Extremal paths in graphs: foundations, search strategies, and related topics
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin
Akad.-Verl.
1997
|
| Edition: | 1. ed. |
| Series: | Mathematical topics
10 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 447 - 474 |
| Physical Description: | 480 S. graph. Darst. |
| ISBN: | 3055016580 |
Staff View
MARC
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| 100 | 1 | |a Huckenbeck, Ulrich |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Extremal paths in graphs |b foundations, search strategies, and related topics |c Ulrich Huckenbeck |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Berlin |b Akad.-Verl. |c 1997 | |
| 300 | |a 480 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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| 500 | |a Literaturverz. S. 447 - 474 | ||
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| 650 | 4 | |a Paths and cycles (Graph theory) | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
1 Introduction 13
1.1 Examples of extremal paths in graphs 13
1.2 Basic Terminology 14
2 Path functions 21
2.1 Introduction of path functions and of related terms 21
2.2 Simple methods to define path functions 26
2.3 Order preserving path functions 31
2.3.1 Definition of order preservation and monotonicity 32
2.3.2 Structural properties of order preserving path functions .... 34
2.3.3 Further methods to construct order preserving functions .... 38
2.3.4 Modified versions of order preservation 51
2.3.5 Further remarks and definitions concerning modified order pre¬
serving principles 62
2.4 Bellman principles for paths in graphs 64
2.4.1 Introduction to strong Bellman principles for path functions . . 65
2.4.2 Introduction to weak Bellman principles for path functions . . 70
2.4.3 Comparing Bellman principles with each other 72
2.4.4 Relationships between Bellman properties and principles of or¬
der preservation 81
2.4.5 Relationships between Bellman properties and L minimum preser¬
vation 86
2.4.6 Structural properties of optimal paths if Bellman principles are
given 95
2.4.7 Functional equations and similar structural properties of cost
functions with Bellman properties 99
2.4.8 Decision models with Bellman properties 101
2.5 Further definitions and results related to path functions 101
2.5.1 Prefix, infix, and suffix monotone path functions 101
2.5.2 Introduction to heuristic functions 103
2.5.3 Negative cycles 104
10 Contents
2.5.4 Consistent functions 107
2.5.5 Admissibility of path functions, of heuristic functions, and of
evaluation functions 108
2.5.6 Triangle inequalities 109
2.5.7 Nonmisleading node functions, proper node functions and non
misleading path functions 114
2.5.8 Lexicographic path functions 115
2.5.9 Dioid properties of paths in graphs 119
3 Combinatorial results about paths in graphs 123
3.1 Basic definitions and results 124
3.2 Short cycles and other short paths in directed and half directed graphs 129
3.3 Short cycles and other short paths in undirected graphs 131
3.3.1 Short cycles in undirected graphs 131
3.3.2 Short acyclic paths in undirected graphs 134
3.4 Definitions and results in connexion with paths of arbitrary lengths . . 143
3.5 Long cycles and long acyclic paths in digraphs 146
3.6 Long cycles and other long paths in undirected graphs 153
3.6.1 Relationships between long paths and degrees of nodes 153
3.6.2 Describing the maximal length of cycles and other paths without
using the degrees of nodes 166
3.6.3 Further results about long cycles and other long paths 169
4 Algorithmic search for optimal paths in graphs and for other optimal
discrete objects 171
4.1 Introduction to optimal path problems 173
4.1.1 A simple optimal path problem and Dijkstra s algorithm .... 173
4.1.2 Three well known problems and their relationships to optimal
path problems 175
4.1.3 Measures for the complexity of optimal path algorithms .... 178
4.1.4 Formal descriptions of optimal path algorithms 179
4.2 The Single Source Single Target problem 179
4.2.1 Abstract search strategies for paths in graphs 180
4.2.2 The algorithm BF** 184
4.2.3 Modifications of BF** and BF*** 249
4.2.4 The algorithm GENGOAL: a search strategy whose redirection
function is not order preserving 253
4.2.5 A further algorithm for extremal goal paths: MarkA 259
4.2.6 Further problems 266
4.3 The Single Source All Targets problem 267
4.3.1 The algorithm FORD BELLMAN 1 267
4.3.2 The algorithm FORD BELLMAN 2 282
4.3.3 The algorithm FORD BELLMAN 3 285
4.3.4 Algorithms computing minimal costs of paths 287
Contents ¦.
