## 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 | American English |
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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 |

Externally published | Yes |

### Bibliographical note

Funding Information:·Corresponding author. E-mail addresses: jayram@almaden.ibm.com (T.S. Jayram), khot@cs.princeton.edu (S. Khot), ravi@almade-n.ibm.com (R. Kumar), rabani@cs.technion.ac.il (Y. Rabani). 1Supported by Prof. Sanjeev Arora’s David and Lucile Packard Fellowship, NSF Grant CCR-0098180, and an NSF ITR Grant. Part of this work was done while the author was visiting the IBM Almaden Research Center and IBM T.J. Watson Research Center. 2Work at the Technion supported by BSF Grant number 99-00217, by ISF Grant number 386/99, by IST contract number 32007 (APPOL), and by the Fund for the Promotion of Research at the Technion. Part of this work was done while the author was visiting the IBM Almaden Research Center.

## Keywords

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