John Paul C. Vergara

PhD Dissertation submitted to the Faculty of the Virginia Tech in partial fulfillment of the requirements for the degree of

Ph.D.

in

Computer Science

Approved

Lenwood S. Heath, Chair

Donald C.S. Allison

Ezra A. Brown

Edward L. Green

Clifford A. Shaffer

April 8, 1997

Blacksburg, Virginia

Let

Pbe a predicate applicable to permutations. A permutation that satisfiesPis called agenerator. Given a permutation $\pi$,MinSort_Pis the problem of finding a shortest sequence of generators that, when composed with $\pi$, yields the identity permutation. The length of this sequence is called thePdistance of $\pi$.Diam_Pis the problem of finding the longest such distance for permutations of a given length.MinSort_PandDiam_P, for some choices ofP, have applications in the study of genome rearrangements and in the design of interconnection networks.This dissertation considers generators that are swaps, reversals, or block-moves. Distance bounds on these generators are introduced and the corresponding problems are investigated. Reduction results, graph-theoretic models, exact and approximation algorithms, and heuristics for these problems are presented. Experimental results on the heuristics are also provided.

When the bound is a function of the length of the permutation, there are several sorting problems such as sorting by block-moves and sorting by reversals whose bounded variants are at least as difficult as the corresponding unbounded problems. For some bounded problems, a strong relationship exists between finding optimal sorting sequences and correcting the relative order of individual pairs of elements. This fact is used in investigating

MinSort_PandDiam_Pfor two particular predicates.A short block-move is a generator that moves an element at most two positions away from its original position. Sorting by short block-moves is solvable in polynomial time for two large classes of permutations: woven bitonic permutations and woven double-strip permutations. For general permutations, a polynomial-time (4/3)-approximation algorithm that computes short block-move distance is devised. The short block-move diameter for length-

npermutations is determined.A short swap is a generator that swaps two elements that have at most one element between them. A polynomial-time 2-approximation algorithm for computing short swap distance is devised and a class of permutations where the algorithm computes the exact short swap distance is determined. Bounds for the short swap diameter for length-

npermutations are determined.

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