Randomized Approximate Nearest Neighbor Search with Limited Adaptivity

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(logd)1/k) total number of memory accesses for all k≥1; —an algorithm with O(k+(1/klogd)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(logd)1/k) lower bound for the total number of memory accesses for any randomized algorithm solving the approximate nearest neighbor search within k ≤log logd/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 logd/log log log d) is big enough, our second algorithm matches the Θ(log logd/log log log d) tight bound for fully adaptive algorithms for approximate nearest neighbor search in [11].