Preface 1 Introduction 1.1 Notion of Algorithm Exercises 1.1 1.2 Fundamentals of Algorithmic Problem Solving Understanding the Problem Ascertaining the Capabilities of a Computational Device Choosing between Exact and Approximate Problem Solving Deciding on Appropriate Data Structures Algorithm Design Techniques Methods of Specifying an Algorithm Proving an Algorithm's Correctness Analyzing an Algorithm Coding an Algorithm Exercises 1.2 1.3 Important Problem Types Sorting Searching String Processing Graph Problems Combinatorial Problems Geometric Problems Numerical Problems Exercises 1.3 1.4 Fundamental Data Structures Linear Data Structures Graphs Trees Sets and Dictionaries Exercises 1.4 Summary 2 Fundamentals of the Analysis of Algorithm Efficiency 2.1 Analysis Framework Measuring an Input's Size Units for Measuring Running Time Orders of Growth Worst-Case, Best-Case, and Average-Case Efficiencies Recapitulation of the Analysis Framework Exercises 2.1 2.2 Asymptotic Notations and Basic Efficiency Classes Informal Introduction O-notation -notation O-notation Useful Property Involving the Asymptotic Notations Using Limits for Comparing Orders of Growth Basic Efficiency Classes Exercises 2.2 2.3 Mathematical Analysis of Nonrecursive Algorithms Exercises 2.3 2.4 Mathematical Analysis of Recursive Algorithms Exercises 2.4 2.5 Example: Fibonacci Numbers Explicit Formula for the nth Fibonacci Number Algorithms for Computing Fibonacci Numbers Exercises 2.5 2.6 Empirical Analysis of Algorithms Exercises 2.6 2.7 Algorithm Visualization Summary 3 Brute Force 3.1 Selection Sort and Bubble Sort Selection Sort Bubble Sort Exercises 3.1 3.2 Sequential Search and Brute-Force String Matching Sequential Search Brute-Force String Matching Exercises 3.2 3.3 Closest-Pair and Convex-Hull Problems by Brute Force Closest-Pair Problem Convex-Hull Problem Exercises 3.3 3.4 Exhaustive Search Traveling Salesman Problem Knapsack Problem Assignment Problem Exercises 3.4 Summary 4 Divide-and-Conquer 4.1 Mergesort Exercises 4.1 4.2 Quicksort Exercises 4.2 4.3 Binary Search Exercises 4.3 4.4 Binary Tree Traversals and Related Properties Exercises 4.4 4.5 Multiplication of Large Integers and Strassen's Matrix Multiplication Multiplication of Large Integers Strassen's Matrix Multiplication Exercises 4.5 4.6 Closest-Pair and Convex-Hull Problems by Divide-and-Conquer Closest-Pair Problem Convex-Hull Problem Exercises 4.6 Summary 5 Decrease-and-Conquer 5.1 Insertion Sort Exercises 5.1 5.2 Depth-First Search and Breadth-First Search Depth-First Search Breadth-First Search Exercises 5.2 5.3 Topological Sorting Exercises 5.3 5.4 Algorithms for Generating Combinatorial Objects Generating Permutations Generating Subsets Exercises 5.4 5.5 Decrease-by-a-Constant-Factor Algorithms Fake-Coin Problem Multiplication a la Russe Josephus Problem Exercises 5,5 5.6 Variable-Size-Decrease Algorithms Computing e Median and the Selection Problem Interpolation Search Searching and Insertion in a Binary Search Tree Exercises 5.6 Summary 6 Transform-and-Conquer 6.1 Presorting Exercises 6.1 6.2 Gaussian Elimination LU Decomposition and Other Applications Computing a Matrix Inverse Computing a Determinant Exercises 6.2 6.3 Balanced Search Trees AVL Trees 2-3 Trees Exercises 6.3 6.4 Heaps and Heapsort Notion of the Heap Heapsort Exercises 6,4 6.5 Horner's Rule and Binary Exponentiation Horner's Rule Binary Exponentiation Exercises 6.5 6.6 Problem Reduction Computing the Least Common Multiple Counting Paths in a Graph Reduction of Optimization Problems Linear Programming Reduction to Graph Problems Exercises 6.6 Summary 7 Space and lime Tradeoffs 7.1 Sorting by Counting Exercises 7.1 7.2 Input Enhancement in String Matching Horspool's Algorithm Boyer-Moore Algorithm Exercises 7,2 7,3 Hashing Open Hashing (Separate Chaining) Closed Hashing (Open Addressing) Exercises 7.3 7.4 B-Trees Exercises 7.4 Summary 8 Dynamic Programming 8.1 Computing a Binomial Coefficient Exercises 8.1 8.2 Warshall's and Floyd's Algorithms Warshall's Algorithm Floyd's Algorithm for the Ali-Pairs Shortest-Paths Problem Exercises 8.2 8.3 Optimal Binary Search Trees Exercises 8.3 8.4 The Knapsack Problem and Memory Functions Memory Functions Exercises 8.4 Summary 9 Greedy Technique 9.1 Prim's Algorithm Exercises 9.1 9.2 Kruskal's Algorithm Disjoint Subsets and Union-Find Algorithms Exercises 9.2 9.3 Dijkstra's Algorithm Exercises 9.3 9.4 Huffman Trees Exercises 9.4 Summary 10 Limitations of Algorithm Power 10.1 Lower-Bound Arguments Trivial Lower Bounds Information-Theoretic Arguments Adversary Arguments Problem Reduction Exercises 10.1 10.2 Decision Trees Decision Trees for Sorting Algorithms Decision Trees for Searching a Sorted Array Exercises 10.2 10.3 P, NP, and NP-complete Problems P and NP Problems NP-complete Problems Exercises 10.3 10.4 Challenges of Numerical Algorithms Exercises 10.4 Summary 44 11 Coping with the Limitations of Algorithm Power 11.1 Backtracking n-Queens Problem Hamiltonian Circuit Problem Subset-Sum Problem General Remarks Exercises 11,1 11.2 Branch-and-Bound Assignment Problem Knapsack Problem Traveling Salesman Problem Exercises 11,2 11.3 Approximation Algorithms for NP-hard Problems Approximation Algorithms for the Traveling Salesman Problem Approximation Algorithms for the Knapsack Problem Exercises 11.3 11.4 Algorithms for Solving Nonlinear Equations Bisection Method Method of False Position Newton's Method Exercises 11.4 Summary Epilogue APPENDIX A Useful Formulas for the Analysis of Algorithms Properties of Logarithms Combinatorics Important Summation Formulas Sum Manipulation Rules Approximation of a Sum by a Definite Integral Floor and Ceiling Formulas Miscellaneous APPENDIX B Short Tutorial on Recurrence Relations Sequences arid Recurrence Relations Methods for Solving Recurrence Relations Common Recurrence Types in Algorithm Analysis Bibliography Hints to Exercises Index
鄂公網(wǎng)安備 42010302001612號 