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Abstract. Extending results of Christie and Irving, we examine the action of reversals and transpositions on finite strings over an alphabet of size k. We show that determining reversal, transposition, or signed reversal distance between two strings over a finite alphabet is NP-hard, while for "dense" instances we give a polynomial-time approximation scheme. We also give a number of extremal results, as well as investigating the distance between random strings and the problem of sorting a string over a finite alphabet.Key words. strings, sorting, genome comparison, reversals, transpositions, NP-complete problems, MAX-SNP hardness, approximation algorithms AMS subject classifications. 68R05, 68R15, 68Q17 DOI. 10.1137/S0895480103433550Introduction. As a result of interest in both modelling large-scale genome changes and fundamental questions on the combinatorics of sequences, rearrangement operations, including transpositions, reversals, and signed reversals, have recently been the focus of intense combinatorial, algorithmic, and complexity-theoretic study. These superficially similar sequence operations turn out to have significantly different properties. Most previous work has concentrated on applying sequence operations to permutations. However, the analysis of operations on strings over finite alphabets was raised by Pevzner and Waterman [26] and investigated by Christie and Irving [9]. The study of sequence operations on strings may also be of some practical interest; for a recent example, see, for instance, Skaletsky et al.[27] on the roles played by palindromes and repetitive segments in the Y-chromosome.The operations under consideration all act on strings α = a 1 · · · a n of length |α| = n. The reversal R ij , where i < j, reverses the substring a i · · · a j , so that

The Poincaré and Cheeger bounds are two useful bounds for the second largest eigenvalue of a reversible Markov chain. Diaconis and Stroock and Jerrum and Sinclair develop versions of these bounds which involve choosing paths. This paper studies these path-related bounds and shows that the Poincaré bound is superior to the Cheeger bound for simple random walk on a tree and random walk on a finite group with any symmetric generating set. This partially resolves a question posed by Diaconis and Stroock.

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A k-uniform hypergraph is s-almost intersecting if every edge is disjoint from exactly s other edges. Gerbner, Lemons, Palmer, Patkós and Szécsi conjectured that for every k, and s > s 0 (k), every k-uniform s-almost intersecting hypergraph has at most (s + 1) 2k−2 k−1 edges. We prove a strengthened version of this conjecture and determine the extremal graphs. We also give some related results and conjectures.