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Chapter 1 Introduction These lecture notes cover the key ideas involved in designing algorithms. We shall see how they depend on the design of suitable data structures, and how some structures and algorithms are more efficient than others for the same task. We will concentrate on a few basic tasks...

Chapter 1 Introduction These lecture notes cover the key ideas involved in designing algorithms. We shall see how they depend on the design of suitable data structures, and how some structures and algorithms are more efficient than others for the same task. We will concentrate on a few basic tasks, such as storing, sorting and searching data, that underlie much of computer science, but the techniques discussed will be applicable much more generally. We will start by studying some key data structures, such as arrays, lists, queues, stacks and trees, and then move on to explore their use in a range of different searching and sorting algorithms. This leads on to the consideration of approaches for more efficient storage of data in hash tables. Finally, we will look at graph based representations and cover the kinds of algorithms needed to work efficiently with them. Throughout, we will investigate the computational efficiency of the algorithms we develop, and gain intuitions about the pros and cons of the various potential approaches for each task. We will not restrict ourselves to implementing the various data structures and algorithms in particular computer programming languages (e.g., Java, C , OCaml ), but specify them in simple pseudocode that can easily be implemented in any appropriate language. 1.1 Algorithms as opposed to programs An algorithm for a particular task can be defined as “a finite sequence of instructions, each of which has a clear meaning and can be performed with a finite amount of effort in a finite length of time”. As such, an algorithm must be precise enough to be understood by human beings. However, in order to be executed by a computer, we will generally need a program that is written in a rigorous formal language; and since computers are quite inflexible compared to the human mind, programs usually need to contain more details than algorithms. Here we shall ignore most of those programming details and concentrate on the design of algorithms rather than programs. The task of implementing the discussed algorithms as computer programs is important, of course, but these notes will concentrate on the theoretical aspects and leave the practical programming aspects to be studied elsewhere. Having said that, we will often find it useful to write down segments of actual programs in order to clarify and test certain theoretical aspects of algorithms and their data structures. It is also worth bearing in mind the distinction between different programming paradigms: Imperative Programming describes computation in terms of instructions that change the program/data state, whereas Declarative Programming 5 specifies what the program should accomplish without describing how to do it. These notes will primarily be concerned with developing algorithms that map easily onto the imperative programming approach. Algorithms can obviously be described in plain English, and we will sometimes do that. However, for computer scientists it is usually easier and clearer to use something that comes somewhere in between formatted English and computer program code, but is not runnable because certain details are omitted. This is called pseudocode, which comes in a variety of forms. Often these notes will present segments of pseudocode that are very similar to the languages we are mainly interested in, namely the overlap of C and Java, with the advantage that they can easily be inserted into runnable programs. 1.2 Fundamental questions about algorithms Given an algorithm to solve a particular problem, we are naturally led to ask: 1. What is it supposed to do? 2. Does it really do what it is supposed to do? 3. How efficiently does it do it? The technical terms normally used for these three aspects are: 1. Specification. 2. Verification. 3. Performance analysis. The details of these three aspects will usually be rather problem dependent. The specification should formalize the crucial details of the problem that the algorithm is intended to solve. Sometimes that will be based on a particular representation of the associated data, and sometimes it will be presented more abstractly. Typically, it will have to specify how the inputs and outputs of the algorithm are related, though there is no general requirement that the specification is complete or non-ambiguous. For simple problems, it is often easy to see that a particular algorithm will always work, i.e. that it satisfies its specification. However, for more complicated specifications and/or algorithms, the fact that an algorithm satisfies its specification may not be obvious at all. In this case, we need to spend some effort verifying whether the algorithm is indeed correct. In general, testing on a few particular inputs can be enough to show that the algorithm is incorrect. However, since the number of different potential inputs for most algorithms is infinite in theory, and huge in practice, more than just testing on particular cases is needed to be sure that the algorithm satisfies its specification. We need correctness proofs. Although we will discuss proofs in these notes, and useful relevant ideas like invariants, we will usually only do so in a rather informal manner (though, of course, we will attempt to be rigorous). The reason is that we want to concentrate on the data structures and algorithms. Formal verification techniques are complex and will normally be left till after the basic ideas of these notes have been studied. Finally, the efficiency or performance of an algorithm relates to the resources required by it, such as how quickly it will run, or how much computer memory it will use. This will 6 usually depend on the problem instance size, the choice of data representation, and the details of the algorithm. Indeed, this is what normally drives the development of new data structures and algorithms. We shall study the general ideas concerning efficiency in Chapter 5, and then apply them throughout the remainder of these notes. 