The Communication Complexity of the Universal Relation

• Authors: Gabor Tardos;Uri Zwick
• Venue: CCC '97 Proceedings of the 12th Annual IEEE Conference on Computational Complexity
• Year: 1997

Consider the following communication problem. Alice gets a word x\in \{0,1\}^n and Bob gets a word y\in\{0,1\}^n. Alice and Bob are told that x\ne y. Their goal is to find an index 1\le i\le n such that x_i\ne y_i (the index i should be known to {\em 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 first protocol is the simplest. It always finds the {\em first\/} difference between~x and~y. It uses, however, about n rounds of communication. The second protocol is the most complicated one. It finds the first difference between~x and~y by exchanging at most n+2 bits in about \log^* n rounds of communication. The third protocol is the most surprising one. It finds a difference, not necessarily the first one, between~x and~y by exchanging at most n+2 bits in at most~3 rounds of communication. The third protocol uses the Hamming error-correcting code. We next consider protocols for finding the first difference using a limited number of rounds. For every c\ge 2, we present an {\em oblivious\/} protocol that finds the first difference by exchanging n+\lceil\log^{(c-1)}n\rceil+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+\lceil\log^{(c-1)}n\rceil-2 bits. Thses 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-\flog{n}+2 bits, in only 2 rounds, after which Alice will know and index i such that x_i\ne y_i. Alice and Bob can exchange at most n-\flog{n}+5 bits, in at most 4 rounds, after which Alice will know and index i such that x_i\ne y_i and Bob will know and index~j such that x_j\ne y_j. Furthermore, i=j unless~x and~y differ in exactly two places.