We present an algorithm for determining the shortest path between a source and a destination on an arbitrary (possibly nonconvex) polyhedral surface. The path is constrained to lie on the surface, and distances are measured according to the Euclidean metric. Our algorithm runs in time O(n log n) and requires O(n2) space, where n is the number of edges of the surface. After we run our algorithm, the distance from the source to any other destination may be determined using standard techniques in time O(log n) by locating the destination in the subdivision created by the algorithm. The actual shortest path from the source to a destination can be reported in time O(k + log n), where k is the number of faces crossed by the path. The algorithm generalizes to the case of multiple source points to build the Voronoi diagram on the surface, where n is now the maximum of the number of vertices and the number of sources.

-Collision detection is of paramount importance for many applications in computer graphics and visualization. Typically, the input to a collision detection algorithm is a large number of geometric objects comprising an environment, together with a set of objects moving within the environment. In addition to determining accurately the contacts that occur between pairs of objects, one needs also to do so at real-time rates. Applications such as haptic force-feedback can require over 1000 collision queries per second.In this paper, we develop and analyze a method, based on bounding-volume hierarchies, for efficient collision detection for objects moving within highly complex environments. Our choice of bounding volume is to use a "discrete orientation polytope" (" -dop"), a convex polytope whose facets are determined by halfspaces whose outward normals come from a small fixed set of orientations. We compare a variety of methods for constructing hierarchies ("BVtrees") of bounding -dops. Further, we propose algorithms for maintaining an effective BV-tree of -dops for moving objects, as they rotate, and for performing fast collision detection using BV-trees of the moving objects and of the environment.Our algorithms have been implemented and tested. We provide experimental evidence showing that our approach yields substantially faster collision detection than previous methods.

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We show that any polygonal subdivision in the plane can be converted into an "mguillotine" subdivision whose length is at most (1 + c m) times that of the original subdivision, for a small constant c. "m-Guillotine" subdivisions have a simple recursive structure that allows one to search for the shortest of such subdivisions in polynomial time, using dynamic programming. In particular, a consequence of our main theorem is a simple polynomial-time approximation scheme for geometric instances of several network optimization problems, including the Steiner minimum spanning tree, the traveling salesperson problem (TSP), and the k-MST problem.

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In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. As a generalization of the classical Euclidean TSP, TSPN is also NP-hard. In this paper, we present new approximation results for the TSPN, including (1) a constant-factor approximation algorithm for the case of arbitrary connected neighborhoods having comparable diameters; and (2) a PTAS for the important special case of disjoint unit disk neighborhoods (or nearly disjoint, nearly-unit disks). Our methods also yield improved approximation ratios for various special classes of neighborhoods, which have previously been studied. Further, we give a linear-time O(1)-approximation algorithm for the case of neighborhoods that are (infinite) straight lines.Comment: Manuscript that was published in J. Algorithms (2003), with brief clarifying remarks added (2014

AND CHRISTOSH. PAPADIMITRIOU b'ni~'er.sit] of Cul@wu at San Dw,w, La Jollu, Ca[lfomu Abstract. The problem of determining shortest paths through a weighted planar polygonal subdivision with n vertices 1s considered. Distances are measured according to a weighted Euchdean metric: The length of a path 1sdefined to be the weighted sum of (Euchdean) lengths of the subpaths wlthm each region. An algorithm that constructs a (restricted) "shortest path map" with respect to a glt en source point N presented. The output is a partitioning of each edge of the subdlvmon mto intervals of c-ophmahty, allowing an~-optimal path to be traced from the source to any query point along any edge. The algorithm runs m worst-case time O(ES) and requmes O(E) space. where E is the number of "events" in the algorithm and S is the time it takes to run a numencal search procedure. In the worst ewe. E is bounded above by O(nq) (and we gwe an Q(ni) lower bound). but It 1s hkel} that h" will be much smaller m practice. We also show that S 1sbounded by 0(n4L ), where L 1s the preclslon of the problem instance (including the number of bits in the user-specified tolerance c). Agdm, the \alue of S should be much smaller m practice. The algorithm applies the "continuous DIJ kstra" paradigm and explolts the fact that shortest paths obey Snell's Law of RefractIon at region boundanes. a local optlrnahty property of shortest paths that is well known from the analogous optics model. The algorithm generalizes to the multi-source case to compute Voronol diagrams.