Finding Cycles and Trees in Sublinear Time
Abstract
We present sublineartime (randomized) algorithms for finding simple cycles of length at least $k\geq 3$ and treeminors in boundeddegree graphs. The complexity of these algorithms is related to the distance of the graph from being $C_k$minorfree (resp., free from having the corresponding treeminor). In particular, if the graph is far (i.e., $\Omega(1)$far) {from} being cyclefree, i.e. if one has to delete a constant fraction of edges to make it cyclefree, then the algorithm finds a cycle of polylogarithmic length in time $\tildeO(\sqrt{N})$, 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 {\em onesided error} property testing algorithms in the boundeddegree graphs model. For example, we show that cyclefreeness of $N$vertex graphs can be tested with onesided error within time complexity $\tildeO(\poly(1/\e)\cdot\sqrt{N})$. This matches the known $\Omega(\sqrt{N})$ query lower bound, and contrasts with the fact that any minorfree property admits a {\em twosided error} tester of query complexity that only depends on the proximity parameter $\e$. For any constant $k\geq3$, 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 $T$minorfreeness has a onesided error tester of query complexity that only depends on the proximity parameter $\e$. Our algorithm for finding cycles in boundeddegree 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 treeminors in $o(\sqrt{N})$ complexity.
 Publication:

arXiv eprints
 Pub Date:
 July 2010
 arXiv:
 arXiv:1007.4230
 Bibcode:
 2010arXiv1007.4230C
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Discrete Mathematics
 EPrint:
 Keywords: SublinearTime Algorithms, Property Testing, BoundedDegree Graphs, OneSided vs TwoSided Error Probability Updated version