### 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 multi-pass 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 (FOCS, 2003), it had been conjectured that the naive 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 C. Borell (1985). 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) |
---|---|

Title of host publication | STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing |

Pages | 51-60 |

Number of pages | 10 |

DOIs | |

State | Published - 2011 |

Event | 43rd ACM Symposium on Theory of Computing, STOC'11 - San Jose, CA, United States Duration: Jun 6 2011 → Jun 8 2011 |

### Other

Other | 43rd ACM Symposium on Theory of Computing, STOC'11 |
---|---|

Country | United States |

City | San Jose, CA |

Period | 6/6/11 → 6/8/11 |

### Fingerprint

### Keywords

- communication complexity
- corruption
- data streams
- gap-hamming-distance
- gaussian noise correlation
- lower bounds

### ASJC Scopus subject areas

- Software

### Cite this

*STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing*(pp. 51-60) https://doi.org/10.1145/1993636.1993644

**An optimal lower bound on the communication complexity of Gap-Hamming-Distance.** / Chakrabarti, Amit; Regev, Oded.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing.*pp. 51-60, 43rd ACM Symposium on Theory of Computing, STOC'11, San Jose, CA, United States, 6/6/11. https://doi.org/10.1145/1993636.1993644

}

TY - GEN

T1 - An optimal lower bound on the communication complexity of Gap-Hamming-Distance

AU - Chakrabarti, Amit

AU - Regev, Oded

PY - 2011

Y1 - 2011

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 multi-pass 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 (FOCS, 2003), it had been conjectured that the naive 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 C. Borell (1985). 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 multi-pass 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 (FOCS, 2003), it had been conjectured that the naive 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 C. Borell (1985). 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=79959768278&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79959768278&partnerID=8YFLogxK

U2 - 10.1145/1993636.1993644

DO - 10.1145/1993636.1993644

M3 - Conference contribution

SN - 9781450306911

SP - 51

EP - 60

BT - STOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing

ER -