Sparse Euclidean Spanners with Tiny Diameter

• Authors: Shay Solomon
• Venue: ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
• Year: 2013

In STOC’95, Arya et al. [1995] showed that for any set of n points in Rd, a (1 + ε)-spanner with diameter at most 2 (respectively, 3) and O(n log n) edges (respectively, O(n log log n) edges) can be built in O(n log n) time. Moreover, it was shown in Arya et al. [1995] and Narasimhan and Smid [2007] that for any k ≥ 4, one can build in O(n(log n)2kαk(n)) time a (1 + ε)-spanner with diameter at most 2k and O(n2kαk(n)) edges. The function αk is the inverse of a certain function at the k/2 th level of the primitive recursive hierarchy, where α0(n) = n/2 , α1(n) = √n , α2(n) = log n , α3(n) = log log n , α4(n) = log* n, α5(n) = 12 log* n , ..., etc. It is also known [Narasimhan and Smid 2007] that if one allows quadratic time, then these bounds can be improved. Specifically, for any k ≥ 4, a (1 + ε)-spanner with diameter at most k and O(nkαk(n)) edges can be constructed in O(n2) time [Narasimhan and Smid 2007]. A major open question in this area is whether one can construct within time O(n log n + nkαk(n)) a (1 + ε)-spanner with diameter at most k and O(nkαk(n)) edges. In this article, we answer this question in the affirmative. Moreover, in fact, we provide a stronger result. Specifically, we show that for any k ≥ 4, a (1 + ε)-spanner with diameter at most k and O(nαk(n)) edges can be built in optimal time O(n log n).