Xuehou Tan

Sweeping simple polygons with the minimum number of chain guards

We study the problem of detecting a moving target using a group of k+1 (k is a positive integer) mobile guards inside a simple polygon. Our guards always form a simple polygonal chain within the polygon such that consecutive guards along the chain are mutually visible. In this paper, we introduce the notion of the link-k diagram of a polygon, which records the pairs of points on the polygon boundary such that the link distance between any of these pairs is at most k and a transition relation among minimum-link (=

Approximation algorithms for cutting out polygons with lines and rays

This paper studies the problem of cutting out a given polygon, drawn on a convex piece of paper, in the cheapest possible way. For the problems of cutting out convex polygons with line cuts and ray cuts, we present a 7.9-approximation algorithm and a 6-approximation algorithm, respectively. For the problem of cutting out ray-cuttable polygons, an O(log n)-approximation algorithm is given.

A unified and efficient solution to the room search problem

We study the problem of searching for a mobile intruder in a polygonal region P with a door d (called a room) by a mobile searcher. The objective is to decide whether there exists a search schedule for the searcher to detect the intruder without allowing him to exit P through d, no matter how fast he moves, and if so, generate a search schedule. A searcher is called the k-searcher if he holds k flashlights and can see only along the rays of the flashlights emanating from his position, or two guards if two endpoints of the 1-searchers flashlight move on the polygon boundary continuously. In this paper, we develop a simple, unified solution to the room search problem. The characterizations of the k-searchable and two-guard walkable rooms are all given in terms of components and deadlocks. A study on the structure of non-redundant components and deadlocks gives critical visibility events which occur in any search schedule, and a vertex of P at which our search schedule ends. Our characterizations are not only simple but also lead to efficient algorithms for all decision problems and schedule reporting problems. Particularly, we present optimal O(n) time algorithms for determining the 1-searchability and the two-guard walkability of a room, and an O(nlogn+m) time and O(n) space algorithm for generating a search schedule, if it exists, where n is the number of vertices of P and m(=

A linear-time 2-approximation algorithm for the watchman route problem for simple polygons

Given a simple polygon P of n vertices, the watchman route problem asks for a shortest (closed) route inside P such that each point in the interior of P can be seen from at least one point along the route. In this paper, we present a simple, linear-time algorithm for computing a watchman route of length at most two times that of the shortest watchman route. The best known algorithm for computing a shortest watchman route takes O (n 4 log n) time, which is too complicated to be suitable in practice. This paper also involves an optimal O(n) time algorithm for computing the set of so-called essential cuts, which are the line segments inside the polygon P such that any route visiting them is a watchman route. It solves an intriguing open problem by improving the previous O (n log n) time result, and is thus of interest in its own right.

Searching a Polygonal Region by a Boundary Searcher

This paper considers the problem of planning the motion of a searcher in a polygonal region to eventually “see” an intruder that is unpredictable and capable of moving arbitrarily fast. A searcher is called the boundary searcher if he continuously moves on the polygon boundary and can see only along the rays of the flashlights he holds at a time. We present necessary and sufficient conditions for an n-sided polygon to be searchable by a boundary searcher. Based on our characterization, the equivalence of the ability of the searchers having only one flashlight and the one of the searchers having full 360° vision is simply established, and moreover, an optimal O(n) time and space algorithm for determining the searchability of simple polygons is obtained. We also give an O(n log n + I) time algorithm for generating a search schedule if it exists, where I (<3n2) is the number of search instructions reported. Our results improve upon the previously known O(n2) time and space bounds.

Searching a Polygonal Region by Two Guards

We study the problem of searching for a mobile intruder in a polygonal region P by two guards. The objective is to decide whether there should exist a search schedule for the two guards to detect the intruder, no matter how fast the intruder moves, and if so, generate a search schedule. During the search, the two guards are required to walk on the boundary of P continuously and be mutually visible all the time. We present a characterization of the class of polygons searchable by two guards in terms of non-redundant components, and thus solve a long-standing open problem in computational geometry. Also, we give an optimal O(n) time algorithm to determine the two-guard searchability in a polygon, and an O(n log n + m) time algorithm to generate a search schedule, if it exists, where n is the number of vertices of P and m (≤ n2) is the number of search instructions reported.

Linear-Time 2-Approximation Algorithm for the Watchman Route Problem

Given a simple polygon P of n vertices, the watchman route problem asks for a shortest (closed) route inside P such that each point in the interior of P can be seen from at least one point along the route. We present a simple, linear-time algorithm for computing a watchman route of length at most 2 times that of the shortest watchman route. The best known algorithm for computing a shortest watchman route takes O(n4 log n) time, which is too complicated to be suitable in practice. This paper also involves an optimal O(n) time algorithm for computing the set of so-called essential cuts, which are the line segments inside the polygon P such that any route visiting them is a watchman route. It solves an intriguing open problem by improving the previous O(n log n) time result, and is thus of interest in its own right.

Complexity of projected images of convex subdivisions

Abstract Let S be a subdivision of R d into n convex regions. We consider the combinatorial complexity of the image of the (k - 1)-skeleton of S orthogonally projected into a k-dimensional subspace. We give an upper bound of the complexity of the projected image by reducing it to the complexity of an arrangement of polytopes. If k = d − 1, we construct a subdivision whose projected image has Ω(n⌊(3d−2)/2⌋) complexity, which is tight when d ⩽ 4. We also investigate the number of topological changes of the projected image when a three-dimensional subdivision is rotated about a line parallel to the projection plane.

Two-Guarding a Rectilinear Polygon

In this paper, we present an O(n3 log n) time and O(n) space algorithm to solve the problem of two-guarding a simple rectilinear polygon. The complexity of our algorithm is much faster than the O(n4) time and O(n2 log n) space algorithm for general simple polygons [Bel92]. It also sheds light on solving the open problem of computing the two-watchmen routes of a simple rectilinear polygon.

Searching a Circular Corridor with Two Flashlights

We consider the problem of searching for a mobile intruder in a circular corridor (a polygon with one polygonal hole) by two searchers, who hold a flashlight. Both searchers move on the outer boundary, directing their flashlights at the inner boundary. The objective is to decide whether there exists a search schedule for the searchers to detect the intruder, no matter how fast he moves. We give a characterization of the class of circular corridors, which are searchable with two flashlights. Based on our characterization, an O (n logn ) time algorithm is then presented to determine the searchability of a circular corridor with two flashlights, where n denotes the total number of vertices of the outer and inner boundaries. Moreover, a search schedule can be output in time linear in its size, if it exists. Our result gives the first efficient solution to the polygon search problem for two searchers.