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Table of Contents

Algorithm Design

Jon Kleinberg and Eva Tardos

TTable of Contents

1 Introduction: Some Representative Problems

1.1 A First Problem: StTable Matching

1.2 Five Representative Problems

Solved Exercises

Excercises

Notes and Further Reading

2 Basics of Algorithms Analysis

2.1 Computational Tractability

2.2 Asymptotic Order of Growth Notation

2.3 Implementing the StTable Matching Algorithm using Lists and Arrays

2.4 A Survey of Common Running Times

2.5 A More Complex Data Structure: Priority Queues

Solved Exercises

Exercises

Notes and Further Reading

3 Graphs

3.1 Basic Definitions and Applications

3.2 Graph Connectivity and Graph Traversal

3.3 Implementing Graph Traversal using Queues and Stacks

3.4 Testing Bipartiteness: An Application of Breadth-First Search

3.5 Connectivity in Directed Graphs

3.6 Directed Acyclic Graphs and Topological Ordering

Solved Exercises

Exercises

Notes and Further Reading

4 Greedy Algorithms

4.1 Interval Scheduling: The Greedy Algorithm Stays Ahead

4.2 Scheduling to Minimize Lateness: An Exchange Argument

4.3 Optimal Caching: A More Complex Exchange Argument

4.4 Shortest Paths in a Graph

4.5 The Minimum Spanning Tree Problem

4.6 Implementing Kruskal’s Algorithm: The Union-Find Data Structure

4.7 Clustering

4.8 Huffman Codes and the Problem of Data Compression

*4.9 Minimum-Cost Arborescences: A Multi-Phase Greedy Algorithm

Solved Exercises

Excercises

Notes and Further Reading

5 Divide and Conquer

5.1 A First Recurrence: The Mergesort Algorithm

5.2 Further Recurrence Relations

5.3 Counting Inversions

5.4 Finding the Closest Pair of Points

5.5 Integer Multiplication

5.6 Convolutions and The Fast Fourier Transform

Solved Exercises

Exercises

Notes and Further Reading

6 Dynamic Programming

6.1 Weighted Interval Scheduling: A Recursive Procedure

6.2 Weighted Interval Scheduling: Iterating over Sub-Problems

6.3 Segmented Least Squares: Multi-way Choices

6.4 Subset Sums and Knapsacks: Adding a VariTable

6.5 RNA Secondary Structure: Dynamic Programming Over Intervals

6.6 Sequence Alignment

6.7 Sequence Alignment in Linear Space

6.8 Shortest Paths in a Graph

6.9 Shortest Paths and Distance Vector Protocols

*6.10 Negative Cycles in a Graph

Solved Exercises

Exercises

Notes and Further Reading

7 Network Flow

7.1 The Maximum Flow Problem and the Ford-Fulkerson Algorithm

7.2 Maximum Flows and Minimum Cuts in a Network

7.3 Choosing Good Augmenting Paths

*7.4 The Preflow-Push Maximum Flow Algorithm

7.5 A First Application: The Bipartite Matching Problem

7.6 Disjoint Paths in Directed and Undirected Graphs

7.7 Extensions to the Maximum Flow Problem

7.8 Survey Design

7.9 Airline Scheduling

7.10 Image Segmentation

7.11 Project Selection

7.12 Baseball Elimination

*7.13 A Further Direction: Adding Costs to the Matching Problem

Solved Exercises

Exercises

Notes and Further Reading

8 NP and Computational Intractability

8.1 Polynomial-Time Reductions

8.2 Reductions via “Gadgets”: The Satisfiability Problem

8.3 Efficient Certification and the Definition of NP

8.4 NP-Complete Problems

8.5 Sequencing Problems

8.6 Partitioning Problems

8.7 Graph Coloring

8.8 Numerical Problems

8.9 Co-NP and the Asymmetry of NP

8.10 A Partial Taxonomy of Hard Problems

Solved Exercises

Exercises

Notes and Further Reading

9 PSPACE: A Class of Problems Beyond NP

9.1 PSPACE

9.2 Some Hard Problems in PSPACE

9.3 Solving Quantified Problems and Games in Polynomial Space

9.4 Solving the Planning Problem in Polynomial Space

9.5 Proving Problems PSPACE-Complete

Solved Exercises

Exercises

Notes and Further Reading

10 Extending the Limits of Tractability

10.1 Finding Small Vertex Covers

10.2 Solving NP-Hard Problem on Trees

10.3 Coloring a Set of Circular Arcs

*10.4 Tree Decompositions of Graphs

*10.5 Constructing a Tree Decomposition

Solved Exercises

Exercises

Notes and Further Reading

11 Approximation Algorithms

11.1 Greedy Algorithms and Bounds on the Optimum: A Load Balancing Problem

11.2 The Center Selection Problem

11.3 Set Cover: A General Greedy Heuristic

11.4 The Pricing Method: Vertex Cover

11.5 Maximization via the Pricing method: The Disjoint Paths Problem

11.6 Linear Programming and Rounding: An Application to Vertex Cover

*11.7 Load Balancing Revisited: A More Advanced LP Application

11.8 Arbitrarily Good Approximations: the Knapsack Problem

Solved Exercises

Exercises

Notes and Further Reading

12 Local Search

12.1 The Landscape of an Optimization Problem

12.2 The Metropolis Algorithm and Simulated Annealing

12.3 An Application of Local Search to Hopfield Neural Networks

12.4 Maximum Cut Approximation via Local Search

12.5 Choosing a Neighbor Relation

*12.6 Classification via Local Search

12.7 Best-Response Dynamics and Nash Equilibria

Solved Exercises

Exercises

Notes and Further Reading

13 Randomized Algorithms

13.1 A First Application: Contention Resolution

13.2 Finding the Global Minimum Cut

13.3 Random VariTables and their Expectations

13.4 A Randomized Approximation Algorithm for MAX 3-SAT

13.5 Randomized Divide-and-Conquer: Median-Finding and Quicksort

13.6 Hashing: A Randomized Implementation of Dictionaries

13.7 Finding the Closest Pair of Points: A Randomized Approach

13.8 Randomized Caching

13.9 Chernoff Bounds

13.10 Load Balancing

*13.11 Packet Routing

13.12 Background: Some Basic Probability Definitions

Solved Exercises

Exercises

Notes and Further Reading

Epilogue: Algorithms that Run Forever

References

Index