Anti-concentration and the exact gap-Hamming problem

Anup Rao, Amir Yehudayoff

Research output: Contribution to journalArticlepeer-review

Abstract

We prove anticoncentration bounds for the inner product of two independent random vectors and use these bounds to prove lower bounds in communication complexity. We show that if A, B are subsets of the cube \{ \pm 1\} n with | A| \cdot | B| \geq 21.01n, and X \in A and Y \in B are sampled independently and uniformly, then the inner product \langle X, Y \rangle takes on any fixed value with probability at most O(1/\surd n). In fact, we prove the following stronger ``smoothness"" statement: maxk\bigm| \bigm| Pr[\langle X, Y \rangle = k] - Pr[\langle X, Y \rangle = k + 4]\bigm| \bigm| \leq O(1/n). We use these results to prove that the exact gap-hamming problem requires linear communication, resolving an open problem in communication complexity. We also conclude anticoncentration for structured distributions with low entropy. If x \in \BbbZ n has no zero coordinates, and B \subseteq \{ \pm 1\} n corresponds to a subspace of \BbbF n2 of dimension 0.51n, then maxk Pr[\langle x, Y \rangle = k] \leq O(\sqrt{} ln(n)/n).

Original languageEnglish
Article number2
Pages (from-to)1071-1092
Number of pages22
JournalSIAM Journal on Discrete Mathematics
Volume36
Issue number2
DOIs
StatePublished - 2022

Keywords

  • anticoncentration
  • communication complexity
  • probability

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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