Spanners for geometric intersection graphs with applications

Lecture Notes in Computer Science

Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R k , a (1 + ǫ)-spanner with O(nǫ −k+1) edges is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.

Discrete & Computational Geometry, 1994

Given an undirected edge-weighted graph G = (V, E), a subgraph G' = (IT, E') is a t-spanner of G if, for every u, v ~ V, the weighted distance between u and v in G' is at most t times the weighted distance between u and v in G. We consider the problem of approximating the distances among points of a Euclidean metric space: given a finite set V of points in ~a, we want to construct a sparse t-spanner of the complete weighted graph induced by V. The weight of an edge in these graphs is the Euclidean distance between the endpoints of the edge. We show by a simple greedy argument that, for any t > 1 and any V c R a, a t-spanner G of V exists such that G has degree bounded by a function of d and r The analysis of our bounded degree spanners improves over previously known upper bounds on the minimum number of edges of Euclidean t-spanners, even compared with spanners of bounded average degree. Our results answer two open problems, one proposed by Vaidya and the other by Keil and Gutwin. The main result of the paper concerns the case of dimension d = 2. It is fairly easy to see that, for some t (t > 7.6), t-spanners of maximum degree 6 exist for any set of points in the Euclidean plane, but it was not known that degree 5 would suffice. We prove that for some (fixed) t, t-spanners of degree 5 exist for any set of points in the plane. We do not know if 5 is the best possible upper bound on the degree. * This research was supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnol6gico, Proc 203039/87.4 (Brazil). 214 J. Soares shortest path between x and y. We say that a subgraph G' = (V, E') (with the same weights on E') is a t-spanner of G if, for every x, y 6 V, dG,(x, y) < t" da(x, y). The number t is a measure of how well G' approximates G with respect to the distances. The construction of t-spanners has received recent attention in several works: [2], [3], [5], [8], [9], [11], and [18], among others. Given a set V ~ •a the complete Euclidean graph on V is the complete graph on V where each edge weight is the Euclidean distance I[x-Y]I. In this paper we consider the problem of constructing bounded degree spanners of complete Euclidean graphs. For brevity we write t-spanner of V instead of t-spanner of the complete Euclidean graph on V. Let A(G) denote the maximum degree of a graph G. Dobkin et al. [5] mention that Feder and others had shown that, for some fixed t and for any set V of points in the Euclidean plane, a t-spanner G of V exists such that A(G) < 7. Then they ask what would be the minimum A for which such a result is possible? This paper has a partial answer to this question. Our main result (Section 4) is that, for some fixed t, t-spanners with A < 5 exist. Nisan [10] has proved the same for A < 6. Section 2 contains the basic algorithm used to construct bounded degree t-spanners. Although the algorithm has been used before by Althrfer et al. [1] and Soares [16] to construct t-spanners for arbitrary graphs, it was not known that the algorithm also constructs bounded degree spanners for complete Euclidean graphs. Section 3 contains a brief analysis of the problem when V is in d-dimensional Euclidean space. We show that, for any t > 1 and any V c ~d, a t-spanner G of V exists where A(G) is bounded by a function that depends only on d and t. This answers a question proposed by Keil and Gutwin in [8]. This bound on the maximum degree implies an improvement on the previously known upper bounds on the number of edges sufficient to build Euclidean spanners. Then we show that, for each dimension d, the least A(G) for which our algorithm constructs Od(1)spanners coincides with the kissing number in dimension d. (Od(1) denotes some function of d, i.e., a constant for each d.) Section 4 contains our main result, the construction of O(1)-spanners of degree 5 for any set of points in the Euclidean plane.

Data & Knowledge Engineering, 2007

The problem of Proximity Searching in Metric Spaces consists in finding the elements of a set which are close to a given query under some similarity criterion. In this paper we present a new methodology to solve this problem, which uses a t-spanner G′(V, E) as the representation of the metric database. A t-spanner is a subgraph G′(V, E) of a graph G (V, A), such that E⊆ A and G′ approximates the shortest path costs over G within a precision factor t. Our key idea is to regard the t-spanner as an approximation to the complete graph ...

We present an approximate distance oracle for a point set S with n points and doubling dimension λ. For every ε > 0, the oracle supports (1 + ε)-approximate distance queries in (universal) constant time, occupies space [ε −O(λ) + 2 O(λ log λ) ]n, and can be constructed in [2 O(λ) log 3 n + ε −O(λ) + 2 O(λ log λ) ]n expected time. This improves upon the best previously known constructions, presented by . Furthermore, the oracle can be made fully dynamic with expected O(1) query time and only 2 O(λ) log n + ε −O(λ) + 2 O(λ log λ) update time. This is the first fully dynamic (1 + ε)-distance oracle.

Lecture Notes in Computer Science, 2002

A t-spanner, a subgraph that approximates graph distances within a precision factor t, is a well known concept in graph theory. In this paper we use it in a novel way, namely as a data structure for searching metric spaces. The key idea is to consider the t-spanner as an approximation of the complete graph of distances among the objects, and use it as a compact device to simulate the large matrix of distances required by successful search algorithms like AESA ]. The t-spanner provides a time-space tradeoff where full AESA is just one extreme. We show that the resulting algorithm is competitive against current approaches, e.g., 1.5 times the time cost of AESA using only 3.21% of its space requirement, in a metric space of strings; and 1.09 times the time cost of AESA using only 3.83 % of its space requirement, in a metric space of documents. We also show that t-spanners provide better space-time tradeoffs than classical alternatives such as pivot-based indexes. Furthermore, we show that the concept of t-spanners has potential for large improvements.

Data & Knowledge Engineering, 2007

The problem of Proximity Searching in Metric Spaces consists in finding the elements of a set which are close to a given query under some similarity criterion. In this paper we present a new methodology to solve this problem, which uses a t-spanner G ′ (V, E) as the representation of the metric database. A t-spanner is a subgraph G ′ (V, E) of a graph G(V, A), such that E ⊆ A and G ′ approximates the shortest path costs over G within a precision factor t.

Lecture Notes in Computer Science

Let G = (V, E) be an weighted undirected graph on n vertices and m edges, and let d G be its shortest path metric. We present two simple deterministic algorithms for approximating allpairs shortest paths in G. Our first algorithm runs in Õ(n 2 ) time, and for any u, v ∈ V reports distance no greater than 2d G (u, v)+h(u, v). Here, h(u, v) is the largest edge weight on a shortest path between u and v. The previous algorithm, due to Baswana and Kavitha that achieved the same result was randomized. Our second algorithm for the all-pairs shortest path problem uses Boolean matrix multiplications and for any u, v ∈ V reports distance no greater than (1+ǫ)d G (u, v)+2h(u, v). The currently best known algorithm for Boolean matrix multiplication yields an O(n 2.24+o(1) ǫ -3 log(nǫ -1 )) time bound for this algorithm. The previously best known result of Elkin with a similar multiplicative factor had a much bigger additive error term. We also consider approximating the diameter and the radius of a graph. For the problem of estimating the radius, we present an almost 3/2-approximation algorithm which runs in Õ(m √ n + n 2 ) time. Aingworth, Chekuri, Indyk, and Motwani used a similar approach and obtained analogous results for diameter approximation. Additionally, we show that if the graph has a small separator decomposition a 3/2-approximation of both the diameter and the radius can be obtained more efficiently.

2007

We present an experimental evaluation of an approximate distance oracle recently suggested by Thorup [1] for undirected planar graphs. The oracle uses the existence of graph separators for planar graphs, discovered by Lipton and Tarjan [2], in order to divide the graph into smaller subgraphs. For a planar graph with n nodes, the algorithmic variant considered uses O(n(log n) 3 / ) preprocessing time and O(n(log n) 2 / ) space to answer factor (1 + ) distance queries in O((log n) 2 / ) time. By performing experiments on randomly generated planar graphs and on planar graphs derived from real world road networks, we investigate some key characteristics of the oracle, such as preprocessing time, query time, precision, and characteristics related to the underlying data structure, including space consumption. For graphs with one million nodes, the average query time is less than 20μs.

Journal of the ACM ( …, 1997

A collection of n balls in d dimensions forms a k-ply system if no point in the space is covered by more than k balls. We show that for every k-ply system ⌫, there is a sphere S that intersects at most O(k 1/d n 1Ϫ1/d) balls of ⌫ and divides the remainder of ⌫ into two parts: those in the interior and those in the exterior of the sphere S, respectively, so that the larger part contains at most (1 Ϫ 1/(d ϩ 2))n balls. This bound of O(k 1/d n 1Ϫ1/d) is the best possible in both n and k. We also present a simple randomized algorithm to find such a sphere in O(n) time. Our result implies that every k-nearest neighbor graphs of n points in d dimensions has a separator of size O(k 1/d n 1Ϫ1/d). In conjunction with a result of Koebe that every triangulated planar graph is isomorphic to the intersection graph of a disk-packing, our result not only gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. The separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space.

2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06), 2006

Let G = (V, E) be a weighted undirected graph with |V | = n and |E| = m. An estimateδ (u, v) of the distance δ (u, v)