Search for: All records

Let d be a (well-behaved) shortest-path metric defined on a path-connected subset of ℝ² and let 𝒟 = {D_1,…,D_n} be a set of geodesic disks with respect to the metric d. We prove that 𝒢^×(𝒟), the intersection graph of the disks in 𝒟, has a clique-based separator consisting of O(n^{3/4+ε}) cliques. This significantly extends the class of objects whose intersection graphs have small clique-based separators. Our clique-based separator yields an algorithm for q-Coloring that runs in time 2^O(n^{3/4+ε}), assuming the boundaries of the disks D_i can be computed in polynomial time. We also use our clique-based separator to obtain a simple, efficient, and almost exact distance oracle for intersection graphs of geodesic disks. Our distance oracle uses O(n^{7/4+ε}) storage and can report the hop distance between any two nodes in 𝒢^×(𝒟) in O(n^{3/4+ε}) time, up to an additive error of one. So far, distance oracles with an additive error of one that use subquadratic storage and sublinear query time were not known for such general graph classes. 

An eight-partition of a finite set of points (respectively, of a continuous mass distribution) in ℝ³ consists of three planes that divide the space into 8 octants, such that each open octant contains at most 1/8 of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in ℝ³ admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument. We prove the following variant of this result: Any mass distribution (or point set) in ℝ³ admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction. Moreover, we present an efficient algorithm for calculating an eight-partition of a set of n points in ℝ³ (with prescribed normal direction of one of the planes) in time O^*(n^{5/2}). 

It is unlikely that the discrete Fréchet distance between two curves of length n can be computed in strictly subquadratic time. We thus consider the setting where one of the curves, P, is known in advance. In particular, we wish to construct data structures (distance oracles) of near-linear size that support efficient distance queries with respect to P in sublinear time. Since there is evidence that this is impossible for query curves of length Θ(n^α), for any α > 0, we focus on query curves of (small) constant length, for which we are able to devise distance oracles with the desired bounds. We extend our tools to handle subcurves of the given curve, and even arbitrary vertex-to-vertex subcurves of a given geometric tree. That is, we construct an oracle that can quickly compute the distance between a short polygonal path (the query) and a path in the preprocessed tree between two query-specified vertices. Moreover, we define a new family of geometric graphs, t-local graphs (which strictly contains the family of geometric spanners with constant stretch), for which a similar oracle exists: we can preprocess a graph G in the family, so that, given a query segment and a pair u,v of vertices in G, one can quickly compute the smallest discrete Fréchet distance between the segment and any (u,v)-path in G. The answer is exact, if t = 1, and approximate if t > 1. 

Given a function f from the set [N] to a d-dimensional integer grid, we consider data structures that allow efficient orthogonal range searching queries in the image of f, without explicitly storing it. We show that, if f is of the form [N] → [2w]d for some w = polylog(N) and is computable in constant time, then, for any 0 < α < 1, we can obtain a data structure using Õ(N1-α/3) words of space such that, for a given d-dimensional axis-aligned box B, we can search for some x ∈ [N] such that f (x) ∈ B in time Õ(Nα). This result is obtained simply by combining integer range searching with the Fiat-Naor function inversion scheme, which was already used in data-structure problems previously. We further obtain • data structures for range counting and reporting, predecessor, selection, ranking queries, and combinations thereof, on the set f ([N]), • data structures for preimage size and preimage selection queries for a given value of f, and • data structures for selection and ranking queries on geometric quantities computed from tuples of points in d-space. These results unify and generalize previously known results on 3SUM-indexing and string searching, and are widely applicable as a black box to a variety of problems. In particular, we give a data structure for a generalized version of gapped string indexing, and show how to preprocess a set of points on an integer grid in order to efficiently compute (in sublinear time), for points contained in a given axis-aligned box, their Theil-Sen estimator, the kth largest area triangle, or the induced hyperplane that is the kth furthest from the origin. 

Let L be a set of n axis-parallel lines in R3. We are are interested in partitions of R3 by a set H of three planes such that each open cell in the arrangement A(H) is intersected by as few lines from L as possible. We study such partitions in three settings, depending on the type of splitting planes that we allow. We obtain the following results. * There are sets L of n axis-parallel lines such that, for any set H of three splitting planes, there is an open cell in A(H)  that intersects at least ⌊n/3⌋ - 1 ≈ n/3 lines. * If we require the splitting planes to be axis-parallel, then there are sets L of n axis-parallel lines such that, for any set H of three splitting planes, there is an open cell in A(H) that intersects at least (3/2) ⌊n/3⌋ - 1 ≈ (1/3 + 1/24) n lines. Furthermore, for any set L of n axis-parallel lines, there exists a set H of three axis-parallel splitting planes such that each open cell in A(H) intersects at most (7/18) n = (1/3 + 1/18) n lines. * For any set L of n axis-parallel lines, there exists a set H of three axis-parallel and mutually orthogonal splitting planes, such that each open cell in A(H) intersects at most ⌈5/12 n⌉ ≈ (1/3 + 1/12) n lines. 

Let T be a set of n planar semi-algebraic regions in R^3 of constant complexity (e.g., triangles, disks), which we call _plates_. We wish to preprocess T into a data structure so that for a query object gamma, which is also a plate, we can quickly answer various intersection queries, such as detecting whether gamma intersects any plate of T, reporting all the plates intersected by gamma, or counting them. We also consider two simpler cases of this general setting: (i) the input objects are plates and the query objects are constant-degree algebraic arcs in R^3 (arcs, for short), or (ii) the input objects are arcs and the query objects are plates in R^3. Besides being interesting in their own right, the data structures for these two special cases form the building blocks for handling the general case. By combining the polynomial-partitioning technique with additional tools from real algebraic geometry, we obtain a variety of results with different storage and query-time bounds, depending on the complexity of the input and query objects. For example, if T is a set of plates and the query objects are arcs, we obtain a data structure that uses O^*(n^(4/3)) storage (where the O^*(...) notation hides subpolynomial factors) and answers an intersection query in O^*(n^(2/3)) time. Alternatively, by increasing the storage to O^*(n^(3/2)), the query time can be decreased to O^*(n^(rho)), where rho = (2t-3)/(3(t-1)) < 2/3 and t≤3 is the number of parameters needed to represent the query arcs. 

We describe a dynamic data structure for approximate nearest neighbor (ANN) queries with respect to multiplicatively weighted distances with additive offsets. Queries take polylogarithmic time, while the cost of updates is amortized polylogarithmic. The data structure requires near-linear space and construction time. The approach works not only for the Euclidean norm, but for other norms in ℝ^d, for any fixed d. We employ our ANN data structure to construct a faster dynamic structure for approximate SINR queries, ensuring polylogarithmic query and polylogarithmic amortized update for the case of non-uniform power transmitters, thus closing a gap in previous state of the art. To obtain the latter result, we needed a data structure for dynamic approximate halfplane range counting in the plane. Since we could not find such a data structure in the literature, we also show how to dynamize one of the known static data structures.