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A Lower Bound on the AverageCase Complexity of Shellsort
, 1999
"... We give a general lower bound on the averagecase complexity of Shellsort: the average number of datamovements (and comparisons) made by a ppass Shellsort for any incremental sequence is \Omega\Gamma pn 1+1=p ) for every p. The proof is an example of the use of Kolmogorov complexity (the incompr ..."
Abstract

Cited by 10 (6 self)
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We give a general lower bound on the averagecase complexity of Shellsort: the average number of datamovements (and comparisons) made by a ppass Shellsort for any incremental sequence is \Omega\Gamma pn 1+1=p ) for every p. The proof is an example of the use of Kolmogorov complexity (the incompressibility method) in the analysis of algorithms. 1 Introduction The question of a nontrivial general lower bound (or upper bound) on the average complexity of Shellsort (due to D.L. Shell [14]) has been open for about four decades [5, 13]. We present such a lower bound for ppass Shellsort for every p. Shellsort sorts a list of n elements in p passes using a sequence of increments h 1 ; : : : ; h p . In the kth pass the main list is divided in h k separate sublists of length dn=h k e, where the ith sublist consists of the elements at positions j, where j mod h k = i \Gamma 1, of the main list (i = 1; : : : ; h k ). Every sublist is sorted using a straightforward insertion sort. The effi...
The averagecase area of Heilbronntype triangles
 RANDOM STRUCTURES AND ALGORITHMS
, 2002
"... From among � � n triangles with vertices chosen from n points in the unit square, 3 let T be the one with the smallest area, and let A be the area of T. Heilbronn’s triangle problem asks for the maximum value assumed by A over all choices of n points. We consider the averagecase: If the n points ..."
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Cited by 6 (2 self)
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From among � � n triangles with vertices chosen from n points in the unit square, 3 let T be the one with the smallest area, and let A be the area of T. Heilbronn’s triangle problem asks for the maximum value assumed by A over all choices of n points. We consider the averagecase: If the n points are chosen independently and at random (with a uniform distribution), then there exist positive constants c and C such that c/n3 <µ n < C/n3 for all large enough values of n, where µ n is the expectation of A. Moreover, c/n3 <A<C/n3, with probability close to one. Our proof uses the incompressibility method based on Kolmogorov complexity; it actually determines the area of the smallest triangle for an arrangement in