## Visibility-Based Pursuit-Evasion in a Polygonal Environment (1997)

Venue: | International Journal of Computational Geometry and Applications |

Citations: | 85 - 25 self |

### BibTeX

@INPROCEEDINGS{Guibas97visibility-basedpursuit-evasion,

author = {Leonidas J. Guibas and Jean-claude Latombe and Steven M. Lavalle and David Lin and Rajeev Motwani},

title = {Visibility-Based Pursuit-Evasion in a Polygonal Environment},

booktitle = {International Journal of Computational Geometry and Applications},

year = {1997},

pages = {17--30},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper addresses the problem of planning the motion of one or more pursuers in a polygonal environment to eventually "see" an evader that is unpredictable, has unknown initial position, and is capable of moving arbitrarily fast. This problem was first introduced by Suzuki and Yamashita. Our study of this problem is motivated in part by robotics applications, such as surveillance with a mobile robot equipped with a camera that must find a moving target in a cluttered workspace. A few bounds are introduced, and a complete algorithm is presented for computing a successful motion strategy for a single pursuer. For simplyconnected free spaces, it is shown that the minimum number of pursuers required is \Theta(lg n). For multiply-connected free spaces, the bound is \Theta( p h + lg n) pursuers for a polygon that has n edges and h holes. A set of problems that are solvable by a single pursuer and require a linear number of recontaminations is shown. The complete algorithm searches a f...

### Citations

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(Show Context)
Citation Context ..., and has been closely related to other graph properties such as cutwidth [15, 17]. Pursuit-evasion scenarios in continuous spaces have arisen in a variety of applications such as air traffic control =-=[1]-=-, military strategy [10], and trajectory tracking [9]. Although interesting decision problems arise through the differential motion models, geometric free-space constraints are usually not considered ... |

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Citation Context ...and executed on an SGI Indigo2 workstation with a 200 Mhz MIPS R4400 processor. Most problems we encountered were solved in a few seconds or less. The implementation uses the quad-edge structure from =-=[7]-=- to maintain the topological ordering of the conservative cells. The search strategy is Dijkstra's shortest path algorithm, in which the distance is measured from the adjacent cell centroids. Figure 5... |

392 |
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Citation Context ...-sight view from a pursuer. A moving visibility polygon in a polygonal environment adds geometric information that must be utilized, and also leads to connections with the static art gallery problems =-=[18, 21]-=-. In the limiting case, art gallery results serve as a loose upper bound on the number of pursuers by allowing a covering of the free space by static guards, guaranteeing that any evader will be immed... |

388 | Tarjan, A Separator Theorem for Planar Graphs
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Citation Context ...n, which has O(n) edges and vertices. Take the subgraph induced by taking only the vertices that correspond to trichromatic triangles in the original triangulation. The planar graph separator theorem =-=[14]-=- implies that at most O( p h) edges can be chosen to partition the graph into two portions with at least one third of the edges on each side of the partition. Each edge in the induced subgraph corresp... |

134 |
Recent results in art galleries
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(Show Context)
Citation Context ...-sight view from a pursuer. A moving visibility polygon in a polygonal environment adds geometric information that must be utilized, and also leads to connections with the static art gallery problems =-=[18, 21]-=-. In the limiting case, art gallery results serve as a loose upper bound on the number of pursuers by allowing a covering of the free space by static guards, guaranteeing that any evader will be immed... |

128 |
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Citation Context ...ts. Interesting results have been obtained for pursuit-evasion in a graph, in which the pursuers and evader can move from vertex to vertex until eventually a pursuer and evader lie in the same vertex =-=[16, 19]-=-. The search number of a graph refers to the minimum number of pursers needed to solve a pursuit-evasion problem, and has been closely related to other graph properties such as cutwidth [15, 17]. Purs... |

121 | Searching for mobile intruders in a polygonal region by a group of mobile searchers
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Citation Context ...l burden of ensuring that an evader cannot "sneak" to a portion of the environment that has already been explored. The problem that we consider and other variations have been considered prev=-=iously in [5, 22]-=-. It was stated in [21] that it remained an interesting challenge to determine if a polygon is searchable by a single pursuer. Several applications can be envisioned for problems and motion strategies... |

113 |
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Citation Context ...related to other graph properties such as cutwidth [15, 17]. Pursuit-evasion scenarios in continuous spaces have arisen in a variety of applications such as air traffic control [1], military strategy =-=[10]-=-, and trajectory tracking [9]. Although interesting decision problems arise through the differential motion models, geometric free-space constraints are usually not considered in classical pursuitevas... |

97 |
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Citation Context ...= 1, it is shown that the same region can require recontamination as many as\Omega (n) times. This result is surprising because pursuit-evasion in a graph is known not to require any recontaminations =-=[11]-=-. Consider the problem of determining the minimum number of pursuers, H(F ), required to find an evader in a given free space F . This number will generally depend on both the topological and geometri... |

87 |
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Citation Context ...ts. Interesting results have been obtained for pursuit-evasion in a graph, in which the pursuers and evader can move from vertex to vertex until eventually a pursuer and evader lie in the same vertex =-=[16, 19]-=-. The search number of a graph refers to the minimum number of pursers needed to solve a pursuit-evasion problem, and has been closely related to other graph properties such as cutwidth [15, 17]. Purs... |

