Randomized Approximate Nearest Neighbor Search with Limited Adaptivity

Abstract

We study the complexity of parallel data structures for approximate nearest neighbor search in d -dimensional Hamming space {0,1} d . A classic model for static data structures is the cell-probe model [27]. We consider a cell-probe model with limited adaptivity , where given a k ≥1, a query is resolved by making at most k rounds of parallel memory accesses to the data structure. We give two randomized algorithms that solve the approximate nearest neighbor search using k rounds of parallel memory accesses: —a simple algorithm with O ( k (log d ) 1/ k ) total number of memory accesses for all k ≥1; —an algorithm with O ( k +(1/ k log d ) O (1/ k ) ) total number of memory accesses for all sufficiently large k . Both algorithms use data structures of polynomial size. We prove an Ω(1/ k (log d ) 1/ k ) lower bound for the total number of memory accesses for any randomized algorithm solving the approximate nearest neighbor search within k ≤log log d /2log log log d rounds of parallel memory accesses on any data structures of polynomial size. This lower bound shows that our first algorithm is asymptotically optimal when k = O (1). And our second algorithm achieves the asymptotically optimal tradeoff between number of rounds and total number of memory accesses. In the extremal case, when k = O (log log d /log log log d ) is big enough, our second algorithm matches the Θ(log log d /log log log d ) tight bound for fully adaptive algorithms for approximate nearest neighbor search in [11].