Geometric Spanners for Points Inside a Polygonal Domain

Let P be a set of n points inside a polygonal domain D. A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set P with respect to the geodesic distance function π where for any two points p and q, π(p, q) is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., h = 0), we construct a ( √ 10 + )-spanner that has O(n log2 n) edges. For a case where there are h holes, our construction gives a (5 + )-spanner with the size of O(n √ h log2 n). Moreover, we study t-spanners for the visibility graph of P (V G(P), for short) with respect to a hole-free polygonal domain D. The graph V G(P) is not necessarily a complete graph or even connected. In this case, we propose an algorithm that constructs a (3 + )-spanner of size O(n4/3+δ). In addition, we show that there is a set P of n points such that any (3− ε)-spanner of V G(P) must contain Ω(n2) edges. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems