Lecture Notes for
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Data Structures and Algorithms
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Revised each year by John Bullinaria
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School of Computer Science
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University of Birmingham
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Birmingham, UK
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Version of 27 March 2019ThesenotesarecurrentlyrevisedeachyearbyJohnBullinaria. Theyincludesectionsba
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sedon
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notes originally written by Mart´ın Escard´o and revised by Manfred Kerber. All are members
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of the School of Computer Science, University of Birmingham, UK.
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(cid:13)c School of Computer Science, University of Birmingham, UK, 2018
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1Contents
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1 Introduction 5
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1.1 Algorithms as opposed to programs . . . . . . . . . . . . . . . . . . . . . . . . . 5
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1.2 Fundamental questions about algorithms . . . . . . . . . . . . . . . . . . . . . . 6
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1.3 Data structures, abstract data types, design patterns . . . . . . . . . . . . . . . 7
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1.4 Textbooks and web-resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
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1.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
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2 Arrays, Iteration, Invariants 9
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2.1 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
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2.2 Loops and Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
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2.3 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
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3 Lists, Recursion, Stacks, Queues 12
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3.1 Linked Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
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3.2 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
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3.3 Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
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3.4 Queues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
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3.5 Doubly Linked Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
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3.6 Advantage of Abstract Data Types . . . . . . . . . . . . . . . . . . . . . . . . . 20
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4 Searching 21
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4.1 Requirements for searching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
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4.2 Specification of the search problem . . . . . . . . . . . . . . . . . . . . . . . . . 22
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4.3 A simple algorithm: Linear Search . . . . . . . . . . . . . . . . . . . . . . . . . 22
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4.4 A more efficient algorithm: Binary Search . . . . . . . . . . . . . . . . . . . . . 23
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5 Efficiency and Complexity 25
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5.1 Time versus space complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
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5.2 Worst versus average complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 25
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5.3 Concrete measures for performance . . . . . . . . . . . . . . . . . . . . . . . . . 26
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5.4 Big-O notation for complexity class . . . . . . . . . . . . . . . . . . . . . . . . . 26
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5.5 Formal definition of complexity classes . . . . . . . . . . . . . . . . . . . . . . . 29
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6 Trees 31
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6.1 General specification of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
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6.2 Quad-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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6.3 Binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
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26.4 Primitive operations on binary trees . . . . . . . . . . . . . . . . . . . . . . . . 34
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6.5 The height of a binary tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
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6.6 The size of a binary tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
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6.7 Implementation of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
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6.8 Recursive algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
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7 Binary Search Trees 40
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7.1 Searching with arrays or lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
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7.2 Search keys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
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7.3 Binary search trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
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7.4 Building binary search trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
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7.5 Searching a binary search tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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7.6 Time complexity of insertion and search . . . . . . . . . . . . . . . . . . . . . . 43
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7.7 Deleting nodes from a binary search tree . . . . . . . . . . . . . . . . . . . . . . 44
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7.8 Checking whether a binary tree is a binary search tree . . . . . . . . . . . . . . 46
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7.9 Sorting using binary search trees . . . . . . . . . . . . . . . . . . . . . . . . . . 47
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7.10 Balancing binary search trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
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7.11 Self-balancing AVL trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
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7.12 B-trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
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8 Priority Queues and Heap Trees 51
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8.1 Trees stored in arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
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8.2 Priority queues and binary heap trees . . . . . . . . . . . . . . . . . . . . . . . 52
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8.3 Basic operations on binary heap trees . . . . . . . . . . . . . . . . . . . . . . . 53
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8.4 Inserting a new heap tree node . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
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8.5 Deleting a heap tree node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
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8.6 Building a new heap tree from scratch . . . . . . . . . . . . . . . . . . . . . . . 56
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8.7 Merging binary heap trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
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8.8 Binomial heaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
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8.9 Fibonacci heaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
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8.10 Comparison of heap time complexities . . . . . . . . . . . . . . . . . . . . . . . 62
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9 Sorting 63
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9.1 The problem of sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
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9.2 Common sorting strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
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9.3 How many comparisons must it take? . . . . . . . . . . . . . . . . . . . . . . . 64
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9.4 Bubble Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
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9.5 Insertion Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
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9.6 Selection Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
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9.7 Comparison of O(n2) sorting algorithms . . . . . . . . . . . . . . . . . . . . . . 70
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9.8 Sorting algorithm stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
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9.9 Treesort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
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9.10 Heapsort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
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9.11 Divide and conquer algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
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9.12 Quicksort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
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9.13 Mergesort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
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9.14 Summary of comparison-based sorting algorithms . . . . . . . . . . . . . . . . . 81
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39.15 Non-comparison-based sorts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
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9.16 Bin, Bucket, Radix Sorts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
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10 Hash Tables 85
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10.1 Storing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
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10.2 The Table abstract data type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
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10.3 Implementations of the table data structure . . . . . . . . . . . . . . . . . . . . 87
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10.4 Hash Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
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10.5 Collision likelihoods and load factors for hash tables . . . . . . . . . . . . . . . 88
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10.6 A simple Hash Table in operation . . . . . . . . . . . . . . . . . . . . . . . . . . 89
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10.7 Strategies for dealing with collisions . . . . . . . . . . . . . . . . . . . . . . . . 90
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10.8 Linear Probing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
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10.9 Double Hashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
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10.10Choosing good hash functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
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10.11Complexity of hash tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
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