ALGORITHMICPROBLEM SOLVING

Roland Backhouse

ContentsPreface xi

PART I Algorithmic Problem Solving 1

C H A P T E R 1 – Introduction 31.1 Algorithms 31.2 Algorithmic Problem Solving 41.3 Overview 51.4 Bibliographic Remarks 6

C H A P T E R 2 – Invariants 72.1 Chocolate Bars 10

2.1.1 The Solution 102.1.2 The Mathematical Solution 11

2.2 Empty Boxes 162.2.1 Review 19

2.3 The Tumbler Problem 222.3.1 Non-deterministic Choice 23

2.4 Tetrominoes 242.5 Summary 302.6 Bibliographic Remarks 34

C H A P T E R 3 – Crossing a River 353.1 Problems 363.2 Brute Force 37

3.2.1 Goat, Cabbage and Wolf 373.2.2 State-Space Explosion 393.2.3 Abstraction 41

3.3 Nervous Couples 423.3.1 What Is the Problem? 423.3.2 Problem Structure 433.3.3 Denoting States and Transitions 443.3.4 Problem Decomposition 453.3.5 A Review 48

3.4 Rule of Sequential Composition 503.5 The Bridge Problem 543.6 Conditional Statements 633.7 Summary 653.8 Bibliographic Remarks 65

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C H A P T E R 4 – Games 674.1 Matchstick Games 674.2 Winning Strategies 69

4.2.1 Assumptions 694.2.2 Labelling Positions 704.2.3 Formulating Requirements 72

4.3 Subtraction-Set Games 744.4 Sums of Games 78

4.4.1 A Simple Sum Game 794.4.2 Maintain Symmetry! 814.4.3 More Simple Sums 824.4.4 Evaluating Positions 834.4.5 Using the Mex Function 87

4.5 Summary 914.6 Bibliographic Remarks 92

C H A P T E R 5 – Knights and Knaves 955.1 Logic Puzzles 955.2 Calculational Logic 96

5.2.1 Propositions 965.2.2 Knights and Knaves 975.2.3 Boolean Equality 985.2.4 Hidden Treasures 1005.2.5 Equals for Equals 101

5.3 Equivalence and Continued Equalities 1025.3.1 Examples of the Associativity of Equivalence 1045.3.2 On Natural Language 105

5.4 Negation 1065.4.1 Contraposition 1095.4.2 Handshake Problems 1125.4.3 Inequivalence 113

5.5 Summary 1175.6 Bibliographic Remarks 117

C H A P T E R 6 – Induction 1196.1 Example Problems 1206.2 Cutting the Plane 1236.3 Triominoes 1266.4 Looking for Patterns 1286.5 The Need for Proof 1296.6 From Verification to Construction 130

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6.7 Summary 1346.8 Bibliographic Remarks 134

C H A P T E R 7 – Fake-Coin Detection 1377.1 Problem Formulation 1377.2 Problem Solution 139

7.2.1 The Basis 1397.2.2 Induction Step 1397.2.3 The Marked-Coin Problem 1407.2.4 The Complete Solution 141

7.3 Summary 1467.4 Bibliographic Remarks 146

C H A P T E R 8 – The Tower of Hanoi 1478.1 Specification and Solution 147

8.1.1 The End of the World! 1478.1.2 Iterative Solution 1488.1.3 Why? 149

8.2 Inductive Solution 1498.3 The Iterative Solution 1538.4 Summary 1568.5 Bibliographic Remarks 156

C H A P T E R 9 – Principles of Algorithm Design 1579.1 Iteration, Invariants and Making Progress 1589.2 A Simple Sorting Problem 1609.3 Binary Search 1639.4 Sam Loyd’s Chicken-Chasing Problem 166

9.4.1 Cornering the Prey 1709.4.2 Catching the Prey 1749.4.3 Optimality 176

9.5 Projects 1779.6 Summary 1789.7 Bibliographic Remarks 180

C H A P T E R 10 – The Bridge Problem 18310.1 Lower and Upper Bounds 18310.2 Outline Strategy 18510.3 Regular Sequences 18710.4 Sequencing Forward Trips 18910.5 Choosing Settlers and Nomads 193

