Finding cycles and trees in sublinear time.

Czumaj, Artur; Goldreich, Oded; Seshadhri, Comandur; Sohler, Christian; Shapira, Asaf; Ron, Dana

Title: Finding cycles and trees in sublinear time.

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Abstract

We present sublinear-time (randomized) algorithms for finding simple cycles of length at least k {ge} 3 and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being C{sub k}-minor-free (resp., free from having the corresponding tree-minor). In particular, if the graph is far (i.e., {Omega}(1)-far) from being cycle-free, i.e. if one has to delete a constant fraction of edges to make it cycle-free, then the algorithm finds a cycle of polylogarithmic length in time {tilde O}({radical}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 {tilde O}(poly(1/{epsilon}) {center_dot} {radical}N). This matches the known {Omega}({radical}N) query lower bound, and contrasts with the fact that any minor-free property admits a two-sided error tester of query complexity that only depends on the proximity parameter {epsilon}. For any constant k {ge} 3, we extend this result to testing whether the input graph has a simple cycle ofmore »« less

Authors:: Czumaj, Artur; Goldreich, Oded; Seshadhri, Comandur; Sohler, Christian; Shapira, Asaf; Ron, Dana

Publication Date:: Fri Oct 01 00:00:00 EDT 2010

Research Org.:: Sandia National Laboratories (SNL), Albuquerque, NM, and Livermore, CA (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1030364

Report Number(s):: SAND2010-7914C
TRN: US201124%%149

DOE Contract Number:: AC04-94AL85000

Resource Type:: Conference

Resource Relation:: Conference: Proposed for presentation at the STOC 2011 Conference held June 6-8, 2011 in San Jose, CA.

Country of Publication:: United States

Language:: English

Subject:: 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ALGORITHMS; TESTING; COMPUTERS

Citation Formats


                    Czumaj, Artur, Goldreich, Oded, Seshadhri, Comandur, Sohler, Christian, Shapira, Asaf, and Ron, Dana. Finding cycles and trees in sublinear time..  United States: N. p., 2010. 
        Web.

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                    Czumaj, Artur, Goldreich, Oded, Seshadhri, Comandur, Sohler, Christian, Shapira, Asaf, & Ron, Dana. Finding cycles and trees in sublinear time..  United States.

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                    Czumaj, Artur, Goldreich, Oded, Seshadhri, Comandur, Sohler, Christian, Shapira, Asaf, and Ron, Dana. 2010.  
        "Finding cycles and trees in sublinear time.".  United States.

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@article{osti_1030364,

  title        = {Finding cycles and trees in sublinear time.},

  author       = {Czumaj, Artur and Goldreich, Oded and Seshadhri, Comandur and Sohler, Christian and Shapira, Asaf and Ron, Dana},

  abstractNote = {We present sublinear-time (randomized) algorithms for finding simple cycles of length at least k {ge} 3 and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being C{sub k}-minor-free (resp., free from having the corresponding tree-minor). In particular, if the graph is far (i.e., {Omega}(1)-far) from being cycle-free, i.e. if one has to delete a constant fraction of edges to make it cycle-free, then the algorithm finds a cycle of polylogarithmic length in time {tilde O}({radical}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 {tilde O}(poly(1/{epsilon}) {center_dot} {radical}N). This matches the known {Omega}({radical}N) query lower bound, and contrasts with the fact that any minor-free property admits a two-sided error tester of query complexity that only depends on the proximity parameter {epsilon}. For any constant k {ge} 3, we extend this result to testing whether the input graph has a simple cycle of length at least k. 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 {epsilon}. 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({radical}N) complexity.},

  doi          = {},

  url          = {https://www.osti.gov/biblio/1030364},
  journal      = {},
number       = ,

  volume       = ,

  place        = {United States},

  year         = {Fri Oct 01 00:00:00 EDT 2010},

  month        = {Fri Oct 01 00:00:00 EDT 2010}

}

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