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Some Bounds on Communication Complexity of Gap Hamming Distance

In this paper we obtain some bounds on communication complexity of Gap Hamming Distance problem ($\mathsf{GHD}^n_{L, U}$): Alice and Bob are given binary string of length $n$ and they are guaranteed that Hamming distance between their inputs is either $\le L$ or $\ge U$ 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 $\mathsf{GHD}^n_{L, U}$ for probabilistic protocols with one-sided error and for deterministic protocols. Our first result is a protocol which communicates $O\left(\left(\frac{s}{U}\right)^\frac{1}{3} \cdot n\log n\right)$ bits and has one-sided error probability $e^{-s}$ provided $s \ge \frac{(L + \frac{10}{n})^3}{U^2}$. Our second result is about deterministic communication complexity of $\mathsf{GHD}^n_{0,\, t}$. Surprisingly, it can be computed with logarithmic precision: $$\mathrm{D}(\mathsf{GHD}^n_{0,\, t}) = n - \log_2 V_2\left(n, \left\lfloor\frac{t}{2}\right\rfloor\right) + O(\log n),$$ where $V_2(n, r)$ denotes the size of Hamming ball of radius $r$. As an application of this result for every $c < 2$ we prove a $Ω\left(\frac{n(2 - c)^2}{p}\right)$ lower bound on the space complexity of any $c$-approximate deterministic $p$-pass streaming algorithm for computing the number of distinct elements in a data stream of length $n$ with tokens drawn from the universe $U = \{1, 2, \ldots, n\}$. Previously that lower bound was known for $c < \frac{3}{2}$ and for $c < 2$ but with larger $|U|$.