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10.6 The Algorithm 19610.7 Summary 19910.8 Bibliographic Remarks 200

C H A P T E R 11 – Knight’s Circuit 20111.1 Straight-Move Circuits 20211.2 Supersquares 20611.3 Partitioning the Board 20911.4 Summary 21611.5 Bibliographic Remarks 218

PART II Mathematical Techniques 219

C H A P T E R 12 – The Language of Mathematics 22112.1 Variables, Expressions and Laws 22212.2 Sets 224

12.2.1 The Membership Relation 22412.2.2 The Empty Set 22412.2.3 Types/Universes 22412.2.4 Union and Intersection 22512.2.5 Set Comprehension 22512.2.6 Bags 227

12.3 Functions 22712.3.1 Function Application 22812.3.2 Binary Operators 23012.3.3 Operator Precedence 230

12.4 Types and Type Checking 23212.4.1 Cartesian Product and Disjoint Sum 23312.4.2 Function Types 235

12.5 Algebraic Properties 23612.5.1 Symmetry 23712.5.2 Zero and Unit 23812.5.3 Idempotence 23912.5.4 Associativity 24012.5.5 Distributivity/Factorisation 24112.5.6 Algebras 243

12.6 Boolean Operators 24412.7 Binary Relations 246

12.7.1 Reflexivity 24712.7.2 Symmetry 24812.7.3 Converse 24912.7.4 Transitivity 24912.7.5 Anti-symmetry 25112.7.6 Orderings 252

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12.7.7 Equality 25512.7.8 Equivalence Relations 256

12.8 Calculations 25712.8.1 Steps in a Calculation 25912.8.2 Relations between Steps 26012.8.3 ‘‘If’’ and ‘‘Only If’’ 262

12.9 Exercises 264

C H A P T E R 13 – Boolean Algebra 26713.1 Boolean Equality 26713.2 Negation 26913.3 Disjunction 27013.4 Conjunction 27113.5 Implication 274

13.5.1 Definitions and Basic Properties 27513.5.2 Replacement Rules 276

13.6 Set Calculus 27913.7 Exercises 281

C H A P T E R 14 – Quantifiers 28514.1 DotDotDot and Sigmas 28514.2 Introducing Quantifier Notation 286

14.2.1 Summation 28714.2.2 Free and Bound Variables 28914.2.3 Properties of Summation 29114.2.4 Warning 297

14.3 Universal and Existential Quantification 29714.3.1 Universal Quantification 29814.3.2 Existential Quantification 300

14.4 Quantifier Rules 30114.4.1 The Notation 30214.4.2 Free and Bound Variables 30314.4.3 Dummies 30314.4.4 Range Part 30314.4.5 Trading 30414.4.6 Term Part 30414.4.7 Distributivity Properties 304

14.5 Exercises 306

C H A P T E R 15 – Elements of Number Theory 30915.1 Inequalities 30915.2 Minimum and Maximum 31215.3 The Divides Relation 31515.4 Modular Arithmetic 316

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15.4.1 Integer Division 31615.4.2 Remainders and Modulo Arithmetic 320

15.5 Exercises 322

C H A P T E R 16 – Relations, Graphs and Path Algebras 32516.1 Paths in a Directed Graph 32516.2 Graphs and Relations 328

16.2.1 Relation Composition 33016.2.2 Union of Relations 33216.2.3 Transitive Closure 33416.2.4 Reflexive Transitive Closure 338

16.3 Functional and Total Relations 33916.4 Path-Finding Problems 341

16.4.1 Counting Paths 34116.4.2 Frequencies 34316.4.3 Shortest Distances 34416.4.4 All Paths 34516.4.5 Semirings and Operations on Graphs 347

16.5 Matrices 35116.6 Closure Operators 35316.7 Acyclic Graphs 354

16.7.1 Topological Ordering 35516.8 Combinatorics 357

16.8.1 Basic Laws 35816.8.2 Counting Choices 35916.8.3 Counting Paths 361

16.9 Exercises 366

Solutions to Exercises 369

References 405

Index 407

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