Some Bounds on Communication Complexity of Gap Hamming Distance

In this paper we obtain some bounds on communication complexity of Gap Hamming Distance problem (GHDnL,U ): Alice and Bob are given binary string of length n and they are guaranteed that Hamming distance between their inputs is either ≤ L or ≥ U for some L < U . They have to output 0, if the first inequality holds, and 1, if the second inequality holds. In this paper we study the communication complexity of GHDnL,U for probabilistic protocols with one-sided error and for deterministic protocols. Our first result is a protocol which communicates O ( ( s U ) 1 3 · n log n ) bits and has one-sided error probability e provided s ≥ (L+ 10 n ) U2 . Our second result is about deterministic communication complexity of GHD n 0, t. Surprisingly, it can be computed with logarithmic precision: D(GHDn0, t) = n− log2 V2 (