Quickest Visibility Queries in Polygonal Domains

Let s be a point in a polygonal domain P of h-1 holes and n vertices. We consider the following quickest visibility query problem. Given a query point q in P, the goal is to find a shortest path in P to move from s to see q as quickly as possible. Previously, Arkin et al. (SoCG 2015) built a data structure of size O(n^2 2^alpha(n) log n) that can answer each query in O(K log^2 n) time, where alpha(n) is the inverse Ackermann function and K is the size of the visibility polygon of q in P (and K can be Theta(n) in the worst case). In this paper, we present a new data structure of size O(n log h + h^2) that can answer each query in O(h log h log n) time. Our result improves the previous work when h is relatively small. In particular, if h is a constant, then our result even matches the best result for the simple polygon case (i.e., h = 1), which is optimal. As a by-product, we also have a new algorithm for the following shortest-path-to-segment query problem. Given a query line segment tau in P, the query seeks a shortest path from s to all points of tau. Previously, Arkin et al. gave a data structure of size O(n^2 2^alpha(n) log n) that can answer each query in O(log^2 n) time, and another data structure of size O(n^3 log n) with O(log n) query time. We present a data structure of size O(n) with query time O(h log n/h), which favors small values of h and is optimal when h = O(1)

Query-points visibility constraint minimum link paths in simple polygons

We study the query version of constrained minimum link paths between two points inside a simple polygon $P$ with $n$ vertices such that there is at least one point on the path, visible from a query point. The method is based on partitioning $P$ into a number of faces of equal link distance from a point, called a link-based shortest path map (SPM). Initially, we solve this problem for two given points $s$, $t$ and a query point $q$. Then, the proposed solution is extended to a general case for three arbitrary query points $s$, $t$ and $q$. In the former, we propose an algorithm with $O(n)$ preprocessing time. Extending this approach for the latter case, we develop an algorithm with $O(n^3)$ preprocessing time. The link distance of a $q$-$visible$ path between $s$, $t$ as well as the path are provided in time $O(\log n)$ and $O(m+\log n)$, respectively, for the above two cases, where $m$ is the number of links

Visualizing Quickest Visibility Maps

Non peer reviewe

On Romeo and Juliet Problems: Minimizing Distance-to-Sight

We introduce a variant of the watchman route problem, which we call the quickest pair-visibility problem. Given two persons standing at points s and t in a simple polygon P with no holes, we want to minimize the distance these persons travel in order to see each other in P. We solve two variants of this problem, one minimizing the longer distance the two persons travel (min-max) and one minimizing the total travel distance (min-sum), optimally in linear time. We also consider a query version of this problem for the min-max variant. We can preprocess a simple n-gon in linear time so that the minimum of the longer distance the two persons travel can be computed in O(log^2 n) time for any two query positions where the two persons lie

The Visibility Center of a Simple Polygon

We introduce the visibility center of a set of points inside a polygon - a point c_V such that the maximum geodesic distance from c_V to see any point in the set is minimized. For a simple polygon of n vertices and a set of m points inside it, we give an O((n+m) log (n+m)) time algorithm to find the visibility center. We find the visibility center of all points in a simple polygon in O(n log n) time. Our algorithm reduces the visibility center problem to the problem of finding the geodesic center of a set of half-polygons inside a polygon, which is of independent interest. We give an O((n+k) log (n+k)) time algorithm for this problem, where k is the number of half-polygons

Geometric Secluded Paths and Planar Satisfiability

We consider paths with low exposure to a 2D polygonal domain, i.e., paths which are seen as little as possible; we differentiate between integral exposure (when we care about how long the path sees every point of the domain) and 0/1 exposure (just counting whether a point is seen by the path or not). For the integral exposure, we give a PTAS for finding the minimum-exposure path between two given points in the domain; for the 0/1 version, we prove that in a simple polygon the shortest path has the minimum exposure, while in domains with holes the problem becomes NP-hard. We also highlight connections of the problem to minimum satisfiability and settle hardness of variants of planar min- and max-SAT

Algorithms for Optimizing Search Schedules in a Polygon

In the area of motion planning, considerable work has been done on guarding problems, where "guards", modelled as points, must guard a polygonal space from "intruders". Different variants of this problem involve varying a number of factors. The guards performing the search may vary in terms of their number, their mobility, and their range of vision. The model of intruders may or may not allow them to move. The polygon being searched may have a specified starting point, a specified ending point, or neither of these. The typical question asked about one of these problems is whether or not certain polygons can be searched under a particular guarding paradigm defined by the types of guards and intruders. In this thesis, we focus on two cases of a chain of guards searching a room (polygon with a specific starting point) for mobile intruders. The intruders must never be allowed to escape through the door undetected. In the case of the two guard problem, the guards must start at the door point and move in opposite directions along the boundary of the polygon, never crossing the door point. At all times, the guards must be able to see each other. The search is complete once both guards occupy the same spot elsewhere on the polygon. In the case of a chain of three guards, consecutive guards in the chain must always be visible. Again, the search starts at the door point, and the outer guards of the chain must move from the door in opposite directions. These outer guards must always remain on the boundary of the polygon. The search is complete once the chain lies entirely on a portion of the polygon boundary not containing the door point. Determining whether a polygon can be searched is a problem in the area of visibility in polygons; further to that, our work is related to the area of planning algorithms. We look for ways to find optimal schedules that minimize the distance or time required to complete the search. This is done by finding shortest paths in visibility diagrams that indicate valid positions for the guards. In the case of the two-guard room search, we are able to find the shortest distance schedule and the quickest schedule. The shortest distance schedule is found in O(n^2) time by solving an L_1 shortest path problem among curved obstacles in two dimensions. The quickest search schedule is found in O(n^4) time by solving an L_infinity shortest path problem among curved obstacles in two dimensions. For the chain of three guards, a search schedule minimizing the total distance travelled by the outer guards is found in O(n^6) time by solving an L_1 shortest path problem among curved obstacles in two dimensions

Engineering Algorithms for Route Planning in Multimodal Transportation Networks

Practical algorithms for route planning in transportation networks are a showpiece of successful Algorithm Engineering. This has produced many speedup techniques, varying in preprocessing time, space, query performance, simplicity, and ease of implementation. This thesis explores solutions to more realistic scenarios, taking into account, e.g., traffic, user preferences, public transit schedules, and the options offered by the many modalities of modern transportation networks