BALANCING DEGREE, DIAMETER, AND WEIGHT IN EUCLIDEAN SPANNERS

In a seminal paper from 1995, Arya et al. [Euclidean spanners: Short, thin, and lanky, in Proceedings of the 27th Annual ACM Symposium on Theory of Computing, ACM, New York, 1995, pp. 489-498] devised a construction that, for any set S of n points in R-d and any c > 0, provides a (1 + is an element of)-spanner with diameter O(log n), weight O(log(2) n) . w(MST(S)), and constant maximum degree. Another construction from the same work provides a (1 + is an element of)-spanner with O(n) edges and diameter O(alpha(n)), where a stands for the inverse Ackermann function. There are also a few other known constructions of (1 + is an element of)-spanners. Das and Narasimhan [A fast algorithm for constructing sparse Euclidean spanners, in Proceedings of the 10th Annual ACM Symposium on Computational Geometry (SOCG), ACM, New York, 1994, pp. 132-139] devised a construction with constant maximum degree and weight O(w(MST(S))), but the diameter may be arbitrarily large. In another construction by Arya et al., there is diameter O(log n) and weight O(log n) . w(MST(S)), but this construction may have arbitrarily large maximum degree. While these constructions address some important practical scenarios, they fail to address situations in which we are prepared to compromise on one of the parameters but cannot afford for this parameter to be arbitrarily large. In this paper we devise a novel unified construction that trades gracefully among the maximum degree, diameter, and weight. For a positive integer k our construction provides a (1 + is an element of)-spanner with maximum degree O(k), diameter O(log(k) n + alpha(k)), weight O(k . log(k) n . log n) . w(MST(S)), and O(n) edges. Note that for k = O(1) this gives rise to maximum degree O(1), diameter O(log n), and weight O(log(2) n) . w(MST(S)), which is one of the aforementioned results of Arya et al. For k = n(1/alpha)(n) this gives rise to diameter O(alpha(n)), weight O(n(1/alpha)(n) . log n . alpha(n)) . w(MST(S)), and maximu