85 |
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Citation Context ...ortion of the free space a linear number of times. This result is surprising because for Parsons' problem it was shown in [11] that no recontamination is necessary (a shorter proof of this appears in =-=[2]-=-). In [22] a free space was given that requires two recontaminations, which at least established that recontamination is generally necessary for visibility-based pursuit evasion. Theorem 7 establishes... |

76 |
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Citation Context ...uers. Recall that for any simple polygon, a pair of vertices can always be connected so that polygon is partitioned into two regions, each with at least one third of the edges of the original polygon =-=[3]-=-. This implies that F can be recursively partitioned until a triangulation is constructed, and each triangular region only requires O(lg n) recombinations before F is obtained (i.e., the recursion dep... |

64 | Finding an unpredictable target in a workspace with obstacles
- LaValle, Lin, et al.
- 1997
(Show Context)
Citation Context ...ent algorithm can be determined for the general problem. A complete algorithm for H(F ) = 1 is detailed in this paper, and the general techniques apply to the case in which H(F ) ? 1. In related work =-=[13]-=- we have developed a greedy algorithm that efficiently solves many multiple-pursuer problems. Because the position of the evader is unknown, one does not have direct access to the state at a given tim... |

63 |
Optimum watchman routes
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Citation Context ...are needed when they are allowed to move and search for an evader; however, the required motion strategies can become quite complex. A closely related art gallery variant is the watchman tour problem =-=[4]-=-. In this case a minimum-length closed path is computed such that any point in the polygon is visible from some point along the path. In our case, however, the pursuers have the additional burden of e... |

63 |
On the piano movers’ problem: III. Coordinating the motion of several independent bodies: The special case of circular bodies moving amidst polygonal barriers
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Citation Context ...by a standard approach used in motion planning, which is to preserve completeness by using a decomposition of the configuration space that is constructed by analyzing critical events. For example, in =-=[20]-=- a cell decomposition is determined by analyzing the contact manifolds in a composite configuration space that is generated by the positions of several disks in the plane. The next definition describe... |

57 | The robot localization problem
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Citation Context ... cells. The idea is to partition the free space into convex cells by identifying critical places at which edge visibility changes. A decomposition of this type has been used for robot localization in =-=[8, 23]-=-, and generates O(n 3 ) cells in the worst case for a simple polygon (which is always true if H(F ) = 1). The free space can be sufficiently partitioned in our case by extending rays in the three gene... |

41 |
Minimizing width in linear layouts
- Makedon, Sudborough
(Show Context)
Citation Context ...ertex [16, 19]. The search number of a graph refers to the minimum number of pursers needed to solve a pursuit-evasion problem, and has been closely related to other graph properties such as cutwidth =-=[15, 17]-=-. Pursuit-evasion scenarios in continuous spaces have arisen in a variety of applications such as air traffic control [1], military strategy [10], and trajectory tracking [9]. Although interesting dec... |

32 | A game-theoretic framework for robot motion planning
- LaValle
- 1995
(Show Context)
Citation Context ...et of possible information states. The information space is a standard representational tool for problems that have imperfect state information, and has been useful for other motion planning problems =-=[6, 12]-=-. For a fixed strategy, fl, a path in the information space will be obtained by j(t) = (fl 1 ; : : : ; fl N ; S(t)) in which S(t) can be determined from an initial S(0) and the trajectories ffl i (t 0... |

27 | Mobile robot self-location using model-image feature correspondence - Talluri, Aggarwal - 1996 |

22 |
Randomization for robot tasks: Using dynamic programming in the space of knowledge states
- Erdmann
- 1993
(Show Context)
Citation Context ...et of possible information states. The information space is a standard representational tool for problems that have imperfect state information, and has been useful for other motion planning problems =-=[6, 12]-=-. For a fixed strategy, fl, a path in the information space will be obtained by j(t) = (fl 1 ; : : : ; fl N ; S(t)) in which S(t) can be determined from an initial S(0) and the trajectories ffl i (t 0... |

20 |
Searching for a mobile intruder in a corridor – The open edge variant of the polygon search problem
- Crass, Suzuki, et al.
- 1995
(Show Context)
Citation Context ...l burden of ensuring that an evader cannot "sneak" to a portion of the environment that has already been explored. The problem that we consider and other variations have been considered prev=-=iously in [5, 22]-=-. It was stated in [21] that it remained an interesting challenge to determine if a polygon is searchable by a single pursuer. Several applications can be envisioned for problems and motion strategies... |

15 |
Pursuit Games
- Hájek
- 1975
(Show Context)
Citation Context ...ies such as cutwidth [15, 17]. Pursuit-evasion scenarios in continuous spaces have arisen in a variety of applications such as air traffic control [1], military strategy [10], and trajectory tracking =-=[9]-=-. Although interesting decision problems arise through the differential motion models, geometric free-space constraints are usually not considered in classical pursuitevasion games. Once these constra... |

4 |
Min cut is NP-complete for edge weighted graphs
- Monien, Sudborough
- 1988
(Show Context)
Citation Context ...ertex [16, 19]. The search number of a graph refers to the minimum number of pursers needed to solve a pursuit-evasion problem, and has been closely related to other graph properties such as cutwidth =-=[15, 17]-=-. Pursuit-evasion scenarios in continuous spaces have arisen in a variety of applications such as air traffic control [1], military strategy [10], and trajectory tracking [9]. Although interesting dec... |