Analysis of Algorithms

Size: px

Start display at page:

Download "Analysis of Algorithms"

Derek French
5 years ago
Views:

1 Second Edition Design and Analysis of Algorithms Prabhakar Gupta Vineet Agarwal Manish Varshney

2 Design and Analysis of ALGORITHMS SECOND EDITION PRABHAKAR GUPTA Professor, Computer Science and Engineering Shri Ram Murti Smarak College of Engineering and Technology (SRMSCET) Bareilly VINEET AGARWAL Professor, Computer Science and Engineering Rakshpal Bahadur College of Engineering and Technology Bareilly MANISH VARSHNEY Associate Professor and Head Computer Science and Information Technology Shri Siddhi Vinayak Institute of Technology Bareilly New Delhi

3 DESIGN AND ANALYSIS OF ALGORITHMS, Second Edition Prabhakar Gupta, Vineet Agarwal, and Manish Varshney 2012 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN The export rights of this book are vested solely with the publisher. Third Printing (Second Edition) September, 2012 Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi and Printed by Meenakshi Art Printers, Delhi

4 Contents Preface Preface to the First Edition Acknowledgements xi xiii xv 1. INTRODUCTION TO ALGORITHMS Definition Expression and Implementation of Algorithms Characteristics of Algorithms Analysis of Algorithms Algorithm Complexity Space Complexity Time Complexity Determination of Complexities Cases of Complexity Importance of Constants during the Analysis of Algorithms Growth of Functions Asymptotic Notations Types of Asymptotic Notations Summary of Asymptotic Notations Designing of Algorithms Approaches for Designing Algorithm Difference between Incremental and Divide-and-Conquer Approach Recurrence Relation Solving Recurrence Relations 24 Summary 32 Exercises 33 iii

5 iv Contents 2. SORTING AND ORDER STATISTICS Introduction Insertion Sort Algorithm: The Insertion Sort Code in C Language Complexity of Insert Sort Advantage of Insertion Sort Disadvantage of Insertion Sort Bubble Sort Procedure: Bubble Sort Code in C Language Complexity of Bubble Sort Advantages of Bubble Sort Disadvantages of Bubble Sort Selection Sort Procedure: Selection Sort Code in C Language Complexity of Selection Sort Advantages of Selection Sort Disadvantages of Selection Sort Shell Sort/Diminishing Increment Sort/Comb Sort Procedure: Shell Sort Code in C Language Complexity of Shell Sort Advantages of Shell Sort Disadvantages of Shell Sort Merge Sort Procedure: Merge Sort Code in C Language Complexity of Merge Sort Different Kinds of Merge Sort Heap Sort Heap Description Operations on Heap Analysis of Heap Sort Graphical Representation of Heap Sort Algorithm Application of Heaps Quick Sort Pictorial Representation of Quick Sort Features of Quick Sort Algorithm Analysis of Quick Sort Graphical Representation of Quick Sort Algorithm Implementation of Quick Sort Algorithm in C Language 77

6 Contents v 2.9 Sorting in Linear Time Counting Sort Radix Sort Bucket Sort Medians and Order Statistics Selection Problem 85 Summary 86 Exercises ELEMENTARY DATA STRUCTURE Stack Operations on a Stack Program to Implement the Push, Pop and Display Function of Stack Using Array Program to Implement the Stack with the Help of Linked List Queue Operations on a Queue Program to Implement the Queue with the Help of Array Program to Implement the Queue with the Help of Linked List Linked List Classification of Linked List Applications of Linked Lists Difference between Linked List and Array Difference between Doubly-linked List and Linear-linked List Binary Tree Types of Binary Trees Binary Search Tree Operations 132 Summary 141 Exercises ADVANCED DATA STRUCTURE Red-Black Tree Properties of Red-Black Tree Graphical Representation of Red-Black Tree Operations on Red-Black Tree Application of Red-Black Tree AVL Tree Balance Factor Rotations in AVL Tree AVL Tree Operations Difference between AVL Tree and Red-Black Tree 178

