Improved upper bounds on Shellsort
Journal of Computer and System Sciences
Extremal problems in combinatorial geometry
Handbook of combinatorics (vol. 1)
Shellsort with three increments
Random Structures & Algorithms - Special issue: average-case analysis of algorithms
An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
The art of computer programming, volume 3: (2nd ed.) sorting and searching
The art of computer programming, volume 3: (2nd ed.) sorting and searching
A lower bound for Hellbronn's triangle problem in d dimensions
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
New applications of the incompressibility method: part II
Theoretical Computer Science - Selected papers in honor of Manuel Blum
A high-speed sorting procedure
Communications of the ACM
Quantum computation and quantum information
Quantum computation and quantum information
Average-Case Complexity of Shellsort
ICAL '99 Proceedings of the 26th International Colloquium on Automata, Languages and Programming
New Applications of the Incompressibility Method
ICAL '99 Proceedings of the 26th International Colloquium on Automata, Languages and Programming
An Algorithm for Heilbronn's Problem
COCOON '97 Proceedings of the Third Annual International Conference on Computing and Combinatorics
Analysis of Shellsort and Related Algorithms
ESA '96 Proceedings of the Fourth Annual European Symposium on Algorithms
The Expected Size of Heilbronn's Triangles
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
Three Approaches to the Quantitative Definition of Information in an Individual Pure Quantum State
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Shellsort and sorting networks
Shellsort and sorting networks
Improved lower bounds for Shellsort
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
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Kolmogorov complexity is a modern notion of randomness dealing with the quantity of information in individual objects; that is, pointwise randomness rather than average randomness as produced by a random source. It was proposed by A.N. Kolmogorov in 1965 to quantify the randomness of individual objects in an objective and absolute manner. This is impossible for classical probability theory. Kolmogorov complexity is known variously as 'algorithmic information', 'algorithmic entropy', 'Kolmogorov-Chaitin complexity', 'descriptional complexity', 'shortest program length', 'algorithmic randomness', and others. Using it, we developed a new mathematical proof technique, now known as the 'incompressibility method'. The incompressibility method is a basic general technique such as the 'pigeon hole' argument, 'the counting method' or the 'probabilistic method'. The new method has been quite successful and we present recent examples. The first example concerns a "static" problem in combinatorial geometry. From among (n 3)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. Heilbronn's triangle problem asks for the maximum value assumed by A over all choices of n points. We consider the average-case: If the n points are chosen independently and at random (uniform distribution) then there exist positive c and C such that c/n3 n C/n3 for all large enough n, where µn is the expectation of A. Moreover, c/n3 A C/n3 for almost all A, that is, almost all A are close to the expectation value so that we determine the area of the smallest triangle for an arrangement in "general position". Our second example concerns a "dynamic" problem in average-case running time of algorithms. The question of a nontrivial general lower bound (or upper bound) on the average-case complexity of Shellsort has been open for about forty years. We obtain the first such lower bound.