8 research outputs found
Computing a rectilinear shortest path amid splinegons in plane
We reduce the problem of computing a rectilinear shortest path between two
given points s and t in the splinegonal domain \calS to the problem of
computing a rectilinear shortest path between two points in the polygonal
domain. As part of this, we define a polygonal domain \calP from \calS and
transform a rectilinear shortest path computed in \calP to a path between s and
t amid splinegon obstacles in \calS. When \calS comprises of h pairwise
disjoint splinegons with a total of n vertices, excluding the time to compute a
rectilinear shortest path amid polygons in \calP, our reduction algorithm takes
O(n + h \lg{n}) time. For the special case of \calS comprising of concave-in
splinegons, we have devised another algorithm in which the reduction procedure
does not rely on the structures used in the algorithm to compute a rectilinear
shortest path in polygonal domain. As part of these, we have characterized few
of the properties of rectilinear shortest paths amid splinegons which could be
of independent interest
Bicriteria Rectilinear Shortest Paths among Rectilinear Obstacles in the Plane
Given a rectilinear domain P of h pairwise-disjoint rectilinear obstacles with a total of n vertices in the plane, we study the problem of computing bicriteria rectilinear shortest paths between two points s and t in P. 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 h is relatively small. For example, for computing a minimum-link shortest s-t path, the previous algorithm runs in O(n log^{3/2} n) time while the time of our new algorithm is O(n + h log^{3/2} h)
Two-point L1 shortest path queries in the plane
Let be a set of pairwise-disjoint polygonal obstacles with a total of vertices in the plane. We consider the problem of building a data structure that can quickly compute an shortest obstacle-avoiding path between any two query points and . Previously, a data structure of size was constructed in time that answers each two-point query in time, i.e., the shortest path length is reported in time and an actual path is reported in additional time, where is the number of edges of the output path. In this paper, we build a new data structure of size in time that answers each query in time. (In contrast, for the Euclidean version of this two-point query problem, the best known algorithm uses space to achieve an query time.) Further, we extend our techniques to the weighted rectilinear version in which the ``obstacles" of are rectilinear regions with ``weights" and allow paths to travel through them with weighted costs. Previously, a data structure of size was built in time that answers each query in time. Our new algorithm answers each query in time with a data structure of size that is built in time