● O is a set of operations.
● R is a set of responses.
● ∆ ⊆ (Q × Π × O) × (Q × R) is a relation that associates a state, a process,
and an operation to a set of possible new states and responses.
Processes invoke operations on the object. As a result, they get responses back, and the state of the object is updated to a new value, following from ∆.
Define a sequential object representing Asset Transfer, i.e., an object that allows processes to exchange units of currency. | If an operation op1 completes (gets a response) before op2 begins (invokes), then op1 precedes op2 in σ {\displaystyle \sigma } .In other words: its invocations and responses can be reordered to yield a sequential history; that sequential history is correct according to the sequential definition of the object; if a response preceded an invocation in the original history, it must still precede it in the sequential reordering. (Note that the first two bullet points here match serializability: the operations appear to happen in some order. It is the last point which is unique to linearizability, and is thus the major contribution of Herlihy and Wing. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A sequential object is a tuple T = (Q, q0, O, R, ∆), where:
● Q is a set of states.
● q0 ∈ Q is an initial state.
● O is a set of operations.
● R is a set of responses.
● ∆ ⊆ (Q × Π × O) × (Q × R) is a relation that associates a state, a process,
and an operation to a set of possible new states and responses.
Processes invoke operations on the object. As a result, they get responses back, and the state of the object is updated to a new value, following from ∆.
Define a sequential object representing Asset Transfer, i.e., an object that allows processes to exchange units of currency. | Object state An object state is a particular situation classification of an object at some point during its lifetime. At every point in time, the object is in one of its states or in transition between two of its states—from its input state to its output state. Process A process is an expression of the pattern of transformation of objects in the system. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Devise an algorithm that, without consensus, implements a weaker specification of NBAC by replacing the termination property with very weak termination.
Very weak termination: If no process crashes, then all processes decide. Is a failure detector needed to implement this algorithm? | The following are correctness arguments to satisfy the algorithm of changing a failure detector W to a failure detector S. The failure detector W is weak in completeness, and the failure detector S is strong in completeness. They are both weak in accuracy. It transforms weak completeness into strong completeness. It preserves the perpetual accuracy. It preserves the eventual accuracy.If all arguments above are satisfied, the reduction of a weak failure detector W to a strong failure detector S will agree with the algorithm within the distributed computing system. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Devise an algorithm that, without consensus, implements a weaker specification of NBAC by replacing the termination property with very weak termination.
Very weak termination: If no process crashes, then all processes decide. Is a failure detector needed to implement this algorithm? | The following is an example abstracted from the Department of Computer Science at Yale University. It functions by boosting the completeness of a failure detector. From the example above, if p crashes, then the weak-detector will eventually suspect it. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Explain why any fail-noisy consensus algorithm (one that uses an eventually perfect failure detector ◇P) requires a majority of the processes to be correct. More precisely, provide a “bad run” in the case where the majority of processes is faulty. | Before arguing that the Chandra–Toueg consensus algorithm satisfies the three properties above, recall that this algorithm requires n = 2*f + 1 processes, where at most f of which are faulty. Furthermore, note that this algorithm assumes the existence of eventually strong failure detector (which are accessible and can be used to detect the crash of a node). An eventually strong failure detector is one that never identifies some specific non-faulty (or correct) process as having failed, after some initial period of confusion, and, at the same time, eventually identifies all faulty processes as failed. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Explain why any fail-noisy consensus algorithm (one that uses an eventually perfect failure detector ◇P) requires a majority of the processes to be correct. More precisely, provide a “bad run” in the case where the majority of processes is faulty. | Protocols that solve consensus problems are designed to deal with limited numbers of faulty processes. These protocols must satisfy a number of requirements to be useful. For instance, a trivial protocol could have all processes output binary value 1. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implement a reliable broadcast algorithm without using any failure detector, i.e., using only BestEffort-Broadcast(BEB). | Designing an algorithm for atomic broadcasts is relatively easy if it can be assumed that computers will not fail. For example, if there are no failures, atomic broadcast can be achieved simply by having all participants communicate with one "leader" which determines the order of the messages, with the other participants following the leader. However, real computers are faulty; they fail and recover from failure at unpredictable, possibly inopportune, times. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implement a reliable broadcast algorithm without using any failure detector, i.e., using only BestEffort-Broadcast(BEB). | The ESBT-broadcast (Edge-disjoint Spanning Binomial Tree) algorithm is a pipelined broadcast algorithm with optimal runtime for clusters with hypercube network topology. The algorithm embeds d {\displaystyle d} edge-disjoint binomial trees in the hypercube, such that each neighbor of processing element 0 {\displaystyle 0} is the root of a spanning binomial tree on 2 d − 1 {\displaystyle 2^{d}-1} nodes. To broadcast a message, the source node splits its message into k {\displaystyle k} chunks of equal size and cyclically sends them to the roots of the binomial trees. Upon receiving a chunk, the binomial trees broadcast it. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Byzantine consistent broadcast (BCB) assumes one designated sender S and it satisfies the following properties:
Validity: If S is correct, then every correct process eventually delivers the message.
No duplication: Every correct process delivers at most one message.
Integrity: If a correct process delivers a message and S is correct, then S has previously broadcast the message.
