Contents PREFACE. xiii

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1 Contents PREFACE v 1 INTRODUCTION WHAT IS AN ALGORITHM? ALGORITHM SPECIFICATION Introduction Recursive Algorithms PERFORMANCE ANALYSIS Space Complexity Time Complexity Amortized Complexity Asymptotic Notation (O, Ω, Θ) Practical Complexities Performance Measurement RANDOMIZED ALGORITHMS Basics of Probability Theory Randomized Algorithms: An Informal Description Identifying the Repeated Element Primality Testing Advantages and Disadvantages REFERENCES AND READINGS ELEMENTARY DATA STRUCTURES STACKS AND QUEUES TREES Terminology Binary Trees DICTIONARIES xiii

2 xiv CONTENTS Binary Search Trees PRIORITY QUEUES Heaps Heapsort SETS AND DISJOINT SET UNION Introduction Union and Find Operations GRAPHS Introduction Definitions Graph Representations REFERENCES AND READINGS DIVIDE-AND-CONQUER GENERAL METHOD DEFECTIVE CHESSBOARD BINARY SEARCH FINDING THE MAXIMUM AND MINIMUM MERGE SORT QUICKSORT Performance Measurement Randomized Sorting Algorithms SELECTION A Worst-Case Optimal Algorithm Implementation of Select STRASSEN S MATRIX MULTIPLICATION CONVEX HULL Some Geometric Primitives The QuickHull Algorithm Graham s Scan An O(n log n) Divide-and-Conquer Algorithm REFERENCES AND READINGS ADDITIONAL EXERCISES THE GREEDY METHOD THE GENERAL METHOD CONTAINER LOADING KNAPSACK PROBLEM

3 CONTENTS xv 4.4 TREE VERTEX SPLITTING JOB SEQUENCING WITH DEADLINES MINIMUM-COST SPANNING TREES Prim s Algorithm Kruskal s Algorithm An Optimal Randomized Algorithm ( ) OPTIMAL STORAGE ON TAPES OPTIMAL MERGE PATTERNS SINGLE-SOURCE SHORTEST PATHS REFERENCES AND READINGS ADDITIONAL EXERCISES DYNAMIC PROGRAMMING THE GENERAL METHOD MULTISTAGE GRAPHS ALL PAIRS SHORTEST PATHS SINGLE-SOURCE SHORTEST PATHS: GENERAL WEIGHTS OPTIMAL BINARY SEARCH TREES ( ) STRING EDITING /1-KNAPSACK RELIABILITY DESIGN THE TRAVELING SALESPERSON PROBLEM FLOW SHOP SCHEDULING REFERENCES AND READINGS ADDITIONAL EXERCISES BASIC TRAVERSAL AND SEARCH TECHNIQUES TECHNIQUES FOR BINARY TREES TECHNIQUES FOR GRAPHS Breadth First Search and Traversal Depth First Search and Traversal CONNECTED COMPONENTS AND SPANNING TREES BICONNECTED COMPONENTS AND DFS REFERENCES AND READINGS

4 xvi CONTENTS 7 BACKTRACKING THE GENERAL METHOD THE 8-QUEENS PROBLEM SUM OF SUBSETS GRAPH COLORING HAMILTONIAN CYCLES KNAPSACK PROBLEM REFERENCES AND READINGS ADDITIONAL EXERCISES BRANCH AND BOUND THE METHOD Least Cost (LC) Search The 15-puzzle: An Example Control Abstractions for LC-Search Bounding FIFO Branch-and-Bound LC Branch-and-Bound /1 KNAPSACK PROBLEM LC Branch-and-Bound Solution FIFO Branch-and-Bound Solution TRAVELING SALESPERSON ( ) EFFICIENCY CONSIDERATIONS REFERENCES AND READINGS ALGEBRAIC PROBLEMS THE GENERAL METHOD EVALUATION AND INTERPOLATION THE FAST FOURIER TRANSFORM An In-place Version of the FFT Some Remaining Points MODULAR ARITHMETIC EVEN FASTER EVALUATION AND INTERPOLATION REFERENCES AND READINGS