4.3.5 Finding C minimal s n paths if C : V{s) (R, ) and if is
neither total nor identitive 292
4.3.6 Using Best First Search instead of Ford Bellman strategies . . 293
4.3.7 Further results from literature 294
4.3.8 Finding C maximal s v paths for all nodes v 295
4.4 The All Pairs Optimal Path problem 296
4.4.1 Matrix algorithms simulating Ford Bellman algorithms 299
4.4.2 Fast versions of the previous matrix algorithms 311
4.4.3 Floyd s fast search strategies 318
4.4.4 Complexity theoretical results 321
4.4.5 Further solutions of All Pairs Optimal Path problems 323
4.5 Path problems and related questions in special graphs or in other special
situations 324
4.5.1 Path problems arising from Geometry 324
4.5.2 Extremal paths in planar graphs 328
4.5.3 Path problems in graphs with a special structure or with a spe¬
cial application 330
4.6 Parallel search algorithms for optimal paths 331
4.6.1 Using computers without common memory 331
4.6.2 Using computers with a common memory 332
4.7 Searching for optimal paths in random graphs 333
4.8 Searching for almost optimal paths in graphs 340
4.8.1 Searching for paths with approximately minimal costs 341
4.8.2 Searching for the K best paths 341
4.9 Paths with extremal values Ci(P)/C2(.P) where Ci,C2 have a simple
structure 344
4.9.1 (Ci/C2) extremal paths in acyclic digraphs 344
4.9.2 (Ci/C2) extremal cycles 352
4.9.3 Relationships between approximation theoretical problems and
(Ci/C2) extremal cycles 355
4.10 Path problems with several path functions or with side constraints . . 369
4.10.1 Path problems with several path functions 369
4.10.2 Path problems with rational side constraints 371
4.10.3 Further path problems with side constraints 374
4.10.4 General remarks about path problems with side constraints . . 374
4.11 Hard problems 376
4.11.1 Path problems whose hardness is caused by the underlying cost
function 376
4.11.2 The Traveling Salesman Problem and other hard path problems
with side constraints 382
4.11.3 Further hard problems 390
4.12 Searching for optimal graphs and for other discrete objects similar to
paths in graphs 391
4.12.1 Searching for optimal subgraphs in a given graph 391
12 Contents
4.12.2 Construction of optimal graphs that are not a part of a given
supergraph 397
4.12.3 Further problems in connection with extremal paths in graphs 399
5 Relationships between paths in graphs and continuous curves 401
5.1 Continuous curves: basic definitions, elementary facts, and relationships
to paths in graphs 402
5.2 Additive functionals 406
5.3 Recursively linear functionals 410
5.4 Generalizations of recursive linearity 416
5.5 Functionals arising from initial value problems 419
5.6 Order preservation of functionals 427
5.7 Bellman properties of functionals 430
5.8 Functionals with many relative extrema 434
5.9 Approximation theoretical results 436
Bibliography 447
Index 475
Subject Index 475
Symbol Index 479
|
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| dewey-sort | 3511 15 |
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| discipline | Mathematik |
| edition | 1. ed. |
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| id | DE-604.BV011254477 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:06:38Z |
| institution | BVB |
| isbn | 3055016580 |
| language | German |
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| physical | 480 S. graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
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| publisher | Akad.-Verl. |
| record_format | marc |
| series | Mathematical topics |
| series2 | Mathematical topics |
| spelling | Huckenbeck, Ulrich Verfasser aut Extremal paths in graphs foundations, search strategies, and related topics Ulrich Huckenbeck 1. ed. Berlin Akad.-Verl. 1997 480 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematical topics 10 Literaturverz. S. 447 - 474 Grafentheorie gtt Graphes, Théorie des ram Optimisation combinatoire ram Combinatorial optimization Paths and cycles (Graph theory) Wegeproblem (DE-588)4392068-8 gnd rswk-swf Wegeproblem (DE-588)4392068-8 s DE-604 Mathematical topics 10 (DE-604)BV008671507 10 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007555527&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Huckenbeck, Ulrich Extremal paths in graphs foundations, search strategies, and related topics Mathematical topics Grafentheorie gtt Graphes, Théorie des ram Optimisation combinatoire ram Combinatorial optimization Paths and cycles (Graph theory) Wegeproblem (DE-588)4392068-8 gnd |
| subject_GND | (DE-588)4392068-8 |
| title | Extremal paths in graphs foundations, search strategies, and related topics |
| title_auth | Extremal paths in graphs foundations, search strategies, and related topics |
| title_exact_search | Extremal paths in graphs foundations, search strategies, and related topics |
| title_full | Extremal paths in graphs foundations, search strategies, and related topics Ulrich Huckenbeck |
| title_fullStr | Extremal paths in graphs foundations, search strategies, and related topics Ulrich Huckenbeck |
| title_full_unstemmed | Extremal paths in graphs foundations, search strategies, and related topics Ulrich Huckenbeck |
| title_short | Extremal paths in graphs |
| title_sort | extremal paths in graphs foundations search strategies and related topics |
| title_sub | foundations, search strategies, and related topics |
| topic | Grafentheorie gtt Graphes, Théorie des ram Optimisation combinatoire ram Combinatorial optimization Paths and cycles (Graph theory) Wegeproblem (DE-588)4392068-8 gnd |
| topic_facet | Grafentheorie Graphes, Théorie des Optimisation combinatoire Combinatorial optimization Paths and cycles (Graph theory) Wegeproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007555527&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008671507 |
| work_keys_str_mv | AT huckenbeckulrich extremalpathsingraphsfoundationssearchstrategiesandrelatedtopics |