Approximating Euclidean distances by small degree graphs (original) (raw)

1994, Discrete & Computational Geometry

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.