The Average-Case Area of Heilbronn-Type Triangles

Random Structures and Algorithms

Volumn 20, Issue 2, 2002, Pages 206-219

  Jiang, Tao ^a   Li, Ming ^b   Vitányi, Paul ^c  

^a University of California Riverside   (United States)

^b University of California   (United States)

^c CWI   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036114716     PISSN: 10429832     EISSN: None     Source Type: Journal    
DOI: 10.1002/rsa.10024     Document Type: Article

References (27)

1. G. Barequet, A lower bound for Heilbronn's triangle problem in d dimensions, SIAM J Discrete Math 14(2) (2001), 230-236.
2. J. Beck, Almost collinear triples among N points on the plane, in A Tribute to Paul Erdó;s, A. Baker, B. Bollobas and A. Hajnal (Editors), Cambridge University Press, Cambridge, 1990, pp. 39-57.
3. C. Bertram-Kretzberg, T. Hofmeister, and H. Lefmann (Editors), An algorithm for Heilbronn's problem, SIAM J Comput 30(2) (2000), 383-390.
4. G. Cairns, M. McIntyre, and J. Strantzen, Geometric proofs of some recent results of Yang Lu, Math Magazine 66 (1993), 263-265.
5. P. Erdó;s, "Problems and results in combinatorial geometry," Discrete geometry and convexity, Ann New York Acad Sci 440 (1985), 1-11.
6. P. Erdó;s and G. Purdy, "Extremal problems in combinatorial theory," Handbook of combinatorics, R.L. Graham, M. Grötschel, L. Lovász, (Editors), Elsevier/MIT Press, 1995, pp. 861-862.
7. M. Goldberg, Maximizing the smallest triangle made by N points in a square, Math Magazine 45 (1972), 135-144.
8. R.K. Guy, Unsolved problems in number theory, 2nd ed., Springer-Verlag, Berlin, 1994, pp. 242-244.
9. T. Jiang, J.I. Seiferas, and P.M.B. Vitányi, Two heads are better than two tapes, J Assoc Comput Mach 44(2) (1997), 237-256.
10. T. Jiang, M. Li, and P.M.B. Vitányi, The expected size of Heilbronn's triangles, Proc 14th IEEE Conf Comput Complexity, IEEE Comp Soc, Los Alamitos, CA, 1999, pp. 105-113.
11. T. Jiang, M. Li, and P.M.B. Vitányi, A lower bound on the average-case complexity of Shellsort, J Assoc Comput Mach 47(5) (2000), 905-911.
12. A.N. Kolmogorov, Three approaches to the quantitative definition of information, Problems Inform Transmission 1(1) (1965), 1-7.
13. J. Komlós, J. Pintz, and E. Szemerédi, On Heilbronn's triangle problem, J London Math Soc 24(2) (1981), 385-396.
14. J. Komlós, J. Pintz, and E. Szemerédi, A lower bound for Heilbronn's problem, J London Math Soc 25 (1982), 13-24.
15. M. Li and P.M.B. Vitányi, An introduction to Kolmogorov complexity and its applications, 2nd ed., Springer-Verlag, New York, 1997.
16. D. Mackenzie, On a roll, New Scientist, 6 November (1999), 44-48.
17. A.M. Odlyzko, J. Pintz, and K.B. Stolarsky, Partitions of planar sets into small triangles, Discrete Math 57 (1985), 89-97.
18. K.F. Roth, On a problem of Heilbronn, J London Math Soc 26 (1951), 198-204.
19. K.F. Roth, On a problem of Heilbronn II, Proc. London Math Soc 25(3) (1972), 193-212.
20. K.F. Roth, On a problem of Heilbronn III, Proc. London Math Soc 25(3) (1972), 543-549.
21. K.F. Roth, Estimation of the area of the smallest triangle obtained by selecting three out of n points in a disc of unit area, Proc. Symp. Pure Mathematics, Vol. 24, AMS, Providence, 1973, pp. 251-262.
22. K.F. Roth, Developments in Heilbronn's triangle problem, Adv Math 22 (1976), 364-385.
23. W.M. Schmidt, On a problem of Heilbronn, J London Math Soc 4(2) (1972), 545-550.
24. T. Z. Ping, On the problem of Heilbronn type, Northeast Math J 10 (1994), 215-216.
25. L. Yang, J.Z. Zhang, and Z.B. Zeng, Heilbronn problem for five points, Int'l Centre Theoret Physics preprint IC/91/252 (1991).
26. L. Yang, J.Z. Zhang, and Z.B. Zeng, A conjecture on the first several Heilbronn numbers and a computation, Chinese Ann Math Ser A 13 (1992), 503-515.
27. L. Yang, J.Z. Zhang, and Z.B. Zeng, On the Heilbronn numbers of triangular regions, Acta Math Sinica 37 (1994), 678-689.