Ordering Metro Lines by Block Crossings

A problem that arises in drawings of transportation networks is to minimize the number of crossings between different transportation lines. While this can be done efficiently under specific constraints, not all solutions are visually equivalent. We suggest merging crossings into block crossings, that is, crossings of two neighboring groups of consecutive lines. Unfortunately, minimizing the total number of block crossings is NP-hard even for very simple graphs. We give approximation algorithms for special classes of graphs and an asymptotically worst-case optimal algorithm for block crossings on general graphs. That is, we bound the number of block crossings that our algorithm needs and construct worst-case instances on which the number of block crossings that is necessary in any solution is asymptotically the same as our bound

Mixing times of lozenge tiling and card shuffling Markov chains

We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall, and Sinclair to generate random tilings of regions by lozenges. For an L X L region we bound the mixing time by O(L^4 log L), which improves on the previous bound of O(L^7), and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and Saloff-Coste, by lower bounding the mixing time of various card-shuffling Markov chains. Our lower bounds are within a constant factor of their upper bounds. When we use our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an O(n^3 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markov chain for linear extensions.Comment: 39 pages, 8 figure

Task swapping networks in distributed systems

In this paper we propose task swapping networks for task reassignments by using task swappings in distributed systems. Some classes of task reassignments are achieved by using iterative local task swappings between software agents in distributed systems. We use group-theoretic methods to find a minimum-length sequence of adjacent task swappings needed from a source task assignment to a target task assignment in a task swapping network of several well-known topologies.Comment: This is a preprint of a paper whose final and definite form is published in: Int. J. Comput. Math. 90 (2013), 2221-2243 (DOI: 10.1080/00207160.2013.772985

Lower bounding edit distances between permutations

International audienceA number of fields, including the study of genome rearrangements and the design of interconnection networks, deal with the connected problems of sorting permutations in "as few moves as possible", using a given set of allowed operations, or computing the number of moves the sorting process requires, often referred to as the distance of the permutation. These operations 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, and has proved useful in settling the complexity of many variants of the above problems. In this paper, we present an algebraic reinterpretation of the cycle graph of a permutation π as an even permutation π, 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/4⌋, and investigate a few relations between some statistics on π and π