Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain

We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of $n$ vertices and $h$ holes. We introduce a \emph{graph of oriented distances} to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in $\min \{\,O(n^\omega), O(n^2 + nh \log h + \chi^2)\,\}$ time, where $\omega<2.373$ denotes the matrix multiplication exponent and $\chi\in \Omega(n)\cap O(n^2)$ is the number of edges of the graph of oriented distances. We also provide a faster algorithm for computing the diameter that runs in $O(n^2 \log n)$ time

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)

An O(n^{5/2} log n) Algorithm for the Rectilinear Minimum Link-Distance Problem in Three Dimensions (Extended Abstract)

In this paper we consider the Rectilinear Minimum Link-Distance Problem in Three Dimensions. The problem is well studied in two dimensions, but is relatively unexplored in higher dimensions. We solve the problem in O(B n log n) time, where n is the number of corners among all obstacles, and B is the size of a BSP decomposition of the space containing the obstacles. It has been shown that in the worst case B = Theta(n^{3/2}), giving us an overall worst case time of O(n^{5/2} log n). Previously known algorithms have had worst-case running times of Omega(n^3)

Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain

We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of n vertices and h holes. We introduce a graph of oriented distances to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in O(min(n^omega, n^2 + nh log h + chi^2)) time, where omega<2.373 denotes the matrix multiplication exponent and chi in Omega(n) cap O(n^2) is the number of edges of the graph of oriented distances. We also provide an alternative algorithm for computing the diameter that runs in O(n^2 log n) time

Shortest path queries in rectilinear worlds

Abstract In this paper, a data structure is given for two and higher dimensional shortest path queries. For a set of n axis-parallel rectangles in the plane, or boxes in d-space, and a fixed target, it is possible with this structure to find a shortest rectilinear path avoiding all rectangles or boxes from any point to this target. Alternatively, it is possible to find the length of the path. The metric considered is a generalization of the Ll-metric and the link metric, where the length of a path is its L1-Iength plus some (fixed) constant times the number of turns on the path. The data structure has size 0« n log n )d-l), and a query takes O(logd-l n) time (plus the output size if the path must be reported). As a byproduct, a relatively simple solution to the single shot problem is obtained; the shortest path between two given points can be computed in time O(ndlogn) for d ~ 3, and in time 0(n 2 ) in the plane