On Sorting by 3-Bounded Transpositions

doi:10.1016/S1571-0653(04)00552-9

Electronic Notes in Discrete Mathematics

Volume 15, May 2003, Pages 115-118

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Abstract

Heath and Vergara (2) proved the equivalence between sorting by 3-bounded transpositions and sorting by correcting skips and correcting hops. This paper explores various algorithmic as well as combinatorial aspects of correcting skips/hops, with the aim of understanding 3-bounded transpositions better.

We obtain a complete characterization of the set G_n ⊆ S_n of all correcting-hop-free permutations. We describe an efficient algorithm to optimally sort a permutation in G_n using skips and hops. We study the class H_n of those permutations of S_n which can be sorted using correcting hops alone. We exhibit an interesting relation between G_n and H_n. The notion of correcting skips/hops is extended to correcting moves and it is shown that one can efficiently sort a permutation with a minimum number of correcting moves.

References (4)

V. Bafna et al.
Sorting by transpositions
SIAM J. Discrete Mathematics
(1998)

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Citation Excerpt :
The pairwise interchange cost is a standard measure of the distance between sequences. It is used, for example, in genome sequencing [16]. This situation is studied from the point of view of the manufacturer, and then from that of the supplier.
A critical issue in supply chain management is coordinating the decisions made by decision makers at different stages, for example a supplier and one or several manufacturers. We model this issue by assuming that both the supplier and each manufacturer have an ideal schedule, determined by their own costs and constraints. An interchange cost is incurred by the supplier or a manufacturer whenever the relative order of two jobs in its actual schedule is different from that in its ideal schedule. An intermediate storage buffer is available to resequence the jobs between the two stages. We consider the problems of finding an optimal supplier's schedule, an optimal manufacturer's schedule, and optimal schedules for both. The objective functions we consider are the minimization of total interchange cost, and of total interchange plus buffer storage cost. We describe efficient algorithms for all the supplier's and manufacturers’ problems, as well as for a special case of the joint scheduling problem. The running time of these algorithms is polynomial in both the number of jobs and the number of manufacturers. Finally, we identify conditions under which cooperation between the supplier and a manufacturer reduces their total cost.
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On Sorting by 3-Bounded Transpositions

Abstract

Sorting by transpositions

SIAM J. Discrete Mathematics