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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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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...
Individual Communication Complexity
 In Proc. STACS, LNCS
, 2004
"... We initiate the theory of communication complexity of individual inputs held by the agents, rather than worstcase or averagecase. We consider total, partial, and partially correct protocols, oneway versus twoway, with and without help bits. ..."
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Cited by 7 (5 self)
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We initiate the theory of communication complexity of individual inputs held by the agents, rather than worstcase or averagecase. We consider total, partial, and partially correct protocols, oneway versus twoway, with and without help bits.
P.: The averagecase complexity of Shellsort
 Lecture Notes in Computer Science 1644
, 1999
"... We prove a general lower bound on the averagecase complexity of Shellsort: the average number of datamovements (and comparisons) made by a ppass Shellsort for 1 1+ any incremental sequence is Ω(pn p) for all p ≤ log n. Using similar arguments, we analyze the averagecase complexity of several oth ..."
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Cited by 6 (2 self)
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We prove a general lower bound on the averagecase complexity of Shellsort: the average number of datamovements (and comparisons) made by a ppass Shellsort for 1 1+ any incremental sequence is Ω(pn p) for all p ≤ log n. Using similar arguments, we analyze the averagecase complexity of several other sorting algorithms. 1
The expected size of Heilbronn's triangles
 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
, 1999
"... Heilbronn’s triangle problem asks for the least ∆ such that n points lying in the unit disc necessarily contain a triangle of area at most ∆. Heilbronn initially conjectured ∆ = O(1/n 2). As a result of concerted mathematical effort it is currently known that there are positive constants c and C su ..."
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Cited by 5 (4 self)
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Heilbronn’s triangle problem asks for the least ∆ such that n points lying in the unit disc necessarily contain a triangle of area at most ∆. Heilbronn initially conjectured ∆ = O(1/n 2). As a result of concerted mathematical effort it is currently known that there are positive constants c and C such that c log n/n 2 ≤ ∆ ≤ C/n 8/7−ǫ for every constant ǫ> 0. We resolve Heilbronn’s problem in the expected case: If we uniformly at random put n points in the unit disc then (i) the area of the smallest triangle has expectation Θ(1/n 3); and (ii) the smallest triangle has area Θ(1/n 3) with probability almost one. Our proof uses the incompressibility method based on Kolmogorov complexity. 1
Abstract Author's personal copy
, 2007
"... www.elsevier.com/locate/acha Coordinate and subspace optimization methods for linear least squares with nonquadratic regularization ..."
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Cited by 2 (0 self)
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www.elsevier.com/locate/acha Coordinate and subspace optimization methods for linear least squares with nonquadratic regularization
Analysis of Sorting Algorithms by Kolmogorov Complexity  A Survey
, 2003
"... Recently, many results on the computational complexity of sorting algorithms were obtained using Kolmogorov complexity (the incompressibility method). Especially, the usually hard averagecase analysis is ammenable to this method. Here we survey such results about Bubblesort, Heapsort, Shellsort, ..."
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Recently, many results on the computational complexity of sorting algorithms were obtained using Kolmogorov complexity (the incompressibility method). Especially, the usually hard averagecase analysis is ammenable to this method. Here we survey such results about Bubblesort, Heapsort, Shellsort, Dobosiewiczsort, Shakersort, and sorting with stacks and queues in sequential or parallel mode. Especially in the case of Shellsort the uses of Kolmogorov complexity surprisingly easily resolved problems that had stayed open for a long time despite strenuous attacks.
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,...
Algorithms and Limits for Compact Plan Representations
"... Compact representations of objects is a common concept in computer science. Automated planning can be viewed as a case of this concept: a planning instance is a compact implicit representation of a graph and the problem is to find a path (a plan) in this graph. While the graphs themselves are repres ..."
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Cited by 1 (1 self)
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Compact representations of objects is a common concept in computer science. Automated planning can be viewed as a case of this concept: a planning instance is a compact implicit representation of a graph and the problem is to find a path (a plan) in this graph. While the graphs themselves are represented compactly as planning instances, the paths are usually represented explicitly as sequences of actions. Some cases are known where the plans always have compact representations, for example, using macros. We show that these results do not extend to the general case, by proving a number of bounds for compact representations of plans under various criteria, like efficient sequential or random access of actions. In addition to this, we show that our results have consequences for what can be gained from reformulating planning into some other problem. As a contrast to this we also prove a number of positive results, demonstrating restricted cases where plans do have useful compact representations, as well as proving that macro plans have favourable access properties. Our results are finally discussed in relation to other relevant contexts. 1.
AverageCase Complexity of Shellsort (Preliminary Version)
, 1999
"... Abstract. We prove 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 Ω(pn 1+1/p) for every p. The proof method is an incompressibility argument based on Kolmogorov compl ..."
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Abstract. We prove 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 Ω(pn 1+1/p) for every p. The proof method is an incompressibility argument based on Kolmogorov complexity. Using similar techniques, the averagecase complexity of several other sorting algorithms is analyzed. 1