Abstract
We consider the problem of sampling simple paths between two given vertices in a planar graph and propose a natural Markov chain exploring such paths by means of “local” modifications. This chain can be tuned so that the probability of sampling a path depends on its length (for instance, output shorter paths with higher probability than longer ones). We show that this chain is always ergodic and thus it converges to the desired sampling distribution for any planar graph. While this chain is not rapidly mixing in general, we prove that a simple restricted variant is. The restricted chain samples paths on a 2D lattice which are monotone in the vertical direction. To the best of our knowledge, this is the first example of a rapidly mixing Markov chain for sampling simple paths with a probability that depends on their lengths.
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Notes
- 1.
The analysis of non-reversible Markov chains is in general rather difficult and it is considered an interesting problem also for simple chains [4].
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Acknowledgments
We are grateful to Francesco Pasquale for comments on an earlier version of this work. We also wish to thank the anonymous referees for pointing out relations to the class of outerplanar graphs, and the use of dynamic programming for the vertical-monotone paths. Part of this work has been done while the second author was at ETH Zurich. This work is supported by the EU FP7/2007-2013 (DG CONNECT.H5-Smart Cities and Sustainability), under grant agreement no. 288094 (project eCOMPASS), and by the French ANR Project DISPLEXITY.
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Montanari, S., Penna, P. (2015). On Sampling Simple Paths in Planar Graphs According to Their Lengths. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_41
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DOI: https://doi.org/10.1007/978-3-662-48054-0_41
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