CONSTRUCTING DEGREE-3 SPANNERS WITH OTHER SPARSENESS PROPERTIES

Constructing Degree-3 Spanners with Other Sparseness Properties

Let V be any set of n points in k-dimensional Euclidean space. A subgraph of the complete Euclidean graph is a t-spanner if for any ~ and ~ in V, the length of the shortest path from u to v in the spanner is at most t times d(~, ~). We show that for any 5 > 1, there exists a polynomial-time constructihle tspanner (where ~ is a constant that depends only on 5 and k) with the following properties...

Towards Plane Spanners of Degree 3

Let S be a finite set of points in the plane that are in convex position. We present an algorithm that constructs a plane 3+4π 3 -spanner of S whose vertex degree is at most 3. Let Λ be the vertex set of a finite non-uniform rectangular lattice in the plane. We present an algorithm that constructs a plane 3 √ 2-spanner for Λ whose vertex degree is at most 3. For points that are in the plane and...

Lattice Spanners of Low Degree

Let δ0(P, k) denote the degree k dilation of a point set P in the domain of plane geometric spanners. If Λ is the infinite square lattice, it is shown that 1+ √ 2 ≤ δ0(Λ, 3) ≤ (3+2 √ 2) 5−1/2 = 2.6065 . . . and δ0(Λ, 4) = √ 2. If Λ is the infinite hexagonal lattice, it is shown that δ0(Λ, 3) = 1 + √ 3 and δ0(Λ, 4) = 2. All our constructions are planar lattice tilings constrained to degree 3 or 4.

Bounded Degree Planar Geometric Spanners

Given a set P of n points in the plane, we show how to compute in O(n logn) time a subgraph of their Delaunay triangulation that has maximum degree 7 and is a strong planar t-spanner of P with t = (1 + √ 2) ∗ δ, where δ is the spanning ratio of the Delaunay triangulation. Furthermore, given a Delaunay triangulation, we show a distributed algorithm that computes the same bounded degree planar sp...

Spanners of bounded degree graphs