The Communication Complexity of the Universal Relation

Consider the following communication problem. Alice gets a word x 2 f0; 1gn and Bob gets a word y 2 f0; 1gn. Alice and Bob are told that x 6= y. Their goal is to find an index 1 i n such that xi 6= yi (the index i should be known to both of them). This problem is one of the most basic communication problems. It arises naturally from the correspondence between circuit depth and communication complexity discovered by Karchmer and Wigderson. We present three protocols using which Alice and Bob can solve the problem by exchanging at most n+ 2 bits. One of this protocols is due to Rudich and Tardos. These protocols improve the previous upper bound of n + log n, obtained by Karchmer. We also show that any protocol for solving the problem must exchange, in the worst case, at least n+ 1 bits. This improves a simple lower bound of n 1 obtained by Karchmer. Our protocols, therefore, are at most one bit away from optimality. The three n + 2 bit protocols use two completely different ideas and they each have some additional interesting properties. The simplest protocol (SIMPLE) always finds the first difference betweenx and y. It uses, however, aboutn rounds of communication. A more complicated version of this protocol (LOGSTAR) finds the first difference between x and y by exchanging at most n + 2 bits in about log n rounds of communication. Our most surprising protocol (HAM3) finds a difference, not necessarily the first one, between x and y by exchanging at mostn+2 bits in at most 3 rounds of communication. Protocol HAM3 uses the Hamming errorcorrecting code. We next consider protocols for finding the first difference using a limited number of rounds. For every c 2, we Mathematical Institute of the Hungarian Academy of Sciences, Pf. 127, Budapest, H-1364 Hungary. Supported by NSF grants CCR-95-03254 and DMS-9304580, a grant from Fuji Bank and the grant OTKA-F014919. This work was done while the authorwas visiting the Institute for Advanced Study, Princeton, NJ 08540. E-mail address: [email protected]. yDepartment of Computer Science,Tel Aviv University, Tel Aviv 69978, Israel, International Computer Science Institute, 1947 Center Street, Suite 600, Berkeley, CA 94704, U.S.A, and Department of Computer Science, University of California, Berkeley, CA 94720, U.S.A. E-mail address: [email protected]. present an oblivious protocol that finds the first difference by exchanging n + dlog(c 1) ne + 1 bits in c rounds of communication. We also show that any protocol that finds the first difference using at most c rounds must exchange at least n + dlog(c 1) ne 2 bits. These protocols are, therefore, at most 3 bits away from being optimal. Finally, we consider protocols for variants of the above communication problem. Our most surprising results are perhaps the following. Alice and Bob can exchange at most n blognc + 2 bits, in only 2 rounds, after which Alice will know and index i such that xi 6= yi. Alice and Bob can exchange at most n blognc+ 4 bits, in at most 4 rounds, after which Alice will know and index i such thatxi 6= yi and Bob will know and index j such that xj 6= yj. Furthermore, i = j unless x and y differ in exactly two places.