HOW TO THINK ABOUT ALGORITHMS
There are many algorithm texts that provide lots of well-polished code and
proofs of correctness. Instead, this one presents insights, notations, and
analogies to help the novice describe and think about algorithms like an
expert. It is a bit like a carpenter studying hammers instead of houses. Jeff
Edmonds provides both the big picture and easy step-by-step methods for
developing algorithms, while avoiding the comon pitfalls. Paradigms such
as loop invariants and recursion help to unify a huge range of algorithms
into a few meta-algorithms. Part of the goal is to teach students to think
abstractly. Without getting bogged down in formal proofs, the book fosters
deeper understanding so that how and why each algorithm works is trans-
parent. These insights are presented in a slow and clear manner accessible
to second- or third-year students of computer science, preparing them to
ﬁnd on their own innovative ways to solve problems.
Abstraction is when you translate the equations, the rules, and the under-
lying essences of the problem not only into a language that can be commu-
nicated to your friend standing with you on a streetcar, but also into a form
that can percolate down and dwell in your subconscious. Because, remem-
ber, it is your subconscious that makes the miraculous leaps of inspiration,
not your plodding perspiration and not your cocky logic. And remember,
unlike you, your subconscious does not understand Java code.
Bookmarks
Cover
Half-title
Title
Copyright
CONTENTS
PREFACE
Introduction
PART ONE: Iterative Algorithms and Loop Invariants
1 Iterative Algorithms: Measures of Progress and Loop Invariants
1.1 A Paradigm Shift: A Sequence of Actions vs. a Sequence of Assertions
1.2 The Steps to Develop an Iterative Algorithm
1.3 More about the Steps
1.4 Different Types of Iterative Algorithms
1.5 Typical Errors
1.6 Exercises
2 Examples Using More-of-the-Input Loop Invariants
2.1 Coloring the Plane
2.2 Deterministic Finite Automaton
2.3 More of the Input vs. More of the Output
3 Abstract Data Types
3.1 Specifications and Hints at Implementations
3.2 Link List Implementation
3.3 Merging with a Queue
3.4 Parsing with a Stack
4 Narrowing the Search Space: Binary Search
4.1 Binary Search Trees
4.2 Magic Sevens
4.3 VLSI Chip Testing
4.4 Exercises
5 Iterative Sorting Algorithms
5.1 Bucket Sort by Hand
5.2 Counting Sort (a Stable Sort)
5.3 Radix Sort
5.4 Radix Counting Sort
6 Euclid’s GCD Algorithm
7 The Loop Invariant for Lower Bounds
PART TWO: Recursion
8 Abstractions, Techniques, and Theory
8.1 Thinking about Recursion
8.2 Looking Forward vs. Backward
8.3 With a Little Help from Your Friends
8.4 The Towers of Hanoi
8.5 Checklist for Recursive Algorithms
8.6 The Stack Frame
8.7 Proving Correctness with Strong Induction
9 Some Simple Examples of Recursive Algorithms
9.1 Sorting and Selecting Algorithms
9.2 Operations on Integers
9.3 Ackermann's Function
9.4 Exercises
10 Recursion on Trees
10.1 Tree Traversals
10.2 Simple Examples
10.3 Generalizing the Problem Solved
10.4 Heap Sort and Priority Queues
10.5 Representing Expressions with Trees
11 Recursive Images
11.1 Drawing a Recursive Image from a Fixed Recursive and a Base Case Image
11.2 Randomly Generating a Maze
12 Parsing with Context-Free Grammars
PART THREE: Optimization Problems
13 Definition of Optimization Problems
14 Graph Search Algorithms
14.1 A Generic Search Algorithm
14.2 Breadth-First Search for Shortest Paths
14.3 Dijkstra's Shortest-Weighted-Path Algorithm
14.4 Depth-First Search
