One-sided error communication complexity of Gap Hamming Distance

Assume that Alice has a binary string x and Bob a binary string y, both of length n. Their goal is to output 0, if x and y are at least Lclose in Hamming distance, and output 1, if x and y are at least U -far in Hamming distance, where L < U are some integer parameters known to both parties. If the Hamming distance between x and y lies in the interval (L, U), they are allowed to output anything. This problem is called the Gap Hamming Distance. In this paper we study public-coin one-sided error communication complexity of this problem. The error with probability at most 1/2 is allowed only for pairs at Hamming distance at least U . In this paper we establish the upper bound O((L2/U) log L) and the lower bound Ω(L2/U) for this complexity. These bounds differ only by a O(log L) factor. The best upper bounds for communication complexity of GHD known before are the following. The upper bounds O(L log n) for one-sided error complexity from [5] and O(L log L) for two-sided error complexity from [6], which do not depend on U and hold for all U > L. Our communication protocol outperforms those from [5] and [6] in the case when the ration U/L is not bounded by a constant. The other known upper bound O(L2/(U − L)2) holds for two-sided error complexity of GHD [7]. If U is greater than L+ √ L, then the protocol from [7] outperforms ours, however it has two-sided error. It is worth to note that all mentioned protocols run in one round. From technical viewpoint, our achievement is a new protocol to prove that x, y are far on the basis of a large difference between distances from x and y to a randomly chosen string. Our lower bound Ω(L2/U) (for the one-sided error communication complexity of GHD) generalizes the lower bound Ω(U) established in [1], [2] for U = O(L). 1 Communication complexity of GHD Given two strings x = x1 . . . xn ∈ {0, 1}, y = y1 . . . yn ∈ {0, 1}, Hamming distance between x and y is defined as the number of positions, where x and y