Lecture Notes for stringlengths 1 100 | Unnamed: 1 stringclasses 7
values | Unnamed: 2 float64 | Unnamed: 3 float64 |
|---|---|---|---|
11 Graphs 98 | null | null | null |
11.1 Graph terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 | null | null | null |
11.2 Implementing graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 | null | null | null |
11.3 Relations between graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 | null | null | null |
11.4 Planarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 | null | null | null |
11.5 Traversals – systematically visiting all vertices. . . . . . . . . . . . . . . . . . . 104 | null | null | null |
11.6 Shortest paths – Dijkstra’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . 105 | null | null | null |
11.7 Shortest paths – Floyd’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 111 | null | null | null |
11.8 Minimal spanning trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 | null | null | null |
11.9 Travelling Salesmen and Vehicle Routing . . . . . . . . . . . . . . . . . . . . . . 117 | null | null | null |
12 Epilogue 118 | null | null | null |
A Some Useful Formulae 119 | null | null | null |
A.1 Binomial formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 | null | null | null |
A.2 Powers and roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 | null | null | null |
A.3 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 | null | null | null |
A.4 Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 | null | null | null |
A.5 Fibonacci numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 | null | null | null |
4Chapter 1 | null | null | null |
Introduction | null | null | null |
These lecture notes cover the key ideas involved in designing algorithms. We shall see how | null | null | null |
theydependonthedesignofsuitabledata structures,andhowsomestructuresandalgorithms | null | null | null |
are more efficient than others for the same task. We will concentrate on a few basic tasks, | null | null | null |
such as storing, sorting and searching data, that underlie much of computer science, but the | null | null | null |
techniques discussed will be applicable much more generally. | null | null | null |
We will start by studying some key data structures, such as arrays, lists, queues, stacks | null | null | null |
and trees, and then move on to explore their use in a range of different searching and sorting | null | null | null |
algorithms. This leads on to the consideration of approaches for more efficient storage of | null | null | null |
data in hash tables. Finally, we will look at graph based representations and cover the kinds | null | null | null |
of algorithms needed to work efficiently with them. Throughout, we will investigate the | null | null | null |
computational efficiency of the algorithms we develop, and gain intuitions about the pros and | null | null | null |
cons of the various potential approaches for each task. | null | null | null |
We will not restrict ourselves to implementing the various data structures and algorithms | null | null | null |
in particular computer programming languages (e.g., Java, C, OCaml), but specify them in | null | null | null |
simple pseudocode that can easily be implemented in any appropriate language. | null | null | null |
1.1 Algorithms as opposed to programs | null | null | null |
An algorithm for a particular task can be defined as “a finite sequence of instructions, each | null | null | null |
of which has a clear meaning and can be performed with a finite amount of effort in a finite | null | null | null |
length of time”. As such, an algorithm must be precise enough to be understood by human | null | null | null |
beings. However, in order to be executed by a computer, we will generally need a program that | null | null | null |
is written in a rigorous formal language; and since computers are quite inflexible compared | null | null | null |
to the human mind, programs usually need to contain more details than algorithms. Here we | null | null | null |
shall ignore most of those programming details and concentrate on the design of algorithms | null | null | null |
rather than programs. | null | null | null |
The task of implementing the discussed algorithms as computer programs is important, | null | null | null |
of course, but these notes will concentrate on the theoretical aspects and leave the practical | null | null | null |
programming aspects to be studied elsewhere. Having said that, we will often find it useful | null | null | null |
to write down segments of actual programs in order to clarify and test certain theoretical | null | null | null |
aspectsofalgorithmsandtheirdatastructures. Itisalsoworthbearinginmindthedistinction | null | null | null |
betweendifferentprogrammingparadigms: ImperativeProgramming describescomputationin | null | null | null |
terms of instructions that change the program/data state, whereas Declarative Programming | null | null | null |
5specifies what the program should accomplish without describing how to do it. These notes | null | null | null |
will primarily be concerned with developing algorithms that map easily onto the imperative | null | null | null |
programming approach. | null | null | null |
Algorithms can obviously be described in plain English, and we will sometimes do that. | null | null | null |
However, for computer scientists it is usually easier and clearer to use something that comes | null | null | null |
somewhere in between formatted English and computer program code, but is not runnable | null | null | null |
because certain details are omitted. This is called pseudocode, which comes in a variety of | null | null | null |
forms. Often these notes will present segments of pseudocode that are very similar to the | null | null | null |
languages we are mainly interested in, namely the overlap of C and Java, with the advantage | null | null | null |
that they can easily be inserted into runnable programs. | null | null | null |
1.2 Fundamental questions about algorithms | null | null | null |
Given an algorithm to solve a particular problem, we are naturally led to ask: | null | null | null |
1. What is it supposed to do? | null | null | null |
2. Does it really do what it is supposed to do? | null | null | null |
3. How efficiently does it do it? | null | null | null |
The technical terms normally used for these three aspects are: | null | null | null |
1. Specification. | null | null | null |
2. Verification. | null | null | null |
3. Performance analysis. | null | null | null |
The details of these three aspects will usually be rather problem dependent. | null | null | null |
The specification should formalize the crucial details of the problem that the algorithm | null | null | null |
is intended to solve. Sometimes that will be based on a particular representation of the | null | null | null |
associated data, and sometimes it will be presented more abstractly. Typically, it will have to | null | null | null |
specify how the inputs and outputs of the algorithm are related, though there is no general | null | null | null |
requirement that the specification is complete or non-ambiguous. | null | null | null |
For simple problems, it is often easy to see that a particular algorithm will always work, | null | null | null |
i.e. that it satisfies its specification. However, for more complicated specifications and/or | null | null | null |
algorithms, the fact that an algorithm satisfies its specification may not be obvious at all. | null | null | null |
In this case, we need to spend some effort verifying whether the algorithm is indeed correct. | null | null | null |
In general, testing on a few particular inputs can be enough to show that the algorithm is | null | null | null |
incorrect. However, since the number of different potential inputs for most algorithms is | null | null | null |
infinite in theory, and huge in practice, more than just testing on particular cases is needed | null | null | null |
to be sure that the algorithm satisfies its specification. We need correctness proofs. Although | null | null | null |
we will discuss proofs in these notes, and useful relevant ideas like invariants, we will usually | null | null | null |
only do so in a rather informal manner (though, of course, we will attempt to be rigorous). | null | null | null |
The reason is that we want to concentrate on the data structures and algorithms. Formal | null | null | null |
verification techniques are complex and will normally be left till after the basic ideas of these | null | null | null |
notes have been studied. | null | null | null |
Finally, the efficiency or performance of an algorithm relates to the resources required | null | null | null |
by it, such as how quickly it will run, or how much computer memory it will use. This will | null | null | null |
6usuallydependontheprobleminstancesize, thechoiceofdatarepresentation, andthedetails | null | null | null |
of the algorithm. Indeed, this is what normally drives the development of new data structures | null | null | null |
and algorithms. We shall study the general ideas concerning efficiency in Chapter 5, and then | null | null | null |
apply them throughout the remainder of these notes. | null | null | null |
1.3 Data structures, abstract data types, design patterns | null | null | null |
For many problems, the ability to formulate an efficient algorithm depends on being able to | null | null | null |
organize the data in an appropriate manner. The term data structure is used to denote a | null | null | null |
particular way of organizing data for particular types of operation. These notes will look at | null | null | null |
numerous data structures ranging from familiar arrays and lists to more complex structures | null | null | null |
such as trees, heaps and graphs, and we will see how their choice affects the efficiency of the | null | null | null |
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