1.3 Data structures, abstract data types, design patterns For many problems, the ability to formulate an efficient algorithm depends on being able to organize the data in an appropriate manner. The term data structure is used to denote a particular way of organizing data for particular types of operation. These notes will look at numerous data structures ranging from familiar arrays and lists to more complex structures such as trees, heaps and graphs, and we will see how their choice affects the efficiency of the algorithms based upon them. Often we want to talk about data structures without having to worry about all the im- plementational details associated with particular programming languages, or how the data is stored in computer memory. We can do this by formulating abstract mathematical models of particular classes of data structures or data types which have common features. These are called abstract data types, and are defined only by the operations that may be performed on them. Typically, we specify how they are built out of more primitive data types (e.g., integers or strings), how to extract that data from them, and some basic checks to control the flow of processing in algorithms. The idea that the implementational details are hidden from the user and protected from outside access is known as encapsulation. We shall see many examples of abstract data types throughout these notes. At an even higher level of abstraction are design patterns which describe the design of algorithms, rather the design of data structures. These embody and generalize important design concepts that appear repeatedly in many problem contexts. They provide a general structure for algorithms, leaving the details to be added as required for particular problems. These can speed up the development of algorithms by providing familiar proven algorithm structures that can be applied straightforwardly to new problems. We shall see a number of familiar design patterns throughout these notes. 1.4 Textbooks and web-resources To fully understand data structures and algorithms you will almost certainly need to comple- ment the introductory material in these notes with textbooks or other sources of information. The lectures associated with these notes are designed to help you understand them and fill in some of the gaps they contain, but that is unlikely to be enough because often you will need to see more than one explanation of something before it can be fully understood. There is no single best textbook that will suit everyone. The subject of these notes is a classical topic, so there is no need to use a textbook published recently. Books published 10 or 20 years ago are still good, and new good books continue to be published every year. The reason is that these notes cover important fundamental material that is taught in all university degrees in computer science. These days there is also a lot of very useful information to be found on the internet, including complete freely-downloadable books. It is a good idea to go to your library and browse the shelves of books on data structures and algorithms. If you like any of them, download, borrow or buy a copy for yourself, but make sure that most of the 7 topics in the above contents list are covered. Wikipedia is generally a good source of fairly reliable information on all the relevant topics, but you hopefully shouldn’t need reminding that not everything you read on the internet is necessarily true. It is also worth pointing out that there are often many different equally-good ways to solve the same task, different equally-sensible names used for the same thing, and different equally-valid conventions used by different people, so don’t expect all the sources of information you find to be an exact match with each other or with what you find in these notes. 1.5 Overview These notes will cover the principal fundamental data structures and algorithms used in computer science, and bring together a broad range of topics covered elsewhere into a coherent framework. Data structures will be formulated to represent various types of information in such a way that it can be conveniently and efficiently manipulated by the algorithms we develop. Throughout, the recurring practical issues of algorithm specification, verification and performance analysis will be discussed. We shall begin by looking at some widely used basic data structures (namely arrays, linked lists, stacks and queues), and the advantages and disadvantages of the associated abstract data types. Then we consider the ubiquitous problem of searching, and how that leads on to the general ideas of computational efficiency and complexity. That will leave us with the necessary tools to study three particularly important data structures: trees (in particular, binary search trees and heap trees), hash tables, and graphs. We shall learn how to develop and analyse increasingly efficient algorithms for manipulating and performing useful operations on those structures, and look in detail at developing efficient processes for data storing, sorting, searching and analysis. The idea is that once the basic ideas and examples covered in these notes are understood, dealing with more complex problems in the future should be straightforward. 