Consistency: No two correct processes deliver different messages.
Do we need to introduce some constraints on the number of Byzantine processes in non-synchronous environments? If yes, what are the constraints? | Agreement - if a correct process delivers a message, then all correct processes eventually deliver that message. Integrity - every correct process delivers the same message at most once and only if that message has been sent by a process. A reliable broadcast can have sequential, causal or total ordering. Replication Resource allocation Spanning tree generation Symmetry breaking, e.g. vertex coloring | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Byzantine consistent broadcast (BCB) assumes one designated sender S and it satisfies the following properties:
Validity: If S is correct, then every correct process eventually delivers the message.
No duplication: Every correct process delivers at most one message.
Integrity: If a correct process delivers a message and S is correct, then S has previously broadcast the message.
Consistency: No two correct processes deliver different messages.
Do we need to introduce some constraints on the number of Byzantine processes in non-synchronous environments? If yes, what are the constraints? | Validity: if the sender is correct and broadcasts a message m {\displaystyle m} , then every correct process delivers m {\displaystyle m} . Integrity: a process delivers a message at most once, and if it delivers some message m ≠ S F {\displaystyle m\neq \mathrm {SF} } , then m {\displaystyle m} was broadcast by the sender. Agreement: if a correct process delivers a message m {\displaystyle m} , then all correct processes deliver m {\displaystyle m} .The presence of faults in the system makes these properties more difficult to satisfy. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
If process i fails, then eventually all processes j≠i fail
Is the following true? If some process j≠i does not fail, nothing can be said about process i | The algorithm assumes that: the system is synchronous. processes may fail at any time, including during execution of the algorithm. a process fails by stopping and returns from failure by restarting. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
If process i fails, then eventually all processes j≠i fail
Is the following true? If some process j≠i does not fail, nothing can be said about process i | there is a failure detector which detects failed processes. message delivery between processes is reliable. each process knows its own process id and address, and that of every other process. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
What happens in the reliable broadcast algorithm if the completeness property of the failure detector is violated? | The following are correctness arguments to satisfy the algorithm of changing a failure detector W to a failure detector S. The failure detector W is weak in completeness, and the failure detector S is strong in completeness. They are both weak in accuracy. It transforms weak completeness into strong completeness. It preserves the perpetual accuracy. It preserves the eventual accuracy.If all arguments above are satisfied, the reduction of a weak failure detector W to a strong failure detector S will agree with the algorithm within the distributed computing system. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
What happens in the reliable broadcast algorithm if the completeness property of the failure detector is violated? | The protocol provides the following guarantees: Strong Completeness: Full completeness is guaranteed (e.g. the crash-failure of any node in the group is eventually detected by all live nodes). Detection Time: The expected value of detection time (from node failure to detection) is T ′ ˙ 1 1 − e − q f {\displaystyle T'{\dot {}}{\frac {1}{1-e^{-q_{f}}}}} , where T ′ {\displaystyle T'} is the length of the protocol period, and q f {\displaystyle q_{f}} is the fraction of non-faulty nodes in the group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a network that is organized as a 2-dimensional grid, such that every process has up to 4 neighbors. The width of the grid is w and the height is h. The grid is big, meaning that w+h is much smaller than w*h. While there are faulty and correct processes in the network, it is assumed that two correct processes are always connected through at least one path of correct processes. In every round processes may send a message to each of its neighbors, the size of the message is not limited.
Assume there is no faulty process. Write a protocol to reach consensus. Optimize your protocol according to speed. How many rounds does your protocol require? | There is also a condition known as validity in the literature which refers to the property that a message sent by a process must be delivered.A protocol that can correctly guarantee consensus amongst n processes of which at most t fail is said to be t-resilient. In evaluating the performance of consensus protocols two factors of interest are running time and message complexity. Running time is given in Big O notation in the number of rounds of message exchange as a function of some input parameters (typically the number of processes and/or the size of the input domain). Message complexity refers to the amount of message traffic that is generated by the protocol. Other factors may include memory usage and the size of messages. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a network that is organized as a 2-dimensional grid, such that every process has up to 4 neighbors. The width of the grid is w and the height is h. The grid is big, meaning that w+h is much smaller than w*h. While there are faulty and correct processes in the network, it is assumed that two correct processes are always connected through at least one path of correct processes. In every round processes may send a message to each of its neighbors, the size of the message is not limited.
Assume there is no faulty process. Write a protocol to reach consensus. Optimize your protocol according to speed. How many rounds does your protocol require? | For n processes in a partially synchronous system (the system alternates between good and bad periods of synchrony), each process chooses a private value. The processes communicate with each other by rounds to determine a public value and generate a consensus vector with the following requirements: if a correct process sends v {\displaystyle v} , then all correct processes receive either v {\displaystyle v} or nothing (integrity property) all messages sent in a round by a correct process are received in the same round by all correct processes (consistency property).It can be shown that variations of these problems are equivalent in that the solution for a problem in one type of model may be the solution for another problem in another type of model. For example, a solution to the Weak Byzantine General problem in a synchronous authenticated message passing model leads to a solution for Weak Interactive Consistency. An interactive consistency algorithm can solve the consensus problem by having each process choose the majority value in its consensus vector as its consensus value. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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