8,203 research outputs found
Analysis of Algorithms for Permutations Biased by Their Number of Records
The topic of the article is the parametric study of the complexity of
algorithms on arrays of pairwise distinct integers. We introduce a model that
takes into account the non-uniformness of data, which we call the Ewens-like
distribution of parameter for records on permutations: the weight
of a permutation depends on its number of records. We show that
this model is meaningful for the notion of presortedness, while still being
mathematically tractable. Our results describe the expected value of several
classical permutation statistics in this model, and give the expected running
time of three algorithms: the Insertion Sort, and two variants of the Min-Max
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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
Group-theoretic models of the inversion process in bacterial genomes
The variation in genome arrangements among bacterial taxa is largely due to
the process of inversion. Recent studies indicate that not all inversions are
equally probable, suggesting, for instance, that shorter inversions are more
frequent than longer, and those that move the terminus of replication are less
probable than those that do not. Current methods for establishing the inversion
distance between two bacterial genomes are unable to incorporate such
information. In this paper we suggest a group-theoretic framework that in
principle can take these constraints into account. In particular, we show that
by lifting the problem from circular permutations to the affine symmetric
group, the inversion distance can be found in polynomial time for a model in
which inversions are restricted to acting on two regions. This requires the
proof of new results in group theory, and suggests a vein of new combinatorial
problems concerning permutation groups on which group theorists will be needed
to collaborate with biologists. We apply the new method to inferring distances
and phylogenies for published Yersinia pestis data.Comment: 19 pages, 7 figures, in Press, Journal of Mathematical Biolog
Stable Matching with Evolving Preferences
We consider the problem of stable matching with dynamic preference lists. At
each time step, the preference list of some player may change by swapping
random adjacent members. The goal of a central agency (algorithm) is to
maintain an approximately stable matching (in terms of number of blocking
pairs) at all times. The changes in the preference lists are not reported to
the algorithm, but must instead be probed explicitly by the algorithm. We
design an algorithm that in expectation and with high probability maintains a
matching that has at most blocking pairs.Comment: 13 page
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