An O(n2) bound for the relaxation time of a Markov chain on cladograms

Authors:
Jason Schweinsberg
Affiliations:
University of California, Berkeley, California
Venue:
Random Structures & Algorithms
Year:
2002

Citing 6
Cited 1

Approximating the permanent

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Mixing Time for a Markov Chain on Cladograms

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A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries

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Abstract

A cladogram is an unrooted tree with labeled leaves and unlabeled internal branchpoints of degree 3. Aldous has studied a Markov chain on the set of n-leaf cladograms in which each transition consists of removing a random leaf and its incident edge from the tree and then reattaching the leaf to a random edge of the remaining tree. Using coupling methods, Aldous showed that the relaxation time (i.e., the inverse of the spectral gap) for this chain is O(n3). Here, we use a method based on distinguished paths to prove an O(n2) bound for the relaxation time, establishing a conjecture of Aldous.