Balancing degree, diameter and weight in Euclidean spanners

Abstract

In a seminal STOC'95 paper, Arya et al. [4] devised a construction that for any set S of n points in ℝd and any ε > 0, provides a (1 + ε)-spanner with diameter O(logn), weight O(log2 n)w(MST(S)), and constant maximum degree. Another construction of [4] provides a (1 + ε)-spanner with O(n) edges and diameter α(n), where α stands for the inverse-Ackermann function. Das and Narasimhan [12] devised a construction with constant maximum degree and weight O(w(MST(S))), but whose diameter may be arbitrarily large. In another construction by Arya et al. [4] there is diameter O(logn) and weight O(logn) w(MST(S)), but it may have arbitrarily large maximum degree. These constructions fail to address situations in which we are prepared to compromise on one of the parameters, but cannot afford it to be arbitrarily large. In this paper we devise a novel unified construction that trades between maximum degree, diameter and weight gracefully. For a positive integer k, our construction provides a (1 + ε)-spanner with maximum degree O(k), diameter O(logk n + α(k)), weight O(k logk n logn) w(MST(S)), and O(n) edges. For k = O(1) this gives rise to maximum degree O(1), diameter O(logn) and weight O(log2 n) w(MST(S)), which is one of the aforementioned results of [4]. For k = n 1/α(n) this gives rise to diameter O(α(n)), weight O(n1/α(n) (logn) α(n)) w(MST(S)) and maximum degree O(n 1/α(n)). In the corresponding result from [4] the spanner has the same number of edges and diameter, but its weight and degree may be arbitrarily large. Our construction also provides a similar tradeoff in the complementary range of parameters, i.e., when the weight should be smaller than log2 n, but the diameter is allowed to grow beyond logn. © 2010 Springer-Verlag.