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Lecture Notes forData Structures and Algorithms
Revised each year by John Bullinaria
School of Computer Science
University of Birmingham
Birmingham, UK
Version of 27 March 2019
,These notes are currently revised each year by John Bullinaria. They include sections based on
notes originally written by Martı́n Escardó and revised by Manfred Kerber. All are members
of the School of Computer Science, University of Birmingham, UK.
c School of Computer Science, University of Birmingham, UK, 2018
1
,Contents
1 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
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