Tighter bounds for nearest neighbor search and related problems in the cell probe model

Abstract

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.

Randomized Approximate Nearest Neighbor Search with Limited Adaptivity

“…For other cases, the answer can be arbitrary. This problem has been extensively studied in the context of lower bounds for nearest neighbor search [6,7,16,20,21].…”

“…The complexity is measured by both the size of the data structure and the number of cell-probes made by the algorithm to answer a query in the worst case. There is a substantial body of works on the cellprobe complexity of nearest neighbor search in Hamming space [6,7,13,16,[19][20][21].…”

The Cell Probe Complexity of Succinct Data Structures

“…In the cell probe model (e.g., [1,3,4,6,7,9,[18][19][20][21]), a boolean static data structure problem is given by a map f : {0, 1} n × {0, 1} m → {0, 1}, where {0, 1} n is a set of possible data to be stored, {0, 1} m is a set of possible queries and f (x, y) is the answer to question y about data x. For natural problems, we have m n: the question we pose to the database is much shorter than the database itself.…”

“…Substring Search also admits an s = O(n), t = O(m) solution [12] and very recently a solution with s = n + o(n) and t = m O(1) was constructed [13] but no solution with s = n + o(n) and t = O(m) is known. For a problem such as Polynomial Evaluation (and many natural data structure problems, such as partial match type problems [3,4,7]), we know of no solution with s = n O(1) , t = m O(1) . Thus, a main concern is to prove that such solutions do not exist.…”

Lower bounds for high dimensional nearest neighbor search and related problems

“…Subsequent to our conference publication, Barkol and Rabani [8] established a significantly improved lower bound for the λ-neighbor problem. They are able to derive a lower bound for two-sided error randomized asymmetric communication complexity protocols that is quite comparable to our one-sided error bound in Theorem 2.2.…”