### Abstract

We prove an optimal Ω(n) lower bound on the randomized communication complexity of the much-studied gap-hamming-distance problem. As a consequence, we obtain essentially optimal multipass space lower bounds in the data stream model for a number of fundamental problems, including the estimation of frequency moments. The gap-hamming-distance problem is a communication problem, wherein Alice and Bob receive n-bit strings x and y, respectively. They are promised that the Hamming distance between x and y is either at least n/2 + √n or at most n/2 - √n, and their goal is to decide which of these is the case. Since the formal presentation of the problem by Indyk and Woodruff [Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 2003, pp. 283-289], it had been conjectured that the naïve protocol, which uses n bits of communication, is asymptotically optimal. The conjecture was shown to be true in several special cases, e.g., when the communication is deterministic or when the number of rounds of communication is limited. The proof of our aforementioned result, which settles this conjecture fully, is based on a new geometric statement regarding correlations in Gaussian space, related to a result of Borell [Z. Wahrsch. Verw. Gebiete, 70 (1985), pp. 1-13]. To prove this geometric statement, we show that random projections of not-too-small sets in Gaussian space are close to a mixture of translated normal variables.

Original language | English (US) |
---|---|

Pages (from-to) | 1299-1317 |

Number of pages | 19 |

Journal | SIAM Journal on Computing |

Volume | 41 |

Issue number | 5 |

DOIs | |

State | Published - 2012 |

### Fingerprint

### Keywords

- Communication complexity
- Corruption
- Data streams
- Gap-hamming-distance
- Gaussian noise correlation
- Lower bounds

### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science(all)

### Cite this

*SIAM Journal on Computing*,

*41*(5), 1299-1317. https://doi.org/10.1137/120861072

**An optimal lower bound on the communication complexity of gap-hamming-distance.** / Chakrabarti, Amit; Regev, Oded.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 41, no. 5, pp. 1299-1317. https://doi.org/10.1137/120861072

}

TY - JOUR

T1 - An optimal lower bound on the communication complexity of gap-hamming-distance

AU - Chakrabarti, Amit

AU - Regev, Oded

PY - 2012

Y1 - 2012

N2 - We prove an optimal Ω(n) lower bound on the randomized communication complexity of the much-studied gap-hamming-distance problem. As a consequence, we obtain essentially optimal multipass space lower bounds in the data stream model for a number of fundamental problems, including the estimation of frequency moments. The gap-hamming-distance problem is a communication problem, wherein Alice and Bob receive n-bit strings x and y, respectively. They are promised that the Hamming distance between x and y is either at least n/2 + √n or at most n/2 - √n, and their goal is to decide which of these is the case. Since the formal presentation of the problem by Indyk and Woodruff [Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 2003, pp. 283-289], it had been conjectured that the naïve protocol, which uses n bits of communication, is asymptotically optimal. The conjecture was shown to be true in several special cases, e.g., when the communication is deterministic or when the number of rounds of communication is limited. The proof of our aforementioned result, which settles this conjecture fully, is based on a new geometric statement regarding correlations in Gaussian space, related to a result of Borell [Z. Wahrsch. Verw. Gebiete, 70 (1985), pp. 1-13]. To prove this geometric statement, we show that random projections of not-too-small sets in Gaussian space are close to a mixture of translated normal variables.

AB - We prove an optimal Ω(n) lower bound on the randomized communication complexity of the much-studied gap-hamming-distance problem. As a consequence, we obtain essentially optimal multipass space lower bounds in the data stream model for a number of fundamental problems, including the estimation of frequency moments. The gap-hamming-distance problem is a communication problem, wherein Alice and Bob receive n-bit strings x and y, respectively. They are promised that the Hamming distance between x and y is either at least n/2 + √n or at most n/2 - √n, and their goal is to decide which of these is the case. Since the formal presentation of the problem by Indyk and Woodruff [Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 2003, pp. 283-289], it had been conjectured that the naïve protocol, which uses n bits of communication, is asymptotically optimal. The conjecture was shown to be true in several special cases, e.g., when the communication is deterministic or when the number of rounds of communication is limited. The proof of our aforementioned result, which settles this conjecture fully, is based on a new geometric statement regarding correlations in Gaussian space, related to a result of Borell [Z. Wahrsch. Verw. Gebiete, 70 (1985), pp. 1-13]. To prove this geometric statement, we show that random projections of not-too-small sets in Gaussian space are close to a mixture of translated normal variables.

KW - Communication complexity

KW - Corruption

KW - Data streams

KW - Gap-hamming-distance

KW - Gaussian noise correlation

KW - Lower bounds

UR - http://www.scopus.com/inward/record.url?scp=84868220384&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84868220384&partnerID=8YFLogxK

U2 - 10.1137/120861072

DO - 10.1137/120861072

M3 - Article

AN - SCOPUS:84868220384

VL - 41

SP - 1299

EP - 1317

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 5

ER -