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Analysis of Shellsort and related algorithms
 ESA ’96: Fourth Annual European Symposium on Algorithms
, 1996
"... This is an abstract of a survey talk on the theoretical and empirical studies that have been done over the past four decades on the Shellsort algorithm and its variants. The discussion includes: upper bounds, including linkages to numbertheoretic properties of the algorithm; lower bounds on Shellso ..."
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This is an abstract of a survey talk on the theoretical and empirical studies that have been done over the past four decades on the Shellsort algorithm and its variants. The discussion includes: upper bounds, including linkages to numbertheoretic properties of the algorithm; lower bounds on Shellsort and Shellsortbased networks; averagecase results; proposed probabilistic sorting networks based on the algorithm; and a list of open problems. 1 Shellsort The basic Shellsort algorithm is among the earliest sorting methods to be discovered (by D. L. Shell in 1959 [36]) and is among the easiest to implement, as exhibited by the following C code for sorting an array a[l],..., a[r]: shellsort(itemType a[], int l, int r) { int i, j, h; itemType v;
The Coin Exchange Problem for Arithmetic Progressions
 American Mathematical Monthly
, 1994
"... Let a1; a2; : : : ; ak be relatively prime, positive integers arranged in increasing order. Let ¡? denote the positive integers in the set f a1x1 + a2x2 + ¢ ¢ ¢+ akxk: xj ¸ 0 g. Let S?(a1; a2; : : : ; ak): = fn =2 ¡? : n+ ¡? µ ¡? g: We determine S?(a1; a2; : : : ; ak) in the case where the aj' ..."
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Let a1; a2; : : : ; ak be relatively prime, positive integers arranged in increasing order. Let ¡? denote the positive integers in the set f a1x1 + a2x2 + ¢ ¢ ¢+ akxk: xj ¸ 0 g. Let S?(a1; a2; : : : ; ak): = fn =2 ¡? : n+ ¡? µ ¡? g: We determine S?(a1; a2; : : : ; ak) in the case where the aj's are in arithmetic progression. In particular, this determines g(a1; a2; : : : ; ak) in this particular case. 1.
The Frobenius Problem in a Free Monoid
, 2008
"... Given positive integers c1, c2,..., ck with gcd(c1, c2,..., ck) = 1, the Frobenius problem (FP) is to compute the largest integer g(c1, c2,..., ck) that cannot be written as a nonnegative integer linear combination of c1, c2,..., ck. The Frobenius problem in a free monoid (FPFM) is a noncommu ..."
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Given positive integers c1, c2,..., ck with gcd(c1, c2,..., ck) = 1, the Frobenius problem (FP) is to compute the largest integer g(c1, c2,..., ck) that cannot be written as a nonnegative integer linear combination of c1, c2,..., ck. The Frobenius problem in a free monoid (FPFM) is a noncommutative generalization of the Frobenius problem. Given words x1, x2,..., xk such that there are only finitely many words that cannot be written as concatenations of words in {x1, x2,..., xk}, the FPFM is to find the longest such words. Unlike the FP, where the upper bound g(c1, c2,..., ck) ≤ max1≤i≤k c2i is quadratic, the upper bound on the length of the longest words in the FPFM can be exponential in certain measures and some of the exponential upper bounds are tight. For the 2FPFM, where the given words