Extremal problems in combinatorial geometry
Handbook of combinatorics (vol. 1)
Two heads are better than two tapes
Journal of the ACM (JACM)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
A lower bound on the average-case complexity of shellsort
Journal of the ACM (JACM)
An Algorithm for Heilbronn's Problem
SIAM Journal on Computing
A Lower Bound for Heilbronn's Triangle Problem in d Dimensions
SIAM Journal on Discrete Mathematics
The Expected Size of Heilbronn's Triangles
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
Distributions of points in the unit-square and large k-gons
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Large triangles in the d-dimensional unit cube
Theoretical Computer Science - Computing and combinatorics
Distributions of points in the unit square and large k-gons
European Journal of Combinatorics
Point sets in the unit square and large areas of convex hulls of subsets of points
COCOA'07 Proceedings of the 1st international conference on Combinatorial optimization and applications
Distributions of points and large convex hulls of k points
AAIM'06 Proceedings of the Second international conference on Algorithmic Aspects in Information and Management
Distributions of points in d dimensions and large k-point simplices
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
The on-line heilbronn’s triangle problem in d dimensions
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
Distributions of points and large quadrangles
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
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From among ??? triangles with vertices chosen from n points in the unit square, 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 (with a uniform distribution), then there exist positive constants c and C such that c/n3 µn C/n3 for all large enough values of n, where µn 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 "general position."