Cell-probe lower bounds for the partial match problem

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/log²n) 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.