Submodular Functions and Electrical NetworksThere is a strong case for electrical network topologists and submodular function theorists being aware of each other's fields. Presenting a topological approach to electrical network theory, this book demonstrates the strong links that exist between submodular functions and electrical networks. The book contains: . a detailed discussion of graphs, matroids, vector spaces and the algebra of generalized minors, relevant to network analysis (particularly to the construction of efficient circuit simulators) . a detailed discussion of submodular function theory in its own right; topics covered include, various operations, dualization, convolution and Dilworth truncation as well as the related notions of prinicpal partition and principal lattice of partitions. In order to make the book useful to a wide audience, the material on electrical networks and that on submodular functions is presented independently of each other. The hybrid rank problem, the bridge between (topological) electrical network theory and submodular functions, is covered in the final chapter. The emphasis in the book is on low complexity algorithms, particularly based on bipartite graphs. The book is intended for self-study and is recommended to designers of VLSI algorithms. More than 300 problems, almost all of them with solutions, are included at the end of each chapter. |
Contents
Chapter 1 Introduction | 1 |
Chapter 2 Mathematical Preliminaries | 15 |
Chapter 3 Graphs | 31 |
Chapter 4 Matroids | 103 |
Chapter 5 Electrical Networks | 131 |
Chapter 6 Topological Hybrid Analysis | 173 |
Chapter 7 The Implicit Duality Theorem and Its Applications | 213 |
Chapter 8 Multiport Decomposition | 269 |
Chapter 10 Convolution of Submodular Functions | 379 |
Chapter 11 Matroid Union | 453 |
Chapter 12 Dilworth Truncation of Submodular Functions | 481 |
Chapter 13 Algorithms for the PLP of a Submodular Function | 533 |
Chapter 14 The Hybrid Rank Problem | 571 |
| 629 | |
| 644 | |
Chapter 9 Submodular Functions | 325 |