We prove new lower bounds for nearest neighbor search in the Hamming cube. Our lower bounds are for randomized, two-sided error, algorithms in Yao's cell probe model. Our bounds are in the form of a tradeoff among the number of cells, the size of a cell, and the search time. For example, suppose we are searching among n points in the d dimensional cube, we use poly(n, d) cells, each containing poly(d, log n) bits. We get a lower bound of W(d/log n) on the search time, a significant improvement over the recent bound of W(log d) of Borodin et al. This should be contrasted with the upper bound of O(log log d) for approximate search (and O(1) for a decision version of the problem; our lower bounds hold in that case). By previous results, the bounds for the cube imply similar bounds for nearest neighbor search in high dimensional Euclidean space, and for other geometric problems. © 2002 Elsevier Science (USA)

Abstract. Motivated by database search problems such as partial match or nearest neighbor, we present secure multiparty computation protocols for constant-depth circuits. Specifically, for a constant-depth circuit C of size s with an m-bit input x, we obtain the following types of protocols.-In a setting where k ≥ poly log(s) servers hold C and a client holdsx, we obtain a protocol in which the client privately learns C(x) by communicatingÕ(m) bits with each server. -In a setting where x is arbitrarily distributed between k ≥ poly log(s) parties who all know C, we obtain a secure protocol for evaluatingBoth types of protocols tolerate t = k/poly log(s) dishonest parties and their computational complexity is nearly linear in s. In particular, the protocols are optimal "up to polylog factors" with respect to communication, local computation, and minimal number of participating parties. We then apply the above results to obtain sublinear-communication secure protocols for natural database search problems. For instance, for the partial match problem on a database of n points in {0, 1} m we get a protocol with k ≈ 1 2 log n servers,Õ(m) communication, and nearly linear server computation. Applying previous protocols to this problem would either require Ω(nm) communication,Ω(m) servers, or superpolynomial computation.

We prove new lower bounds for nearest neighbor search in the Hamming cube. Our lower bounds are for randomized, two-sided error, algorithms in Yao's cell probe model. Our bounds are in the form of a tradeoff among the number of cells, the size of a cell, and the search time. For example, suppose we are searching among n points in the d dimensional cube, we use poly(n, d) cells, each containing poly(d, log n) bits. We get a lower bound of ~2(d/log n) on the search time, a significant improvement over the recent bound of ft(log d) of Borodin et al. This should be contrasted with the upper bound of O(loglogd) for approximate search (and O(1) for a decision version of the problem; our lower bounds hold in that case). By previous results, the bounds for the cube imply similar bounds for nearest neighbor search in high dimensional Euclidean space, and for other geometric problems.

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