Efficient construction of low weighted bounded degree planar spanner

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 = ρ(α) = max{2 , π sin α 2 + 1} · Cdel, such that the degree of each node is bounded from above by 19 + d 2π α e, where 0 < α < π/2 is an adjustable parameter. Here Cdel is the spanning ratio of the Delaunay triangulation, which is at most 4 √ 3 9 π. We also show, by applying the greedy method in 14, how to construct a low weighted bounded degree planar spanner with spanning ratio ρ(α)2(1 + ) and the same degree bound, where  is any positive real constant. Here, a structure is called low weighted if its total edge length is proportional to the total edge length of the Euclidean minimum spanning tree of V . 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. Previously, only centralized method 6 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(n2) communications in the worst case. Our method can be converted to a localized algorithm where the total number of messages sent by all nodes is at most O(n).