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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.
n points independently at random in 2, according to a prescribed n ≤... be the areas of the triangles
"... thus formed, in nondecreasing order. If µ is absolutely continuous with respect to Lebesgue measure, then, under weak conditions, the set {n3∆n i: i ≥ 1} converges as n → ∞ to a Poisson process with a constant intensity κ(µ). This result, and related conclusions, are proved using standard argument ..."
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thus formed, in nondecreasing order. If µ is absolutely continuous with respect to Lebesgue measure, then, under weak conditions, the set {n3∆n i: i ≥ 1} converges as n → ∞ to a Poisson process with a constant intensity κ(µ). This result, and related conclusions, are proved using standard arguments of Poisson approximation, and may be extended to functionals more general than the area of a triangle. It is proved in addition that, if µ is the uniform probability measure on the region S, then κ(µ) ≤ 2/S, where S  denotes the area of S. Equality holds in that κ(µ) = 2/S  if S is convex, and essentially only then. This work generalizes and extends considerably the conclusions of a recent paper of Jiang, Li, and Vitányi. 1. The problem Drop n points independently and uniformly at random into the unit square [0, 1] 2 of R2. Of the ()
method
"... Background: Although a variety of methods and expensive kits are available, molecular cloning can be a timeconsuming and frustrating process. Results: Here we report a highly simplified, reliable, and efficient PCRbased cloning technique to insert any DNA fragment into a plasmid vector or into a ge ..."
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Background: Although a variety of methods and expensive kits are available, molecular cloning can be a timeconsuming and frustrating process. Results: Here we report a highly simplified, reliable, and efficient PCRbased cloning technique to insert any DNA fragment into a plasmid vector or into a gene (cDNA) in a vector at any desired position. With this method, the vector and insert are PCR amplified separately, with only 18 cycles, using a high fidelity DNA polymerase. The amplified insert has the ends with ~16base overlapping with the ends of the amplified vector. After DpnI digestion of the mixture of the amplified vector and insert to eliminate the DNA templates used in PCR reactions, the mixture is directly transformed into competent E. coli cells to obtain the desired clones. This technique has many advantages over other cloning methods. First, it does not need gel purification of the PCR product or linearized vector. Second, there is no need of any cloning kit or specialized enzyme for cloning. Furthermore, with reduced number of PCR cycles, it also decreases the chance of random mutations. In addition, this method is highly effective and reproducible. Finally, since this cloning method is also sequence independent, we demonstrated that it can be used for chimera construction, insertion, and multiple mutations spanning a stretch of DNA up to 120 bp.
New lower bounds for Heilbronn numbers
, 2002
"... The nth Heilbronn number, H_n, is the largest value such that n points can be placed in the unit square in such a way that all possible triangles defined by any three of the points have area at least H_n. In this note we establish new bounds for the first Heilbronn numbers. These new values have be ..."
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The nth Heilbronn number, H_n, is the largest value such that n points can be placed in the unit square in such a way that all possible triangles defined by any three of the points have area at least H_n. In this note we establish new bounds for the first Heilbronn numbers. These new values have been found by using a simple implementation of simulated annealing to obtain a first approximation and then optimizing the results by finding the nearest exact local maximum.
ON SMALLEST TRIANGLES
"... Abstract. Pick n points independently at random in R2, according to a prescribed probability measure µ, and let ∆n 1 ≤ ∆n 2 ≤... be the areas of the �n � triangles 3 thus formed, in nondecreasing order. If µ is absolutely continuous with respect to Lebesgue measure, then, under weak conditions, the ..."
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Abstract. Pick n points independently at random in R2, according to a prescribed probability measure µ, and let ∆n 1 ≤ ∆n 2 ≤... be the areas of the �n � triangles 3 thus formed, in nondecreasing order. If µ is absolutely continuous with respect to Lebesgue measure, then, under weak conditions, the set {n3∆n i: i ≥ 1} converges as n → ∞ to a Poisson process with a constant intensity κ(µ). This result, and related conclusions, are proved using standard arguments of Poisson approximation, and may be extended to functionals more general than the area of a triangle. It is proved in addition that, if µ is the uniform probability measure on the region S, then κ(µ) ≤ 2/S, where S  denotes the area of S. Equality holds in that κ(µ) = 2/S  if S is convex, and essentially only then. This work generalizes and extends considerably the conclusions of a recent paper of Jiang, Li, and Vitányi. 1. The problem Drop n points independently and uniformly at random into the unit square [0, 1] 2 of R2. Of the � � n
BOLYAI SOCIETY Entropy, Search, Complexity, pp. 209–232. MATHEMATICAL STUDIES, 16 Analysis of Sorting Algorithms by Kolmogorov Complexity
"... 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, Do ..."
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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. 1.