New Similarity Measures between Polylines with Applications to Morphing and Polygon

We introduce two new related metrics, the geodesic width and the link width, for measuring the “distance” between two nonintersecting polylines in the plane. If the two polylines have n vertices in total, we present algorithms to compute the geodesic width of the two polylines in O(n2 log n) time using O(n2) space and the link width in O(n3 log n) time using O(n2) working space where n is the total number of edges of the polylines. ∗ Preliminary versions of this paper appeared in the Proceedings of the 11th Annual ACM–SIAM Symposium on Discrete Algorithms [12] and the Proceedings of the 12th Annual ACM–SIAM Symposium on Discrete Algorithms [13]. The first author did part of this research while affiliated with Stanford University. The second author was partially supported by NSF (CCR-9910633), by U.S. Army Research Office MURI Grant DAAH0496-1-007, and a grant from the Intel Corporation. The third author did part of this research while affiliated with Duke University. The fourth author was partially supported by NSF (CCR-9732220), a DARPA subcontract from HRL Laboratories, NASA Ames Research (NAG2-1325), Northrop-Grumman Corporation, and Sun Microsystems. The fifth author did this research while affiliated with Stanford University and with Compaq Computer Corporation. OF2 A. Efrat, L. J. Guibas, S. Har-Peled, J. S. B. Mitchell, and T. M. Murali Our computation of these metrics relies on two closely related combinatorial strutures: the shortest-path diagram and the link diagram of a simple polygon. The shortest-path (resp., link) diagram encodes the Euclidean (resp., link) shortest path distance between all pairs of points on the boundary of the polygon. We use these algorithms to solve two problems: • Compute a continuous transformation that “morphs” one polyline into another polyline. Our morphing strategies ensure that each point on a polyline moves as little as necessary during the morphing, that every intermediate polyline is also simple and disjoint to any other intermediate polyline, and that no portion of the polylines to be morphed is stretched or compressed by more than a user-defined parameter during the entire morphing. We present an algorithm that computes the geodesic width of the two polylines and utilizes it to construct a corresponding morphing strategy in O(n2 log n) time using O(n2) space. We also give an O(n log n) time algorithm to compute a 2-approximation of the geodesic width and a corresponding morphing scheme. • Locate a continuously moving target using a group of guards moving inside a simple polygon. The guards always determine a simple polygonal chain within the polygon, with consecutive guards along the chain being mutually visible. We compute a strategy that sweeps such a chain of guards through the polygon in order to locate a target. We compute in O(n3) time and O(n2) working space the minimum number r∗ of guards needed to sweep an n-vertex polygon. We also give an approximation algorithm, using O(n log n) time and O(n) space, to compute an integer r such that max(r − 16, 2) ≤ r∗ ≤ r and P can be swept with a chain of r guards.