Given a set V of n points in R d and a real constant t > 1, we present the first O(n log n)-time algorithm to compute a geometric t-spanner on V . A geometric t-spanner on V is a connected graph G = (V, E) with edge weights equal to the Euclidean distances between the endpoints, and with the property that, for all u, v ∈ V , the distance between u and v in G is at most t times the Euclidean distance between u and v. The spanner output by the algorithm has O(n) edges and weight O(1) · wt(MST ), and its degree is bounded by a constant. Introduction.Complete graphs represent ideal communication networks, but they are expensive to build; sparse spanners represent low-cost alternatives. The weight of the spanner network is a measure of its sparseness; other sparseness measures include the number of edges, the maximum degree, and the number of Steiner points. Spanners for complete Euclidean graphs as well as for arbitrary weighted graphs find applications in robotics, network topology design, distributed systems, design of parallel machines, and many other areas and have been a subject of considerable research [1,2,4,8,11].Consider a set V of n points in R d , where the dimension d is a constant. A network on V can be modeled as an undirected graph G with vertex set V and with edges e = (u, v) of weight wt(e). A Euclidean network is a geometric network where the weight of the edge e = (u, v) is equal to the Euclidean distance d(u, v) between its two endpoints u and v. Let t > 1 be a real number. We say that G is a t-spanner for V if, for each pair of points u, v ∈ V , there exists a path in G of weight at most t times the Euclidean distance between u and v. A sparse t-spanner is defined to be a t-spanner of size (number of edges) O(n) and weight (sum of edge weights) O(1) · wt(MST ), where wt(MST ) is the total weight of a minimal spanning tree. Given a geometric network G = (V, E), a (generic) weight function wt defined on its edges, and two vertices u, v ∈ V , we let D {G,wt} (u, v) denote the weight of the shortest path from u to v in G for the weight function wt.The problem of constructing spanners has been investigated by many researchers. Levcopoulos and Lingas [10] presented an O(n log n)-time algorithm that produced a sparse t-spanner for the two-dimensional case. It works by taking any t-spanner which has the form of a (possibly partial) triangulation and achieving almost the same t as that triangulation. However, the problem gets much more difficult in higher *

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Given an arbitrary real constant ε > 0, and a geometric graph G in d-dimensional Euclidean space with n points, O(n) edges, and constant dilation, our main result is a data structure that answers (1 + ε)-approximate shortest-path-length queries in constant time. The data structure can be constructed in O(n log n) time using O(n log n) space. This represents the first data structure that answers (1 + ε)-approximate shortest-path queries in constant time, and hence functions as an approximate distance oracle. The data structure is also applied to several other problems. In particular, we also show that approximate shortest-path queries between vertices in a planar polygonal domain with "rounded" obstacles can be answered in constant time. Other applications include query versions of closest-pair problems, and the efficient computation of the approximate dilations of geometric graphs. Finally, we show how to extend the main result to answer (1 + ε)-approximate shortest-path-length queries in constant time for geometric spanner graphs with m = ω(n) edges. The resulting data structure can be constructed in O(m + n log n) time using O(n log n) space.

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