Lower bounds for high dimensional nearest neighbor search and related problems

Abstract

In spite of extensive and continuing research, for various geometric search problems (such as nearest neighbor search), the best algorithms known have performance that degrades exponentially in the dimension. This phenomenon is sometimes called the curse of dimensionality. Recent results [38,37,40] show that in some sense it is possible to avoid the curse of dimensionality for the approximate nearest neighbor search problem. But must the exact nearest neighbor search problem suffer this curse? We provide some evidence in support of the curse. Specifically we investigate the exact nearest neighbor search problem and the related problem of exact partial match within the asymmetric communication model first used by Miltersen [43] to study data structure problems. We derive non-trivial asymptotic lower bounds for the exact problem that stand in contrast to known algorithms for approximate nearest neighbor search.

Unifying the Landscape of Cell-Probe Lower Bounds

“…The partial match problem is well investigated [36,14,31,43]. The best previous bound [31] for Alice's communication was Ω(d/ lg n) bits, instead of our optimal Ω(d).…”

“…The end of the 80s saw the publication of two landmark papers in the field: Ajtai's static lower bound for predecessor search [1], and the dynamic lower bounds of Fredman and Saks [26]. In the 20 years that have passed, cellprobe complexity has developed into a mature research direction, with a substantial bibliography: we are aware of [1,26,34,37,35,29,5,25,36,14,2,15,6,13,11,12,27,28,16,33,31,44,43,8,41,42,40,45,48].…”

“…The topics being studied cluster into three main categories: lopsided set disjointness reachability oracles in the butterfly loses lg lg n partial match [36,14,31,43] (1 + ε)-ANN in ℓ 1 , ℓ 2 [8] dyn. marked ancestor [5] 2D stabbing 3-ANN in ℓ ∞ [30] NN in ℓ 1 , ℓ 2 [11,14,43] worst-case union-find [26,2] dyn. 1D stabbing partial sums [26,13,28,41,42] 4D range reporting 2D range counting [40] dyn.…”

A Directed Isoperimetric Inequality with application to Bregman Near Neighbor Lower Bounds

“…Lower bounds for near neighbor search in metric spaces have been studied extensively. Borodin, Ostrovsky and Rabani [11] show a lower bound that any randomized cell probe algorithm for the exact match problem that must probe at least Ω(log d) cells. Barkol and Rabani improve this bound to Ω( d log n ) cells [8].…”

“…• A cell probe lower bound of Ω( d log n ) queries for any randomized algorithm that solves exact nearest neighbor search on the Bregman cube in polynomial space and word size polynomial in d, log n via [11].…”