8 Chapter 2 Arrays, Iteration, Invariants Data is ultimately stored in computers as patterns of bits, though these days most program- ming languages deal with higher level objects, such as characters, integers, and floating point numbers. Generally, we need to build algorithms that manipulate collections of such objects, so we need procedures for storing and sequentially processing them. 2.1 Arrays In computer science, the obvious way to store an ordered collection of items is as an array. Array items are typically stored in a sequence of computer memory locations, but to discuss them, we need a convenient way to write them down on paper. We can just write the items in order, separated by commas and enclosed by square brackets. Thus, [1, 4, 17, 3, 90, 79, 4, 6, 81] is an example of an array of integers. If we call this array a, we can write it as: a = [1, 4, 17, 3, 90, 79, 4, 6, 81] This array a has 9 items, and hence we say that its size is 9. In everyday life, we usually start counting from 1. When we work with arrays in computer science, however, we more often (though not always) start from 0. Thus, for our array a, its positions are 0, 1, 2,... , 7, 8. The element in the 8th position is 81, and we use the notation a to denote this element. More generally, for any integer i denoting a position, we write a[i] to denote the element in the ith position. This position i is called an index (and the plural is indices). Then, in the above example, a = 1, a = 4, a = 17, and so on. It is worth noting at this point that the symbol = is quite overloaded. In mathematics, it stands for equality. In most modern programming languages, = denotes assignment, while equality is expressed by ==. We will typically use = in its mathematical meaning, unless it is written as part of code or pseudocode. We say that the individual items a[i] in the array a are accessed using their index i, and one can move sequentially through the array by incrementing or decrementing that index, or jump straight to a particular item given its index value. Algorithms that process data stored as arrays will typically need to visit systematically all the items in the array, and apply appropriate operations on them. 9 2.2 Loops and Iteration The standard approach in most programming languages for repeating a process a certain number of times, such as moving sequentially through an array to perform the same operations on each item, involves a loop. In pseudocode, this would typically take the general form For i = 1,...,N, do something and in programming languages like C and Java this would be written as the for-loop for( i = 0 ; i < N ; i++ ) { // do something } in which a counter i keep tracks of doing “the something” N times. For example, we could compute the sum of all 20 items in an array a using for( i = 0, sum = 0 ; i < 20 ; i++ ) { sum += a[i]; } We say that there is iteration over the index i. The general for-loop structure is for( INITIALIZATION ; CONDITION ; UPDATE ) { REPEATED PROCESS } in which any of the four parts are optional. One way to write this out explicitly is INITIALIZATION if ( not CONDITION ) go to LOOP FINISHED LOOP START REPEATED PROCESS UPDATE if ( CONDITION ) go to LOOP START LOOP FINISHED In these notes, we will regularly make use of this basic loop structure when operating on data stored in arrays, but it is important to remember that different programming languages use different syntax, and there are numerous variations that check the condition to terminate the repetition at different points. 2.3 Invariants An invariant, as the name suggests, is a condition that does not change during execution of a given program or algorithm. It may be a simple inequality, such as “i < 20”, or something more abstract, such as “the items in the array are sorted”. Invariants are important for data structures and algorithms because they enable correctness proofs and verification. In particular, a loop-invariant is a condition that is true at the beginning and end of every iteration of the given loop. Consider the standard simple example of a procedure that finds the minimum of n numbers stored in an array a: 10  minimum(int n, float a[n]) { float min = a; // min equals the minimum item in a,...,a for(int i = 1 ; i != n ; i++) { // min equals the minimum item in a,...,a[i-1] if (a[i] < min) min = a[i]; } // min equals the minimum item in a,...,a[i-1], and i==n return min; } At the beginning of each iteration, and end of any iterations before, the invariant “min equals the minimum item in a,..., a[i − 1]” is true – it starts off true, and the repeated process and update clearly maintain its truth. Hence, when the loop terminates with “i == n”, we know that “min equals the minimum item in a,..., a[n − 1]” and hence we can be sure that min can be returned as the required minimum value. This is a kind of proof by induction: the invariant is true at the start of the loop, and is preserved by each iteration of the loop, therefore it must be true at the end of the loop. As we noted earlier, formal proofs of correctness are beyond the scope of these notes, but identifying suitable loop invariants and their implications for algorithm correctness as we go along will certainly be a useful exercise. We will also see how invariants (sometimes called inductive assertions) can be used to formulate similar correctness proofs concerning properties of data structures that are defined inductively. 11