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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 ..."
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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...
On the Performance of WEAKHEAPSORT
, 2000
"... . Dutton #1993# presents a further HEAPSORT variant called WEAKHEAPSORT, which also contains a new data structure for priority queues. The sorting algorithm and the underlying data structure are analyzed showing that WEAKHEAPSORT is the best HEAPSORT variant and that it has a lot of nice propert ..."
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Cited by 6 (2 self)
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. Dutton #1993# presents a further HEAPSORT variant called WEAKHEAPSORT, which also contains a new data structure for priority queues. The sorting algorithm and the underlying data structure are analyzed showing that WEAKHEAPSORT is the best HEAPSORT variant and that it has a lot of nice properties. It is shown that the worst case number of comparisons is ndlog ne# 2 dlog ne + n #dlog ne#nlog n +0:1nand weak heaps can be generated with n # 1 comparisons. A doubleended priority queue based on weakheaps can be generated in n + dn=2e#2 comparisons. Moreover, examples for the worst and the best case of WEAKHEAPSORT are presented, the number of WeakHeaps on f1;:::;ng is determined, and experiments on the average case are reported. 1
Asymptotic Complexity from Experiments?  A Case Study for Randomized Algorithms
 IN PROCEEDINGS OF THE 4TH WORKSHOP OF ALGORITHMS AND ENGINEERING (WAE'00
, 2000
"... In the analysis of algorithms we are usually interested in obtaining closed form expressions for their complexity, or at least asymptotic expressions in O()notation. Unfortunately, there are fundamental reasons why we cannot obtain such expressions from experiments. This paper explains how we can ..."
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Cited by 4 (2 self)
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In the analysis of algorithms we are usually interested in obtaining closed form expressions for their complexity, or at least asymptotic expressions in O()notation. Unfortunately, there are fundamental reasons why we cannot obtain such expressions from experiments. This paper explains how we can at least come close to this goal using the scientific method. Besides the traditional role of experiments as a source of preliminary ideas for theoretical analysis, experiments can test falsifiable hypotheses obtained by incomplete theoretical analysis. Asymptotic behavior can also be deduced from stronger hypotheses which have been induced from experiments. As long as a complete mathematical analysis is impossible, well tested hypotheses may have to take their place. Several
Kolmogorov Complexity and a Triangle Problem of the Heilbronn Type
"... From among \Gamma n 3 \Delta triangles with vertices chosen from among n points in the unit square, U , let T be the one with the smallest area, and let A be the area of T . If the n points are chosen independently and at random (uniform distribution) then there exist positive c and C such tha ..."
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Cited by 1 (0 self)
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From among \Gamma n 3 \Delta triangles with vertices chosen from among n points in the unit square, U , let T be the one with the smallest area, and let A be the area of T . If the n points are chosen independently and at random (uniform distribution) then there exist positive c and C such that c=n 3 ! n ! C=n 3 for all large enough n, where n is the expectation of A. Moreover, with probability close to one c=n 3 ! A ! C=n 3 . Our proof uses the incompressibility method based on Kolmogorov complexity. The related Heilbronn problem asks for the maximum value assumed by A over all choices of n points. 1 Introduction From among \Gamma n 3 \Delta triangles with vertices chosen from among n points in the unit circle, let T be the one of least area, and let A be the area of T . Let \Delta n be the maximum assumed by A over all choices of n points. H.A. Heilbronn (19081975) 1 asked for the exact value or approximation of \Delta n . The list [1, 2, 3, 5, 8, 9, 10, 11,...
5. Using Finite Experiments to Study Asymptotic Performance
"... In the analysis of algorithms we are interested in obtaining closed form expressions for algorithmic complexity, or at least asymptotic expressions in O(·)notation. It is often possible to use experimental results to make significant progress towards this goal, although there are fundamental reason ..."
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In the analysis of algorithms we are interested in obtaining closed form expressions for algorithmic complexity, or at least asymptotic expressions in O(·)notation. It is often possible to use experimental results to make significant progress towards this goal, although there are fundamental reasons whywe cannot guarantee to obtain such expressions from experiments alone. This paper investigates two approaches relating to problems of developing theoretical analyses based on experimental data. We first consider the scientific method, which views experimentation as part of a cycle alternating with theoretical analysis. This approach has been verysuccessful in the natural sciences. Besides supplying preliminary ideas for theoretical analysis, experiments can test falsifiable hypotheses obtained by incomplete theoretical analysis. Asymptotic behavior can also sometimes be deduced from stronger hypotheses which have been induced from experiments. As long as complete mathematical analyses remains elusive, well tested hypotheses may have to take their place. Several examples