Tight bounds for minimax grid matching with applications to the average case analysis of algorithms

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.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.