5 CONTENTS xvii 10 LOWER BOUND THEORY COMPARISON TREES Ordered Searching Sorting Selection ORACLES AND ADVERSARY ARGUMENTS Merging Largest and Second Largest State Space Method Selection LOWER BOUNDS THROUGH REDUCTIONS Finding the Convex Hull Disjoint Sets Problem On-line Median Finding Multiplying Triangular Matrices Inverting a Lower Triangular Matrix Computing the Transitive Closure TECHNIQUES FOR ALGEBRAIC PROBLEMS ( ) REFERENCES AND READINGS N P-HARD AND N P-COMPLETE PROBLEMS BASIC CONCEPTS Nondeterministic Algorithms The Classes N P-hard and N P-complete COOK S THEOREM ( ) N P-HARD GRAPH PROBLEMS Clique Decision Problem (CDP) Node Cover Decision Problem (NCDP) Chromatic Number Decision Problem (CNDP) Directed Hamiltonian Cycle (DHC) ( ) Traveling Salesperson Decision Problem (TSP) AND/OR Graph Decision Problem (AOG) N P-HARD SCHEDULING PROBLEMS Scheduling Identical Processors Flow Shop Scheduling

6 xviii CONTENTS Job Shop Scheduling N P-HARD CODE GENERATION PROBLEMS Code Generation with Common Subexpressions Implementing Parallel Assignment Instructions SOME SIMPLIFIED N P-HARD PROBLEMS REFERENCES AND READINGS ADDITIONAL EXERCISES APPROXIMATION ALGORITHMS INTRODUCTION ABSOLUTE APPROXIMATIONS Planar Graph Coloring Maximum Programs Stored Problem N P-hard Absolute Approximations ǫ-approximations Scheduling Independent Tasks Bin Packing N P-hard ǫ-approximation Problems POLYNOMIAL TIME APPROXIMATION SCHEMES Scheduling Independent Tasks /1 Knapsack FULLY POLYNOMIAL TIME APPROXIMATION SCHEMES Rounding Interval Partitioning Separation PROBABILISTICALLY GOOD ALGORITHMS ( ) REFERENCES AND READINGS ADDITIONAL EXERCISES PRAM ALGORITHMS INTRODUCTION COMPUTATIONAL MODEL FUNDAMENTAL TECHNIQUES AND ALGORITHMS Prefix Computation

7 CONTENTS xix List Ranking SELECTION Maximal Selection with n 2 Processors Finding the Maximum Using n Processors Maximal Selection Among Integers General Selection Using n 2 Processors A Work-Optimal Randomized Algorithm ( ) MERGING A Logarithmic Time Algorithm Odd-Even Merge A Work-Optimal Algorithm An O(log log m)-time Algorithm SORTING Odd-Even Merge Sort An Alternative Randomized Algorithm Preparata s Algorithm Reischuk s Randomized Algorithm ( ) GRAPH PROBLEMS An Alternative Algorithm for Transitive Closure All-Pairs Shortest Paths COMPUTING THE CONVEX HULL LOWER BOUNDS A lower bound on average-case sorting Finding the maximum REFERENCES AND READINGS ADDITIONAL EXERCISES MESH ALGORITHMS COMPUTATIONAL MODEL PACKET ROUTING Packet Routing on a Linear Array A Greedy Algorithm for PPR on a Mesh A Randomized Algorithm with Small Queues FUNDAMENTAL ALGORITHMS Broadcasting Prefix Computation Data Concentration Sparse Enumeration Sort

8 xx CONTENTS 14.4 SELECTION A Randomized Algorithm for n = p ( ) Randomized Selection for n > p ( ) A Deterministic Algorithm For n > p MERGING Rank Merge on a Linear Array Odd-Even Merge on a Linear Array Odd-Even Merge on a Mesh SORTING Sorting on a Linear Array Sorting on a Mesh GRAPH PROBLEMS An n n Mesh Algorithm for Transitive Closure All-Pairs Shortest Paths COMPUTING THE CONVEX HULL REFERENCES AND READINGS ADDITIONAL EXERCISES HYPERCUBE ALGORITHMS COMPUTATIONAL MODEL The Hypercube The Butterfly Network Embedding of Other Networks PPR ROUTING A Greedy Algorithm A Randomized Algorithm FUNDAMENTAL ALGORITHMS Broadcasting Prefix Computation Data Concentration Sparse Enumeration Sort SELECTION A Randomized Algorithm for n = p ( ) Randomized Selection for n > p ( ) A Deterministic Algorithm for n > p MERGING Odd-Even Merge Bitonic Merge

9 CONTENTS xxi 15.6 SORTING Odd-Even Merge Sort Bitonic Sort GRAPH PROBLEMS COMPUTING THE CONVEX HULL REFERENCES AND READINGS ADDITIONAL EXERCISES INDEX I-1

2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8]

2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8] Code No: R05220502 Set No. 1 1. (a) Describe the performance analysis in detail. (b) Show that f 1 (n)+f 2 (n) = 0(max(g 1 (n), g 2 (n)) where f 1 (n) = 0(g 1 (n)) and f 2 (n) = 0(g 2 (n)). [8+8] 2. (a)