Matching Nuts and Bolts Noga Alon

We describe a procedure which may be helpful to any disorganized carpenter who has a mixed pile of bolts and nuts and wants to nd the corresponding pairs of bolts and nuts. The procedure uses our (and the carpenter's) ability to construct e ciently highly expanding graphs. The problem considered is given a collection of n bolts of distinct widths and n nuts such that there is a 1-1 correspondence between the nuts and bolts. The goal is to nd for each bolt its corresponding nut by comparing nuts to bolts but not nuts to nuts or bolts to bolts. Our objective is to minimize the number of operations of this kind (as well as the total running time). The problem has a randomized algorithm similar to Quicksort. Our main result is an n(logn) O(1) Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel and AT & T Bell Labs, Murray Hill, NJ 07974, USA. e-mail: [email protected]. Research supported in part by a United States Israel BSF Grant y Computer Science Division, University of California at Berkeley, Berkeley, CA 94720, USA. e-mail: [email protected]. Supported by NSF grant CCR92-01092. z Department of Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. e-mail: [email protected]. Research supportedby a grant from the Israeli Academy of Sciences x Department of Computer Science, University of Arizona, USA. e-mail: [email protected]. { Department of Applied Mathematics and Computer Science, TheWeizmann Institute of Science, Rehovot 76100, Israel. e-mail: [email protected]. Supported by an Alon Fellowship. k University of California at Berkeley Computer Science Division, and International Computer Science Institute at Berkeley. e-mail: [email protected]. Supported by NSF postdoctoral fellowship and ICSI. Part of this work was done while visiting Tel Aviv University and Hebrew University of Jerusalem. time deterministic algorithm, based on expander graphs, for matching the bolts and the nuts.