14.5 Recursive Depth-First Search
14.6 Linear Ordering of a Partial Order
14.7 Exercise
15 Network Flows and Linear Programming
15.1 A Hill-Climbing Algorithm with a Small Local Maximum
15.2 The Primal…Dual Hill-Climbing Method
15.3 The Steepest-Ascent Hill-Climbing Algorithm
15.4 Linear Programming
15.5 Exercises
16 Greedy Algorithms
16.1 Abstractions, Techniques, and Theory
16.2 Examples of Greedy Algorithms 16.2.1 Example: The Job/Event Scheduling Problem
16.2.2 Example: The Interval Cover Problem
16.2.3 Example: The Minimum-Spanning-Tree Problem
16.3 Exercises
17 Recursive Backtracking
17.1 Recursive Backtracking Algorithms
17.2 The Steps in Developing a Recursive Backtracking
17.3 Pruning Branches
17.4 Satisfiability
17.5 Exercises
18 Dynamic Programming Algorithms
18.1 Start by Developing a Recursive Backtracking
18.2 The Steps in Developing a Dynamic Programming Algorithm
18.3 Subtle Points
18.3.1 The Question for the Little Bird
18.3.2 Subinstances and Subsolutions
18.3.3 The Set of Subinstances
18.3.4 Decreasing Time and Space
18.3.5 Counting the Number of Solutions
18.3.6 The New Code
19 Examples of Dynamic Programs
19.1 The Longest-Common-Subsequence Problem
19.2 Dynamic Programs as More-of-the-Input Iterative Loop Invariant Algorithms
19.3 A Greedy Dynamic Program: The Weighted Job/Event Scheduling Problem
19.4 The Solution Viewed as a Tree: Chains of Matrix Multiplications
19.5 Generalizing the Problem Solved: Best AVL Tree
19.6 All Pairs Using Matrix Multiplication
19.7 Parsing with Context-Free Grammars
19.8 Designing Dynamic Programming Algorithms via Reductions
20 Reductions and NP-Completeness
20.1 Satisfiability Is at Least as Hard as Any Optimization Problem
20.2 Steps to Prove NP-Completeness
20.3 Example: 3-Coloring Is NP-Complete
20.4 An Algorithm for Bipartite Matching Using the Network Flow Algorithm
21 Randomized Algorithms
21.1 Using Randomness to Hide the Worst Cases
21.2 Solutions of Optimization Problems with a Random Structure
PART FOUR: Appendix
22 Existential and Universal Quantifiers
23 Time Complexity
23.1 The Time (and Space) Complexity of an Algorithm
23.2 The Time Complexity of a Computational Problem
24 Logarithms and Exponentials
25 Asymptotic Growth
25.1 Steps to Classify a Function
25.2 More about Asymptotic Notation
26 Adding-Made-Easy Approximations
26.1 The Technique
26.2 Some Proofs for the Adding-Made-Easy Technique
27 Recurrence Relations
27.1 The Technique
27.2 Some Proofs
28 A Formal Proof of Correctness
PART FIVE: Exercise Solutions
Chapter 1. Iterative Algorithms: Measures of Progress and Loop Invariants
Chapter 2. Examples UsingMore-of-the-Input Loop Invariant
Chapter 3. Abstract Data Types
Chapter 4. Narrowing the Search Space: Binary Search
Chapter 6. Euclid’s GCD Algorithm
Chapter 7. The Loop Invariant for Lower Bounds
Chapter 8. Abstractions, Techniques, and Theory
Chapter 9. Some Simple Examples of Recursive Algorithms
Chapter 10. Recursion on Trees
Chapter 11. Recursive Images
Chapter 12. Parsingwith Context-Free Grammars
Chapter 14. Graph Search Algorithms
Chapter 15. Network Flows and Linear Programming
Chapter 16: Greedy Algorithms
Chapter 17. Recursive Backtracking
Chapter 18. Dynamic Programming Algorithms
Chapter 19. Examples of Dynamic Programs
Chapter 20. Reductions and NP-Completeness
Chapter 22. Existential and Universal Quantifiers
Chapter 23. Time Complexity
Chapter 24. Logarithms and Exponentials
Chapter 25. Asymptotic Growth
Chapter 26. Adding-Made-Easy Approximations
Chapter 27. Recurrence Relations
CONCLUSION
INDEX