Efficient Construction of Low Weight Bounded Degree Planar Spanner

Abstract

Given a set V of n points in a two-dimensional plane, we give an O(n log n)-time centralized algorithm that constructs a planar t-spanner for V, for t ≤ max$\{ \frac{\pi } {2},\pi \sin \frac{\alpha } {2} + 1\}$ · C _del , such that the degree of each node is bounded from above by $19 + [\frac{{2\pi }} {\alpha }]$ , and the total edge length is proportional to the weight of the minimum spanning tree of V, where 0 < α < π / 2 is an adjustable parameter. Here C _del is the spanning ratio of the Delaunay triangulation, which is at most $\frac{{4\sqrt 3 }} {9}\pi$. Moreover, we show that our method can be extended to construct a planar bounded degree spanner for unit disk graphs with the adjustable parameter α satisfying 0 < α < π / 3. This method can be converted to a localized algorithm where the total number of messages sent by all nodes is at most O(n) (under broadcasting communication model). These constants are all worst case constants due to our proofs. Previously, only centralized method 1 of constructing bounded degree planar spanner is known, with degree bound 27 and spanning ratio t ≃ 10.02. The distributed implementation of this centralized method takes O(n ²) communications in the worst case.