Finding cycles and trees in sublinear time
ABSTRACT We present sublinear‐time (randomized) algorithms for finding simple cycles of length at least k≥3 and tree‐minors in bounded‐degree graphs. The complexity of these algorithms is related to the distance of the graph from being Ck‐minor free (resp., free from having the corresponding tree‐mi...
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| Veröffentlicht in: | Random structures & algorithms Jg. 45; H. 2; S. 139 - 184 |
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| Hauptverfasser: | , , , , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Hoboken
Blackwell Publishing Ltd
01.09.2014
Wiley Subscription Services, Inc |
| Schlagworte: | |
| ISSN: | 1042-9832, 1098-2418 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | ABSTRACT
We present sublinear‐time (randomized) algorithms for finding simple cycles of length at least k≥3 and tree‐minors in bounded‐degree graphs. The complexity of these algorithms is related to the distance of the graph from being Ck‐minor free (resp., free from having the corresponding tree‐minor). In particular, if the graph is Ω(1)‐far from being cycle‐free (i.e., a constant fraction of the edges must be deleted to make the graph cycle‐free), then the algorithm finds a cycle of polylogarithmic length in time O˜(N), where N denotes the number of vertices. This time complexity is optimal up to polylogarithmic factors.
The foregoing results are the outcome of our study of the complexity of one‐sided error property testing algorithms in the bounded‐degree graphs model. For example, we show that cycle‐freeness of N‐vertex graphs can be tested with one‐sided error within time complexity O˜(poly(1/ϵ)·N), where ∊ denotes the proximity parameter. This matches the known Ω(N) query lower bound for one‐sided error cycle‐freeness testing, and contrasts with the fact that any minor‐free property admits a two‐sided error tester of query complexity that only depends on ∊. We show that the same upper bound holds for testing whether the input graph has a simple cycle of length at least k, for any k≥3. On the other hand, for any fixed tree T, we show that T‐minor freeness has a one‐sided error tester of query complexity that only depends on the proximity parameter ∊.
Our algorithm for finding cycles in bounded‐degree graphs extends to general graphs, where distances are measured with respect to the actual number of edges. Such an extension is not possible with respect to finding tree‐minors in o(N) complexity. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 139–184, 2014 |
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| Bibliographie: | istex:5A7C1DA196D7CDC1806E3691E880E06429F5BB19 ArticleID:RSA20462 ark:/67375/WNG-1GQBRDQ4-8 Supported by the Israel Science Foundation (grant No.1041/08). Employee of Sandia National Laboratories. Sandia National Laboratories is a multi‐program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energys National Nuclear Security Administration under contract DE‐AC04‐94AL85000. Supported in part by NSF Grant DMS‐0901355, ISF Grant 224/11, and a Marie‐Curie CIG Grant 303320. Supported by DFG grant So 514/3‐2. Supported by the Israel Science Foundation (grant No. 246/08). Research partially supported by EPSRC awards EP/G064679/1and EP/D063191/1. ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 ObjectType-Article-2 ObjectType-Feature-1 content type line 23 |
| ISSN: | 1042-9832 1098-2418 |
| DOI: | 10.1002/rsa.20462 |