Jae-Ha Lee

Visibility-based pursuit-evasion in a polygonal room with a door

Visibility-based pursuit-evasion problems are as follows: given a polygonal region, one or more searchers with visibility, and an unpredictable intruder that is arbitrarily faster than the searcher, plan the motion of the searchers so as to see the intruder. In this paper, we consider several visibility-based pursuit-evasion problems with a single searcher: l Given a simple polygon with a door (i.e., penetrable vertex) d, can a searcher find an intruder within the polygon in such a way that the intruder couldn’t make a dash for the door d? l Given a simple polygon with a door d, can a searcher make no undetected intruder remain in the polygon (that is, find the intruder or evict it from the polygon through d)? l Given a building (represented as a sequence of simple polygons joined by staircases), can the searcher find the intruder within it? For each of the three problems above, we give a characterization of the class of regions that admits a search strategy and present an O(n2)-time algorithm for constructing a search path, if one exists, for an n-sided region. Interestingly, our characterizations imply that each of the above regions searchable by a searcher with omnidirectional vision (i.e., 360’ vision) is also searchable by a searcher with two flashlights (i.e., ray visions). As a by-product, we improves the time complexity of the corridor search problem in [2], by a factor of log n. *This work was partially supported by KOSEF 98-0102-07-01-3. permission lo make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists. requires prior specific permission and/or a fee. SCG’99 Miami Beach Florida Copyright ACM 1999 I-581 13-068-6/99/06...

Simple algorithms for searching a polygon with flashlights

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SEARCHING A ROOM BY TWO GUARDS

The k-searcher is a mobile guard whose visibility is limited to k rays emanating from her position, where the direction of each ray can be changed continuously with bounded angular rotation speed. Given a polygonal region P, is it possible for the k-searcher to eventually see a mobile intruder that is arbitrarily faster than the searcher within P? We present O(n2)-time algorithms for constructing a search schedule of the 1-searcher and the 2-searcher, respectively. Our framework for the 1-searcher can be viewed as a modification of that of LaValle et al. [Proc. 16th ACM Symp. on Computational Geometry, 2000, pp. 260–269] and is naturally extended for the 2-searcher.

Characterization of Rooms Searchable by Two Guards

We consider the problem of searching for mobile intruders in a polygonal region with one door by two guards. Given a simple polygon with a door d, which is called a room , two guards start at d and walk along the boundary of to detect a mobile intruder with a laser beam between the two guards. During the walk, two guards are required to be mutually visible all the time and eventually meet at one point. We give a characterization of the class of rooms searchable by two guards and an O(n log n)-time algorithm to test if a given room admits a walk, where n is the number of the vertices in .

New competitive strategies for searching in unknown star-shaped polygons

We consider the problem of searching for mobile intruders in a polygonal region with one door by two guards. Given a simple polygon P with one door d, which is called a room (P, d), two guards start at d and walk along the boundary of P to detect a mobile intruder with a laser beam between the two guards. During the walk, two guards are required to be mutually visible all the time and eventually meet at one point. We give the characterization of the class of rooms searchable by two guards, which naturally leads to O(n log n)-time algorithm for testing the searchability of an n-sided room.

Tight analysis of a self-approaching strategy for the online kernel-search problem

We consider searching problems in robotics that a robot has to find a path to a target by traveling in an unknown starshaped polygon P. The goal is to minimize the ratio of the distance traveled by the robot to the length of the shortest start-to-target path. Let s be a starting point in P. We first present a competitive strategy to find a path from s to the closest kernel point k of P. The length of the path that the robot generates is less than 1 + 2W (< 3.829) times the distance from s to k, which improves the best previous bound 5.331 [3]. Second, given a specified target t in P, we present a competitive strategy to find a path from s to t whose length does not exceed 17 times the length of the shortest s-t path.

Directed hamiltonian packing in d-dimensional meshes and its application

The online kernel-search problem is for a mobile robot with 360° degree vision to move from a starting point to the closest kernel point within an unknown star-shaped polygon. Icking and Klein (1991, 1997) presented a simple strategy, called CAB, and showed that CAB is 5.331-competitive. In this paper we show that CAB is (π + 1)-competitive, which is a tight analysis of it.

Online Scheduling of Parallel Communications with Individual Deadlines

A digraph G with minimum in-degree d and minimum out-degree d is said to have a directed hamiltonian packing if G has d link-disjoint directed hamiltonian cycles. We show that a d-dimensional N1 × ⋯ × Nd mesh, when Ni ≥ 2d is even, has a directed hamiltonian packing, where an edge (u, v) in G is regarded as two directed links (u, v) and 〈v,u〉. As its application, we design a time-efficient all-to-all broadcasting algorithm in 3-dimensional meshes under the wormhole routing model.

Optimization Algorithms for Sweeping a Polygonal Region with Mobile Guards

We consider the online competitiveness for scheduling a set of communication jobs (best described in terms of a weighted graph where nodes denote the communication agents and edges denote communication jobs and three weights associated with each edge denote its length, release time, and deadline, respectively), where each node can only send or receive one message at a time. A job is accepted if it is scheduled without interruption in the time interval corresponding to its length between release time and deadline. We want to maximize the sum of the length of the accepted jobs. When an algorithm is not able to preempt (i.e., abort) jobs in service in order to make room for better jobs, previous lower bound shows that no algorithm can guarantee any constant competitive ratio. We examine a natural variant in which jobs can be aborted and each aborted job can be rescheduled from start (called restart). We present simple algorithms under the assumptions on job length: 2-competitive algorithm for unit jobs under the discrete model of time and (6+4√2 ≈ 11:656)-competitive algorithm for jobs of arbitrary length. These upper bounds are compensated by the lower bounds 1.5, 8-Ɛ, respectively.

Carrying Umbrellas: An Online Relocation Game on a Graph

We study the problem of sweeping a simple polygon using a chain of mobile guards. The basic question is as follows: Given a simple polygon P in the plane, is it possible for two guards to simultaneously walk along the boundary of P from one point to another point in such a way that two guards are always mutually visible and any target moving continuously inside P should eventually lie on the line segment between two guards? It is known that an O(n2)-time algorithm can decide this question. Our contribution is to present efficient algorithms for the following optimization problems: - Given an n-sided polygon, we present an O(n2 log n)-time algorithm for computing a shortest walk in which the total length of the paths that two guards traverse is minimized. - Given an n-sided polygon, we present an O(n2)-time algorithm for computing a minimum diameter walk in which the maximum distance between two guards is minimized. Finally we allow more than two guards. Here the guards should form a simple chain within the polygon such that any consecutive two guards along the chain are mutually visible and the first and last guard have to move along the boundary but others do not. - We present an O(n2)-time algorithm for computing the minimum number of guards to sweep an n-sided polygon and an O(n3)-time algorithm for computing such a schedule.