On the Complexity of the (Approximate) Nearest Colored Node Problem

Given a graph G=(V,E) where each vertex is assigned a color from the set C={c_1, c_2, .., c_sigma}. In the (approximate) nearest colored node problem, we want to query, given v in V and c in C, for the (approximate) distance dist^(v, c) from v to the nearest node of color c. For any integer 1 <= k <= log n, we present a Color Distance Oracle (also often referred to as Vertex-label Distance Oracle) of stretch 4k-5 using space O(kn sigma^{1/k}) and query time O(log{k}). This improves the query time from O(k) to O(log{k}) over the best known Color Distance Oracle by Chechik [Chechik, 2012]. We then prove a lower bound in the cell probe model showing that even for unweighted undirected paths any static data structure that uses space S requires at least Omega (log (log{sigma} / log(S/n)+log log{n})) query time to give a distance estimate of stretch O(polylog(n)). This implies for the important case when sigma = Theta(n^{epsilon}) for some constant 0 < epsilon < 1, that our Color Distance Oracle has asymptotically optimal query time in regard to k, and that recent Color Distance Oracles for trees [Tsur, 2018] and planar graphs [Mozes and Skop, 2018] achieve asymptotically optimal query time in regard to n. We also investigate the setting where the data structure additionally has to support color-reassignments. We present the first Color Distance Oracle that achieves query times matching our lower bound from the static setting for large stretch yielding an exponential improvement over the best known query time [Chechik, 2014]. Finally, we give new conditional lower bounds proving the hardness of answering queries if edge insertions and deletion are allowed that strictly improve over recent bounds in time and generality

Down the Rabbit Hole: Robust Proximity Search and Density Estimation in Sublinear Space

For a set of $n$ points in $\Re^d$, and parameters $k$ and \eps, we present a data structure that answers (1+\eps,k)-\ANN queries in logarithmic time. Surprisingly, the space used by the data-structure is \Otilde (n /k); that is, the space used is sublinear in the input size if $k$ is sufficiently large. Our approach provides a novel way to summarize geometric data, such that meaningful proximity queries on the data can be carried out using this sketch. Using this, we provide a sublinear space data-structure that can estimate the density of a point set under various measures, including: \begin{inparaenum}[(i)] \item sum of distances of $k$ closest points to the query point, and \item sum of squared distances of $k$ closest points to the query point. \end{inparaenum} Our approach generalizes to other distance based estimation of densities of similar flavor. We also study the problem of approximating some of these quantities when using sampling. In particular, we show that a sample of size \Otilde (n /k) is sufficient, in some restricted cases, to estimate the above quantities. Remarkably, the sample size has only linear dependency on the dimension

The Skip Quadtree: A Simple Dynamic Data Structure for Multidimensional Data

We present a new multi-dimensional data structure, which we call the skip quadtree (for point data in R^2) or the skip octree (for point data in R^d, with constant d>2). Our data structure combines the best features of two well-known data structures, in that it has the well-defined "box"-shaped regions of region quadtrees and the logarithmic-height search and update hierarchical structure of skip lists. Indeed, the bottom level of our structure is exactly a region quadtree (or octree for higher dimensional data). We describe efficient algorithms for inserting and deleting points in a skip quadtree, as well as fast methods for performing point location and approximate range queries.Comment: 12 pages, 3 figures. A preliminary version of this paper appeared in the 21st ACM Symp. Comp. Geom., Pisa, 2005, pp. 296-30

Optimal, scalable forward models for computing gravity anomalies

We describe three approaches for computing a gravity signal from a density anomaly. The first approach consists of the classical "summation" technique, whilst the remaining two methods solve the Poisson problem for the gravitational potential using either a Finite Element (FE) discretization employing a multilevel preconditioner, or a Green's function evaluated with the Fast Multipole Method (FMM). The methods utilizing the PDE formulation described here differ from previously published approaches used in gravity modeling in that they are optimal, implying that both the memory and computational time required scale linearly with respect to the number of unknowns in the potential field. Additionally, all of the implementations presented here are developed such that the computations can be performed in a massively parallel, distributed memory computing environment. Through numerical experiments, we compare the methods on the basis of their discretization error, CPU time and parallel scalability. We demonstrate the parallel scalability of all these techniques by running forward models with up to $10^8$ voxels on 1000's of cores.Comment: 38 pages, 13 figures; accepted by Geophysical Journal Internationa