### Abstract

Given a database of n points in {0,1}
^{d}, the partial match problem is: In response to a query x in {0,1,*}
^{d}, is there a database point y such that for every i whenever x
_{i}≠*, we have x
_{i} = y
_{i}. In this paper we show randomized lower bounds in the cell-probe model for this well-studied problem (Analysis of associative retrieval algorithms, Ph.D. Thesis, Stanford University, 1974; The Art of Computer Programming; Sorting and Searching, Addison-Wesley, Reading, MA, 1973; SIAM J. Comput. 5(1) (1976) 19; J. Comput. System Sci. 57(1) (1998) 37; Proceedings of the 31st Annual ACM Symposium on Theory of Computing, 1999; Proceedings of the 29th International Colloquium on Algorithms, Logic, and Programming, 1999). Our lower bounds follow from a near-optimal asymmetric communication complexity lower bound for this problem. Specifically, we show that either Alice has to send Ω(d/logn) bits or Bob has to send Ω(n
^{1-o(1)}) bits. When applied to the cell-probe model, it means that if the number of cells is restricted to be poly(n,d) where each cell is of size poly(logn,d), then Ω(d/log
^{2}n) probes are needed. This is an exponential improvement over the previously known lower bounds for this problem obtained by Miltersen et al. (1998) and Borodin et al. (1999). Our lower bound also leads to new and improved lower bounds for related problems including a lower bound for the ℓ
_{∞} c-nearest neighbor problem for c<3 and an improved communication complexity lower bound for the exact nearest neighbor problem.

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

Pages (from-to) | 435-447 |

Number of pages | 13 |

Journal | Journal of Computer and System Sciences |

Volume | 69 |

Issue number | 3 SPEC. ISS. |

DOIs | |

State | Published - Nov 2004 |

### Fingerprint

### Keywords

- Asymmetric communication complexity
- Cell-probe model
- Nearest neighbor problem
- Partial match problem

### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Journal of Computer and System Sciences*,

*69*(3 SPEC. ISS.), 435-447. https://doi.org/10.1016/j.jcss.2004.04.006

**Cell-probe lower bounds for the partial match problem.** / Jayram, T. S.; Khot, Subhash; Kumar, Ravi; Rabani, Yuval.

Research output: Contribution to journal › Article

*Journal of Computer and System Sciences*, vol. 69, no. 3 SPEC. ISS., pp. 435-447. https://doi.org/10.1016/j.jcss.2004.04.006

}

TY - JOUR

T1 - Cell-probe lower bounds for the partial match problem

AU - Jayram, T. S.

AU - Khot, Subhash

AU - Kumar, Ravi

AU - Rabani, Yuval

PY - 2004/11

Y1 - 2004/11

N2 - Given a database of n points in {0,1} d, the partial match problem is: In response to a query x in {0,1,*} d, is there a database point y such that for every i whenever x i≠*, we have x i = y i. In this paper we show randomized lower bounds in the cell-probe model for this well-studied problem (Analysis of associative retrieval algorithms, Ph.D. Thesis, Stanford University, 1974; The Art of Computer Programming; Sorting and Searching, Addison-Wesley, Reading, MA, 1973; SIAM J. Comput. 5(1) (1976) 19; J. Comput. System Sci. 57(1) (1998) 37; Proceedings of the 31st Annual ACM Symposium on Theory of Computing, 1999; Proceedings of the 29th International Colloquium on Algorithms, Logic, and Programming, 1999). Our lower bounds follow from a near-optimal asymmetric communication complexity lower bound for this problem. Specifically, we show that either Alice has to send Ω(d/logn) bits or Bob has to send Ω(n 1-o(1)) bits. When applied to the cell-probe model, it means that if the number of cells is restricted to be poly(n,d) where each cell is of size poly(logn,d), then Ω(d/log 2n) probes are needed. This is an exponential improvement over the previously known lower bounds for this problem obtained by Miltersen et al. (1998) and Borodin et al. (1999). Our lower bound also leads to new and improved lower bounds for related problems including a lower bound for the ℓ ∞ c-nearest neighbor problem for c<3 and an improved communication complexity lower bound for the exact nearest neighbor problem.

AB - Given a database of n points in {0,1} d, the partial match problem is: In response to a query x in {0,1,*} d, is there a database point y such that for every i whenever x i≠*, we have x i = y i. In this paper we show randomized lower bounds in the cell-probe model for this well-studied problem (Analysis of associative retrieval algorithms, Ph.D. Thesis, Stanford University, 1974; The Art of Computer Programming; Sorting and Searching, Addison-Wesley, Reading, MA, 1973; SIAM J. Comput. 5(1) (1976) 19; J. Comput. System Sci. 57(1) (1998) 37; Proceedings of the 31st Annual ACM Symposium on Theory of Computing, 1999; Proceedings of the 29th International Colloquium on Algorithms, Logic, and Programming, 1999). Our lower bounds follow from a near-optimal asymmetric communication complexity lower bound for this problem. Specifically, we show that either Alice has to send Ω(d/logn) bits or Bob has to send Ω(n 1-o(1)) bits. When applied to the cell-probe model, it means that if the number of cells is restricted to be poly(n,d) where each cell is of size poly(logn,d), then Ω(d/log 2n) probes are needed. This is an exponential improvement over the previously known lower bounds for this problem obtained by Miltersen et al. (1998) and Borodin et al. (1999). Our lower bound also leads to new and improved lower bounds for related problems including a lower bound for the ℓ ∞ c-nearest neighbor problem for c<3 and an improved communication complexity lower bound for the exact nearest neighbor problem.

KW - Asymmetric communication complexity

KW - Cell-probe model

KW - Nearest neighbor problem

KW - Partial match problem

UR - http://www.scopus.com/inward/record.url?scp=4544261973&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4544261973&partnerID=8YFLogxK

U2 - 10.1016/j.jcss.2004.04.006

DO - 10.1016/j.jcss.2004.04.006

M3 - Article

VL - 69

SP - 435

EP - 447

JO - Journal of Computer and System Sciences

JF - Journal of Computer and System Sciences

SN - 0022-0000

IS - 3 SPEC. ISS.

ER -