Cell-probe lower bounds for the partial match problem

  • Authors:

  • T. S. Jayram;Subhash Khot;Ravi Kumar;Yuval Rabani

  • Affiliations:

  • CS Principles and Methodologies, IBM Almaden Research Center, 650 Harry Road, San Jose, CA;Department of Computer Science, Princeton University, Princeton, NJ and IBM Almaden Research Center and IBM T.J. Watson Research Center;CS Principles and Methodologies, IBM Almaden Research Center, 650 Harry Road, San Jose, CA;Computer Science Department, Technion, Israel Institute of Technology, Technion City, Haifa 32000, Israel and IBM Almaden Research Center

  • Venue:
  • Journal of Computer and System Sciences - Special issue: STOC 2003
  • Year:
  • 2004

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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 xi ≠ *, we have xi = yi. 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/log n) bits or Bob has to send Ω(n1-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(log n,d), then Ω(d/log2 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