The minimax grid matching problem is a fundamental combinatorial problem associated with the average case analysis of algorithms. The problem has arisen in a number of interesting and seemingly unrelated areas, including wafer-scale integration of systolic arrays, two-dimensional discrepancy problems, and testing pseudorandom number generators. However, the minimax grid matching problem is best known for its application to the maximum up-right matching problem. The maximum up-right matching problem was originally defined by Karp, Luby and Marchetti-Spaccamela in association with algorithms for 2-dimensional bin packing. More recently, the up-right matching problem has arisen in the average case analysis of on-line algorithms for 1-dimen-sional bin packing and dynamic allocation.
In this paper, we solve both the minimax grid matching problem and the maximum up-right matching problem. As a direct result, we obtain tight upper bounds on the average case behavior of the best algorithms known for 2-dimensional bin packing, 1-dimensional on-line bin packing and on-line dynamic allocation. The results also solve a long-open question in mathematical statistics.
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M. Ajtai, J. Komlós andG. Tusnády, On optimal matchings,Combinatorica,4, 259–264, 1983.
J. L.Bentley, D. S.Johnson, F. T,Leighton, C. C.McGeoch and L.McGeoch, Some unexpected expected behavior results for bin packing,Proceedings of the 16th ACM Symp. on the Theory of Computing, 279–288, 1984.
J. Blum, On convergence of empirical distribution functions,Annals of Mathematical Statistics,26, 527–529, 1955.
E. G.Coffman, Jr. and F. T.Leighton, A provably efficient algorithm for dynamic storage allocation,Proceedings of the 18th ACM Symp. on Theory of Computing, May 1986,to appear.
P.Diaconis, personal communication, 1985.
R. M. Dudley, A course in empirical processes,École d'Été de Probabilités de Saint-Flour XII, 1982, Lecture Notes in Math. No. 1097, 1–142, Springer Verlag, NY, 1984. (The relevant part is Chapter 8.)
R. M. Dudley, Empirical and Poisson Processes on classes of sets or functions too large for central limit theorems,Z. Wahrsch. verw. Gebiete,61, 355–368, 1982.
A.Fiat and A.Shamir, Polymorphic arrays: a novel VLSI layout for systolic computers,Proceedings of the 25th Symp. on Foundations of Computer Science, 37–45, 1984.
P. Hall, On representatives of subsets,J. London Math. Soc.,10, 26–30, 1935.
R. M.Karp, M.Luby and A.Marchetti-Spaccamela, Probabilistic analysis of multi-dimensional bin packing problems,Proceedings of the 16th ACM Symp. on Theory of Computing-289–298, 1984.
J. Kiefer, On the large deviation of the empiric d.f. of vector chance variables and a law of the iterated logarithm,Pacific J. Math.,11, 649–660, 1961.
F. T.Leighton, 18.419 class notes, 1984.
F. T. Leighton andC. E. Leiserson, Wafer-scale integration of systolic arrays,IEEE Trans. on Computers,C-34, No. 5, 448–461, 1985.
M. Lerch, Question 1547,L'Intermédiare Math.,11, 145–146, 1904.
M.Luby, personal communication, 1985.
W. Philipp, Empirical distribution functions and uniform distribution mod 1,Diophantine Approximation and its Applications, C.F. Osgood, ed., Academic Press, NY, 1973.
W.Philipp, personal communication quoted in .
G. Sawitzki, Another random number generator which should be avoided,Statistical Software Newsletters,11, No. 2, 81–82, 1985.
W. Schmidt,Lectures on Irregularities of Distribution, Tata Institute of Fundamental Research, Bombay, India, 1977.
W. Schmidt, Irregularities of distribution, VII,Acta Arithmetica,21, 45–50, 1972.
P. W.Shor, The average-case analysis of some on-line algorithms for bin packing,Proceedings of the 25th Symp. on Foundations of Computer Science, 193–200, 1984.
P. W.Shor,Random Planar Matching and Bin Packing, Ph. D. Thesis, MIT Math. Dept., 1985.
M. Steele, Limit properties of random variables associated with a partial ordering ofR d ,Annals of Probability,5, No. 3, 395–403, 1977.
J. Van der Corput, Verteilungsfunctionen I.,Proc. Kon. Ned. Akad. v. Wetensch,38, 813–821, 1935.
V. N. Vapnik andA. Ya. Cervonenkis, Necessary and sufficient conditions for the uniform convergences of means to their expectations,Theory of Probability and Applications,26, 532–553, 1981.
F. T. Wright, The empirical discrepancy over lower layers and a related law of large numbers,Annals of Probability,9, 323–329, 1981.
This research was supported by Air Force Contracts AFOSR-82-0326 and AFOSR-86-0078, NSF Grant 8120790, and DARPA contract N00014-80-C-0326. In addition, Tom Leighton was supported by an NSF Presidential Young Investigator Award with matching funds from Xerox and IBM.
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Leighton, T., Shor, P. Tight bounds for minimax grid matching with applications to the average case analysis of algorithms. Combinatorica 9, 161–187 (1989). https://doi.org/10.1007/BF02124678
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