Selecting the Kth largest-area convex polygon
- 675 Downloads
The purpose of this note is to solve an open problem submitted by B. Chazelle : Given a set P of n points in the Euclidean plane, select the k th largest-area convex polygon determined by subsets of P. We show that the decision problem is NP-hard by a reduction from the problem of finding the k th largest m-tuple  determined by m sets X1,X2,...,X m . We also show that the enumeration problem is #P-complete and exhibit a pseudo-polynomial-time algorithm for the decision problem.
KeywordsConvex Hull Convex Polygon Euclidean Plane Good Subset Enumeration Problem
Unable to display preview. Download preview PDF.
- 1.B. M. Chazelle, L. J. Guibas and D. T. Lee, The Power of Geometric Duality, Proc. of the 24th Annual Symp. on Foundations of Computer Science, 1983, pp. 217–225.Google Scholar
- 2.B. Chazelle, New Techniques for Computing Order Statistics in Euclidean Space, Proc. of the 1st Annual Symposium on Computational Geometry, 1985, pp. 125–134.Google Scholar
- 3.R. Cole, J. S. Salowe, W. L. Steiger and E. Szemeredi, Optimal Slope Selection, Proc. of the 15th International Colloquium on Automata, Languages and Programming, 1988, pp. 133–146.Google Scholar
- 7.Z. Galil and N. Megiddo, A Fast Selection Algorithm and the Problem of Optimum Distribution of Effort, J. ACM, 26, 1979, pp. 58–64.Google Scholar
- 8.M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Company, San Francisco, CA, 1979.Google Scholar
- 9.D. B. Johnson and T. Mizoguchi, Selecting the k th Element in X+Y and X 1+X 2+...+X m, Siam J. on Computing, 7, 1978, pp. 147–153.Google Scholar
- 10.N. Megiddo, A. Tamir, E. Zemel and R. Chandrasekaran, An O (n log2 n) Algorithm for the k th Longest Path in a Tree with Applications to Location Problems, Siam J. on Computing, 10 (2), 1981, pp. 328–337.Google Scholar
- 12.F. P. Preparata and M. I. Shamos, Computational Geometry: An Introduction, Springer Verlag, New York, NY, 1985.Google Scholar