Given a rectilinear domain @math of @math pairwise-disjoint rectilinear obstacles with a total of @math vertices in the plane, we study the problem of computing bicriteria rectilinear shortest paths between two points @math and @math in @math . Three types of bicriteria rectilinear paths are considered: minimum-link shortest paths, shortest minimum-link paths, and minimum-cost paths where the cost of a path is a non-decreasing function of both the number of edges and the length of the path. The one-point and two-point path queries are also considered. Algorithms for these problems have been given previously. Our contributions are threefold. First, we find a critical error in all previous algorithms. Second, we correct the error in a not-so-trivial way. Third, we further improve the algorithms so that they are even faster than the previous (incorrect) algorithms when @math is relatively small. For example, for the minimum-link shortest paths, we obtain the following results. Our algorithm computes a minimum-link shortest @math - @math path in @math time. For the one-point queries, we build a data structure of size @math in @math time for a source point @math , such that given any query point @math , a minimum-link shortest @math - @math path can be determined in @math time. For the two-point queries, with @math time and space preprocessing, a minimum-link shortest @math - @math path can be determined in @math time for any two query points @math and @math ; alternatively, with @math time and @math space preprocessing, we can answer each two-point query in @math time.
To find a minimum-link path between two points @math and @math in an arbitrary polygonal domain @math , Mitchell @cite_8 gave an @math time algorithm, where @math is the inverse of Ackermann's function and @math is the size of the visibility graph of @math and @math in the worst case. The one-point query problem was also studied in @cite_8 .
@cite_8
2020449301
Given a polygon @math , for two points @math and @math contained in the polygon, their is the length of the shortest @math -path within @math . A of radius @math centered at a point @math is the set of points in @math whose geodesic distance to @math is at most @math . We present a polynomial time @math -approximation algorithm for finding a densest geodesic unit disk packing in @math . Allowing arbitrary radii but constraining the number of disks to be @math , we present a @math -approximation algorithm for finding a packing in @math with @math geodesic disks whose minimum radius is maximized. We then turn our focus on of @math and present a @math -approximation algorithm for covering @math with @math geodesic disks whose maximal radius is minimized. Furthermore, we show that all these problems are @math -hard in polygons with holes. Lastly, we present a polynomial time exact algorithm which covers a polygon with two geodesic disks of minimum maximal radius. Let @math be a simple polygon of @math vertices. We consider two-point @math shortest path queries in @math . We build a data structure of @math size in @math time such that given any two query points @math and @math , the length of an @math shortest path from @math to @math in @math can be computed in @math time, or in @math time if both @math and @math are vertices of @math , and an actual shortest path can be output in additional linear time in the number of edges of the path. To achieve the result, we propose a mountain decomposition of simple polygons, which may be interesting in its own right. Most importantly, our approach is much simpler than the previous work on this problem. Multiple objects in the plane are called pseudodisks if the boundaries of any two of them intersect transversely at most twice. Given a set of @math (possibly intersecting) convex pseudodisks of @math complexity each and two points @math and @math in the plane, we present an efficient algorithm for computing a shortest @math -to- @math path avoiding the pseudodisks. After the union of the pseudodisks is computed, which can be done in @math randomized time or @math deterministic time, our algorithm runs in @math deterministic time, where @math is the size of the extended visibility graph of the union of the pseudodisks. Note that @math in the worst case. In over two decades, the previously best algorithms for this problem have not improved on the bound of @math time, even when all the pseudodisks are pairwise disjoint disks. Our technique is also applicable to a motion planning problem of finding a shortest path to translate a convex object in the plane from one location to anot... We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math .