Hal Sudborough

A 2-approximation algorithm for genome rearrangements by reversals and transpositions

Recently, a new approach to analyze genomes evolving which is based on comparision of gene orders versus traditional comparision of DNA sequences was proposed (Sankoff et al. 1992). The approach is based on the global rearrangements (e.g., inversions and transpositions of fragments). Analysis of genomes evolving by inversions and transpositions leads to a combinatorial problem of sorting by reversals and transpositions, i.e., sorting of a permutation using reversals and transpositions of arbitrary fragments. We study sorting of signed permutations by reversals and transpositions, a problem which adequately models genome rearrangements, as the genes in DNA are oriented. We establish a lower bound and give a 2-approximation algorithm for the problem.

Sorting circular permutations by bounded transpositions

A k-bounded (k ≥ 2) transposition is an operation that switches two elements that have at most k - 2 elements in between. We study the problem of sorting a circular permutation π of length n for k = 2, i.e., adjacent swaps and k = 3, i.e., short swaps. These transpositions mimic microrearrangements of gene order in viruses and bacteria. We prove a (1/4)n (2) lower bound for sorting by adjacent swaps. We show upper bounds of (5/32)n (2) + O(n log n) and (7/8)n + O(log n) for sequential and parallel sorting, respectively, by short swaps.

The 4-star graph is not a subgraph of any hypercube

Abstract We prove that it is impossible to precisely embed any star graph S n ( n ⩾4) into a hypercube.

Adjacent Swaps on Strings

Transforming strings by exchanging elements at bounded distance is applicable in fields like molecular biology, pattern recognition and music theory. A reversal of length two at position iis denoted by (i i+1). When it is applied to i¾?, where i¾?= i¾? 1 ,i¾? 2 , i¾? 3 ,..., i¾? i ,i¾? i+ 1 , i¾? n , it transforms i¾?to i¾?i¾?, where i¾?i¾? = i¾? 1 ,i¾? 2 , i¾? 3 ,..., i¾? ii¾? 1 ,i¾? i+ 1 , i¾? i , i¾? i+ 1 , ..., i¾? n . We call this operation an adjacent swap. We study the problem of computing the minimum number of adjacent swaps needed to transform one string of size ninto another compatible string over an alphabet i¾?of size k, i.e.adjacent swap distance problem. O(nlog 2 n) time complexity algorithms are known for adjacent swap distance. We give an algorithm with O(nk) time for both signed and unsigned versions of this problem where kis the number of symbols. We also give an algorithm with O(nk) time for transforming signed strings with reversals of length up to 2, i.e.reversals of length 1 or 2.

Parallel Partition and Extension

We give better lower bounds for M(n, d), for various positive integers d and n with d < n, where M(n, d) is the largest number of permutations on n symbols with pairwise Hamming distance at least d. Larger sets of permutations on n symbols with pairwise Hamming distance d is a necessary component of constructing error correcting permutation codes, which have been proposed for power-line communications. Our technique, Parallel Partition and Extension, is universally applicable to constructing such sets for all n and all d, d < n.

Kronecker product and tiling of permutation arrays for hamming distances

We give improved lower bounds for M(n, d), for various positive integers d and n with d < n, where M(n, d) is the largest number of permutations on n symbols with pairwise Hamming distance at least d. Permutation arrays are used for constructing error correcting permutation codes, which have been proposed for power-line communications. We describe two techniques, which use a modified Kronecker product and a tiling operation, called doubling. Our techniques improve the size of permutation arrays, and improve lower bounds on M(n, d), for infinitely many n and d, d < n.

A quadratic lower bound for Topswops

A quadratic lower bound for the topswops function is exhibited. This provides a non-trivial lower bound for a problem posed by J.H. Conway, D.E. Knuth, M. Gardner and others. We describe an infinite family of permutations, each taking a linear number of steps for the topswops process to terminate, and a chaining process that creates from them an infinite family of permutations taking a quadratic number of steps to reach a fixed point with the identity permutation.

Embedding k-D meshes into optimum hypercubes with dilation 2k−1 (extended abstract)

It is shown that, for every k, and for every k-dimensional mesh M, there is a one-to-one embedding of M into its optimum size hypercube with dilation at most 2k-1.