Publications

Pancake flipping, a famous open problem in computer science, can be formalised as the problem of sorting a permutation of positive integers using as few prefix reversals as possible. In that context, a prefix reversal of length k reverses the order of the first k elements of the permutation. The burnt variant of pancake flipping involves permutations of signed integers, and reversals in that case not only reverse the order of elements but also invert their signs. Although three decades have now passed since the first works on these problems, neither their computational complexity nor the maximal number of prefix reversals needed to sort a permutation is yet known. In this work, we prove a new lower bound for sorting burnt pancakes, and show that an important class of permutations, known as “simple permutations”, can be optimally sorted in polynomial time.

A number of fields, including genome rearrangements and interconnection network design, are concerned with sorting permutations in “as few moves as possible”, using a given set of allowed operations. These often act on just one or two segments of the permutation, e.g. by reversing one segment or exchanging two segments. The cycle graph of the permutation to sort is a fundamental tool in the theory of genome rearrangements. In this paper, we present an algebraic reinterpretation of the cycle graph as an even permutation, and show how to reformulate our sorting problems in terms of particular factorisations of the latter permutation. Using our framework, we recover known results in a simple and unified way, and obtain a new lower bound on the prefix transposition distance (where a prefix transposition displaces the initial segment of a permutation), which is shown to outperform previous results. Moreover, we use our approach to improve the best known lower bound on the prefix transposition diameter from 2n/3 to (3n+1)/(4).

Finding a sequence of transpositions that transforms a given permutation into the identity permutation and is of the shortest possible length is an important problem in bioinformatics. Here, a transposition consists in exchanging two contiguous intervals of the permutation. Bafna and Pevzner introduced the cycle graph as a tool for working on this problem. In particular, they took advantage of the decomposition of the cycle graph into so-called alternating cycles. Later, Hultman raised the question of determining the number of permutations with a cycle graph containing a given quantity of alternating cycles. The resulting number is therefore similar to the Stirling number of the first kind. We provide an explicit formula for computing what we call the Hultman numbers, and give a few numerical values. We also derive formulae for related cases, as well as for a much more general problem. Finally, we indicate a counting result related to another operation on permutations called the “block-interchange”.

The problem of sorting by transpositions asks for a sequence of adjacent interval exchanges that sorts a permutation and is of the shortest possible length. The distance of the permutation is defined as the length of such a sequence. Despite the apparently intuitive nature of this problem, introduced in 1995 by Bafna and Pevzner, the complexity of both finding an optimal sequence and computing the distance remains open today. In this paper, we establish connections between two different graph representations of permutations, which allows us to compute the distance of a few nontrivial classes of permutations in linear time and space, bypassing the use of any graph structure. By showing that every permutation can be obtained from one of these classes, we prove a new tight upper bound on the transposition distance. Finally, we give improved bounds on some other families of permutations and prove formulas for computing the exact distance of other classes of permutations, again in polynomial time.

We study the problem of computing the minimal number of adjacent, non-intersecting block interchanges required to transform a permutation into the identity permutation. In particular, we use the graph of a permutation to compute that number for a particular class of permutations in linear time and space, and derive a new tight upper bound on the so-called transposition distance.