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Gap-Hamming Problem

In communication complexity, the Gap-Hamming problem asks, if Alice and Bob are each given a string, what is the minimal number of bits that they need to exchange in order for Alice to approximately compute the Hamming distance between their strings. The solution to the problem roughly states that, if Alice and Bob are each given a string, then any communication protocol used to compute the Hamming distance between their strings does (asymptotically) no better than Bob sending his whole string to Alice. More specifically, if Alice and Bob are each given $n$ -bit strings, there exists no communication protocol that lets Alice compute the hamming distance between their strings to within $n/2\pm {\sqrt {n}}$ using less than $\Omega (n)$ bits.

The Gap-Hamming problem has applications to proving lower bounds for many streaming algorithms, including moment frequency estimation^[1] and entropy estimation^[2].

Formal Statement

In this problem, Alice and Bob each receive a string, $x\in \{\pm 1\}^{n}$ and $y\in \{\pm 1\}^{n}$ , respectively, while Alice is required to compute the (partial) function,

$\mathrm {GHD} (x,y)={\begin{cases}+1&D_{H}(x,y)\geq {\frac {n}{2}}+{\sqrt {n}}\\-1&D_{H}(x,y)\leq {\frac {n}{2}}-{\sqrt {n}}\\*&{\text{otherwise}},\end{cases}}$

using the least amount of communication possible. Here, $*$ indicates that Alice can return any of $\pm 1$ . In other words, Alice needs to return whether Bob's string is significantly similar or significantly different from hers while minimizing the number of bits she exchanges with Bob.

History

The Gap-Hamming problem was originally proposed by Indyk and Woodruff, who initially proved a linear lower bound on the one-way communication complexity of the problem and conjectured a linear lower bound in the general case.^[1] The question of the infinite-round case (in which Alice and Bob are allowed to exchange as many messages as desired) remained open until Chakrabarti and Regev proved, via an anti-concentration argument, that the general problem also has linear lower bound complexity, thus settling the original question completely.^[3] This result was followed by a series of other papers that sought to simplify or find new approaches to proving the desired lower bound, notably first by Vidick

References

^ ^a ^b Indyk, Piotr; Woodruff, David (2005). "Optimal approximations of the frequency moments of data streams". Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC '05. Baltimore, MD, USA: ACM Press: 202. doi:10.1145/1060590.1060621. ISBN 9781581139600.
^ Chakrabarti, Amit; Cormode, Graham; Mcgregor, Andrew (2010). "A Near-optimal Algorithm for Estimating the Entropy of a Stream". ACM Trans. Algorithms. 6 (3): 51:1–51:21. doi:10.1145/1798596.1798604. ISSN 1549-6325.
^ Chakrabarti, Amit; Regev, Oded (2011). "An Optimal Lower Bound on the Communication Complexity of Gap-hamming-distance". Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing. STOC '11. New York, NY, USA: ACM: 51–60. doi:10.1145/1993636.1993644. ISBN 9781450306911.

[:0-1] Indyk, Piotr; Woodruff, David (2005). "Optimal approximations of the frequency moments of data streams". Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC '05. Baltimore, MD, USA: ACM Press: 202. doi:10.1145/1060590.1060621. ISBN 9781581139600.

[2] Chakrabarti, Amit; Cormode, Graham; Mcgregor, Andrew (2010). "A Near-optimal Algorithm for Estimating the Entropy of a Stream". ACM Trans. Algorithms. 6 (3): 51:1–51:21. doi:10.1145/1798596.1798604. ISSN 1549-6325.

[3] Chakrabarti, Amit; Regev, Oded (2011). "An Optimal Lower Bound on the Communication Complexity of Gap-hamming-distance". Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing. STOC '11. New York, NY, USA: ACM: 51–60. doi:10.1145/1993636.1993644. ISBN 9781450306911.

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