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Contents1 Introduction 5
1.1 Algorithms as opposed to programs . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Fundamental questions about algorithms . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Data structures, abstract data types, design patterns . . . . . . . . . . . . . . . 7
1.4 Textbooks and web-resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Arrays, Iteration, Invariants 9
2.1 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Loops and Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Lists, Recursion, Stacks, Queues 12
3.1 Linked Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Queues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 Doubly Linked Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.6 Advantage of Abstract Data Types . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Searching 21
4.1 Requirements for searching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Specification of the search problem . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 A simple algorithm: Linear Search . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4 A more efficient algorithm: Binary Search . . . . . . . . . . . . . . . . . . . . . 23
5 Efficiency and Complexity 25
5.1 Time versus space complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2 Worst versus average complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3 Concrete measures for performance . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.4 Big-O notation for complexity class . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.5 Formal definition of complexity classes . . . . . . . . . . . . . . . . . . . . . . . 29
6 Trees 31
6.1 General specification of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.2 Quad-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.3 Binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2
, 6.4 Primitive operations on binary trees . . . . . . . . . . . . . . . . . . . . . . . . 34
6.5 The height of a binary tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.6 The size of a binary tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.7 Implementation of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.8 Recursive algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7 Binary Search Trees 40
7.1 Searching with arrays or lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7.2 Search keys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7.3 Binary search trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
7.4 Building binary search trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
7.5 Searching a binary search tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.6 Time complexity of insertion and search . . . . . . . . . . . . . . . . . . . . . . 43
7.7 Deleting nodes from a binary search tree . . . . . . . . . . . . . . . . . . . . . . 44
7.8 Checking whether a binary tree is a binary search tree . . . . . . . . . . . . . . 46
7.9 Sorting using binary search trees . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.10 Balancing binary search trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.11 Self-balancing AVL trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.12 B-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
8 Priority Queues and Heap Trees 51
8.1 Trees stored in arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
8.2 Priority queues and binary heap trees . . . . . . . . . . . . . . . . . . . . . . . 52
8.3 Basic operations on binary heap trees . . . . . . . . . . . . . . . . . . . . . . . 53
8.4 Inserting a new heap tree node . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.5 Deleting a heap tree node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.6 Building a new heap tree from scratch . . . . . . . . . . . . . . . . . . . . . . . 56
8.7 Merging binary heap trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8.8 Binomial heaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8.9 Fibonacci heaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8.10 Comparison of heap time complexities . . . . . . . . . . . . . . . . . . . . . . . 62
9 Sorting 63
9.1 The problem of sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
9.2 Common sorting strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
9.3 How many comparisons must it take? . . . . . . . . . . . . . . . . . . . . . . . 64
9.4 Bubble Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
9.5 Insertion Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9.6 Selection Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9.7 Comparison of O(n2 ) sorting algorithms . . . . . . . . . . . . . . . . . . . . . . 70
9.8 Sorting algorithm stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.9 Treesort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.10 Heapsort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
9.11 Divide and conquer algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.12 Quicksort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.13 Mergesort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
9.14 Summary of comparison-based sorting algorithms . . . . . . . . . . . . . . . . . 81
3
, 9.15 Non-comparison-based sorts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9.16 Bin, Bucket, Radix Sorts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
10 Hash Tables 85
10.1 Storing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.2 The Table abstract data type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.3 Implementations of the table data structure . . . . . . . . . . . . . . . . . . . . 87
10.4 Hash Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
10.5 Collision likelihoods and load factors for hash tables . . . . . . . . . . . . . . . 88
10.6 A simple Hash Table in operation . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.7 Strategies for dealing with collisions . . . . . . . . . . . . . . . . . . . . . . . . 90
10.8 Linear Probing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
10.9 Double Hashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
10.10Choosing good hash functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
10.11Complexity of hash tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
11 Graphs 98
11.1 Graph terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
11.2 Implementing graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
11.3 Relations between graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
11.4 Planarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
11.5 Traversals – systematically visiting all vertices . . . . . . . . . . . . . . . . . . . 104
11.6 Shortest paths – Dijkstra’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . 105
11.7 Shortest paths – Floyd’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 111
11.8 Minimal spanning trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
11.9 Travelling Salesmen and Vehicle Routing . . . . . . . . . . . . . . . . . . . . . . 117
12 Epilogue 118
A Some Useful Formulae 119
A.1 Binomial formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.2 Powers and roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.3 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.4 Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.5 Fibonacci numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4
, 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
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