Robust Proximity Search for Balls using Sublinear Space

Given a set of n disjoint balls b1, . . ., bn in IRd, we provide a data structure, of near linear size, that can answer (1 \pm \epsilon)-approximate kth-nearest neighbor queries in O(log n + 1/\epsilon^d) time, where k and \epsilon are provided at query time. If k and \epsilon are provided in advance, we provide a data structure to answer such queries, that requires (roughly) O(n/k) space; that is, the data structure has sublinear space requirement if k is sufficiently large

Neural Distributed Autoassociative Memories: A Survey

Introduction. Neural network models of autoassociative, distributed memory allow storage and retrieval of many items (vectors) where the number of stored items can exceed the vector dimension (the number of neurons in the network). This opens the possibility of a sublinear time search (in the number of stored items) for approximate nearest neighbors among vectors of high dimension. The purpose of this paper is to review models of autoassociative, distributed memory that can be naturally implemented by neural networks (mainly with local learning rules and iterative dynamics based on information locally available to neurons). Scope. The survey is focused mainly on the networks of Hopfield, Willshaw and Potts, that have connections between pairs of neurons and operate on sparse binary vectors. We discuss not only autoassociative memory, but also the generalization properties of these networks. We also consider neural networks with higher-order connections and networks with a bipartite graph structure for non-binary data with linear constraints. Conclusions. In conclusion we discuss the relations to similarity search, advantages and drawbacks of these techniques, and topics for further research. An interesting and still not completely resolved question is whether neural autoassociative memories can search for approximate nearest neighbors faster than other index structures for similarity search, in particular for the case of very high dimensional vectors.Comment: 31 page

New Applications of the Nearest-Neighbor Chain Algorithm

The nearest-neighbor chain algorithm was proposed in the eighties as a way to speed up certain hierarchical clustering algorithms. In the first part of the dissertation, we show that its application is not limited to clustering. We apply it to a variety of geometric and combinatorial problems. In each case, we show that the nearest-neighbor chain algorithm finds the same solution as a preexistent greedy algorithm, but often with an improved runtime. We obtain speedups over greedy algorithms for Euclidean TSP, Steiner TSP in planar graphs, straight skeletons, a geometric coverage problem, and three stable matching models. In the second part, we study the stable-matching Voronoi diagram, a type of plane partition which combines properties of stable matchings and Voronoi diagrams. We propose political redistricting as an application. We also show that it is impossible to compute this diagram in an algebraic model of computation, and give three algorithmic approaches to overcome this obstacle. One of them is based on the nearest-neighbor chain algorithm, linking the two parts together

Sublinear time approximation of the cost of a metric k-nearest neighbor graph

Let (X, d) be an n-point metric space. We assume that (X, d) is given in the distance oracle model, that is, X = {1, …, n} and for every pair of points x, y from X we can query their distance d(x, y) in constant time. A k-nearest neighbor (k-NN) graph for (X, d) is a directed graph G = (V, E) that has an edge to each of v's k nearest neighbors. We use cost(G) to denote the sum of edge weights of G. In this paper, we study the problem of approximating cost(G) in sublinear time, when we are given oracle access to the metric space (X, d) that defines G. Our goal is to develop an algorithm that solves this problem faster than the time required to compute G. We first present an algorithm that in Õ∊(n2/k) time with probability at least approximates cost(G) to within a factor of 1 + ∊. Next, we present a more elaborate sublinear algorithm that in time Õϵ(min{nk3/2, n2/k}) computes an estimate of cost(G) that satisfies with probability at least where mst(X) denotes the cost of the minimum spanning tree of (X, d). Further, we complement these results with near matching lower bounds. We show that any algorithm that for a given metric space (X, d) of size n, with probability at least estimates cost(G) to within a 1 + ∊ factor requires Ω(n2/k) time. Similarly, any algorithm that with probability at least estimates cost(G) to within an additive error term ϵ · (mst(X) + cost(X)) requires Ωϵ(min{nk3/2, n2/k}) time