Algorithms, Graphs, and Computers, Vol. 62

Algorithms Graphs and Computers......Page 4
Copyright Page......Page 5
Contents......Page 12
Preface......Page 6
1. Introduction......Page 18
2. A Commuting Problem......Page 20
3. Enumeration of Paths......Page 21
4. Tree Diagrams......Page 23
5. Timing and the Shortest Route......Page 27
6. An Improved Map......Page 29
7. The Array of Times......Page 33
8. Arbitrary Starting Point......Page 36
9. Minimum Time Functions......Page 38
10. The Minimum Function......Page 39
11. Equations for the fi......Page 42
12. Reduced Tree......Page 44
13. General Map......Page 45
14. Solving the Equations......Page 47
15. Why Not Enumeration Using Computers?......Page 49
16. Graphs......Page 50
17. The Problem of the Königsberg Bridges......Page 52
18. Systems and States......Page 56
19. Critical Path Analysis......Page 60
20. Summing Up......Page 63
Bibliography and Comments......Page 64
1. Introduction......Page 66
2. A Simple Example of Successive Approximations......Page 67
3. A Simple Example of the Failure of the Method......Page 70
4. Algorithms......Page 71
5. Computer Programming......Page 74
6. Algorithms for Solving x = f(x)......Page 77
7. The Newton–Raphson Method: Square Roots......Page 80
8. An Example......Page 83
9. Enter the Digital Computer......Page 84
11. Flow Charts for the Determination of S a......Page 85
12. What Initial Approximation?......Page 86
13. An Approximation Method of Steinhaus......Page 87
14. Successive Approximations for a Simple Routing Problem......Page 88
15. The Map of Fig. 3 of Chapter One......Page 90
16. Determination of Optimal Policy......Page 93
17. A Different Initial Policy......Page 94
18. Successive Approximations for a General Map......Page 96
20. The Connection Matrix and A Second Programming Method......Page 99
21. Fictitious Links......Page 102
22. Paths With a Given Number of Junctions......Page 103
23. A Labelling Algorithm......Page 107
24. Working Back From the End......Page 108
Miscellaneous Exercises......Page 111
Bibliography and Comment......Page 116
1. Introduction......Page 118
2. How to Plan a Vacation......Page 119
4. The Peculiarities of Structure......Page 123
5. Computer Program......Page 126
7. Use of Tapes......Page 127
8. Storage of Sparse Data......Page 128
10. Transformation of Systems......Page 129
11. Stratification......Page 131
12. Acceleration of Convergence......Page 132
13. Design of Electric Power Stations......Page 135
Miscellaneous Exercises......Page 137
Bibliography and Comments......Page 138
1. Will the Method Always Work?......Page 140
3. How Do We Find the Desired Solution?......Page 142
4. Monotonicity......Page 143
5. The Upper Solution......Page 144
6. The Lower Solution......Page 145
8. Uniqueness of Solution......Page 146
9. Determination of Optimal Policy......Page 148
10. Arbitrary Initial Approximation......Page 149
11. Least Time to Move k Steps......Page 150
12. Number of Iterations......Page 152
Miscellaneous Exercises......Page 154
Bibliography and Comment......Page 158
2. Precise Formulation......Page 159
4. Tree Diagrams......Page 160
5. Some Further Observations......Page 164
6. Enumeration by Computer......Page 165
7. A Geometrical Representation......Page 166
8. Cost Matrix......Page 169
9. Connection Matrix......Page 171
11. Powers of the Adjacency Matrix......Page 172
12. Functional Equations......Page 173
13. Successive Approximations......Page 174
14. The Labelling Algorithm......Page 175
15. Avoiding the Cost Matrix......Page 177
17. Getting Close......Page 178
18. Making the Closest Approach......Page 180
19. Calculation of fN(y, z )......Page 181
20. Accessible and Inaccessible States......Page 184
Miscellaneous Exercises......Page 185
Bibliography and Comment......Page 190
1. The Sawyer Graph......Page 191
2. Analysis of the Sawyer Graph......Page 194
3. Connectedness of the Sawyer Graph......Page 197
4. The Spine of the Sawyer Graph......Page 198
5. The Case When c is a Divisor of b......Page 199
6. The General Case......Page 200
7. The Case when b and c Have No Common Factor......Page 202
8. The Billiard Ball Solution......Page 204
9. The Billiard Ball Diagram and Sawyer Graph......Page 205
10. Graphical Determination of Shortest Paths......Page 206
12. Computer Assistance in Drawing State Graphs......Page 209
Miscellaneous Exercises......Page 210
Bibliography and Comment......Page 212
1. Another Classical Puzzle......Page 213
2. Trees and Enumeration......Page 214
3. The State Diagram......Page 215
4. Existence and Uniqueness......Page 217
5. Analytic Methods......Page 221
6. Example......Page 223
Miscellaneous Exercises......Page 226
Bibliography and Comment......Page 228
1. Introduction......Page 230
3. A Map and a Matrix......Page 231
5. A Simplified Version......Page 233
6. Recurrence Relations......Page 234
7. Do All the Errands!......Page 235
9. How Many Errands Could We Do?......Page 236
10. Trading Time for Storage......Page 237
11. How to Build a Tennis Court......Page 238
12. Sequential Selection......Page 239
13. Feasibility......Page 240
14. Simplified and More Realistic Assignment Problem......Page 241
16. Discussion......Page 242
18. A Branching Process......Page 243
19. Recurrence Relations......Page 244
Miscellaneous Exercises......Page 245
Bibliography and Comment......Page 253
Author lndex......Page 256
Subject Index......Page 260
Mathematics in Science and Engineering......Page 264