7 vi Contents 4.4 Augmenting Data Structure Problems Related to Augmented Data Structure B-Trees Graphical Representation of B-Tree Properties of B-Tree Insertion into a B-Tree Representation of B-Tree in Computer Memory Deletion from B-Tree Searching in a B-Tree Applications of B-Tree B+ Tree Graphical Representation of B+ Tree B*-Tree Binomial Tree Properties of Binomial Tree Binomial Heap Properties of a Binomial Heap Operations of a Binomial Heap Explanation of Binomial-Heap-Union Operation Fibonacci Heap Representation of a Fibonacci Heap Properties of Fibonacci Heap Operations of a Fibonacci Heap Algorithm for FIB-HEAP-DELETE Graphical Representation of All Operations Disjoint Set Data Structure Operations of Disjoint Set Applications of Disjoint Set Data Structures The Disjoint Set Representation 219 Summary 219 Exercises DIVIDE AND CONQUER Introduction General Issues in Divide and Conquer Sorting Merge Sort Quick Sort Randomization Quick Sort Matrix Multiplication Binary Search Convex Hull Divide and Conquer Approach 232 Summary 233 Exercises 234

8 Contents vii 6. ADVANCED DESIGN AND ANALYSIS TECHNIQUES PART I Dynamic Programming Definition Approaches of Dynamic Programming Application of Dynamic Programming Steps Involved in the Implementation of Dynamic Programming Elements of Dynamic Programming Greedy Algorithms Functions of Greedy Algorithms Structure of Greedy Algorithm Properties of Greedy Algorithm Problems Solved by Greedy Approach 252 Summary 265 Exercises ADVANCED DESIGN AND ANALYSIS TECHNIQUES PART II Amortized Analysis Aggregate Method Accounting Method Potential Method Backtracking General Method for Backtracking Queens Problem for Backtracking n-queens Problem Sum of Subsets Problem for Backtracking Graph Colouring Problem for Backtracking Hamiltonian Cycles for Backtracking Zero-n Knapsack Problem 285 Summary 286 Exercises GRAPH Graph Representation of Graph Program for Addition and Deletion of Nodes and Edges in a Graph Using Adjacency Matrix Program for Creation of Adjacency Matrix Graph Traversal Breadth-First Search Depth-First Search Topological Sort Strongly-Connected Components 304

9 viii Contents 8.5 Minimum Spanning Tree Kruskal s Algorithm Prim s Algorithm Shortest Path Algorithms Single-Source Shortest Path Dijkstra s Algorithm Bellman-Ford Algorithm All-Pairs-Shortest-Path Problem Optimum Matrix Multiplication or Shortest Path and Matrix Multiplication Floyd-Warshall Algorithm Sparse Graphs Johnson s Algorithm Maximum Flow Ford-Fulkerson Algorithm Travelling Salesman Problem Improved Greedy Solutions Branch-and-Bound Solution Alternative Approaches 344 Summary 345 Exercises PATTERN MATCHING ALGORITHMS Randomized Algorithms Motivation of Randomized Algorithms Complexity Applications of Randomized Algorithm Derandomization Advantages of Randomized Algorithm String-Matching String-Matching Algorithm 351 Summary 366 Exercises NP-COMPLETENESS NP-Complete Importance of NP-Completeness Polynomial Time The Classes P and NP Example of NP-Complete Problems NP-Hard Difference between NP-Hard and NP-Complete 385

10 Contents ix 10.8 Approximation Algorithm Approximating Vertex Cover 388 Summary 389 Exercises 389 GLOSSARY REFERENCES 403 INDEX

11 Design And Analysis Of Algorithms 25% OFF Publisher : PHI Learning ISBN : Author : Gupta And Prabhakar Type the URL : Get this ebook

Thomas H. Cormen Charles E. Leiserson Ronald L. Rivest. Introduction to Algorithms

Thomas H. Cormen Charles E. Leiserson Ronald L. Rivest Introduction to Algorithms Preface xiii 1 Introduction 1 1.1 Algorithms 1 1.2 Analyzing algorithms 6 1.3 Designing algorithms 1 1 1.4 Summary 1 6