Geometric Pursuit Evasion

In this dissertation we investigate pursuit evasion problems set in geometric environments. These games model a variety of adversarial situations in which a team of agents, called pursuers, attempts to catch a rogue agent, called the evader. In particular, we consider the following problem: how many pursuers, each with the same maximum speed as the evader, are needed to guarantee a successful capture? Our primary focus is to provide combinatorial bounds on the number of pursuers that are necessary and sufficient to guarantee capture. The first problem we consider consists of an unpredictable evader that is free to move around a polygonal environment of arbitrary complexity. We assume that the pursuers have complete knowledge of the evader's location at all times, possibly obtained through a network of cameras placed in the environment. We show that regardless of the number of vertices and obstacles in the polygonal environment, three pursuers are always sufficient and sometimes necessary to capture the evader. We then consider several extensions of this problem to more complex environments. In particular, suppose the players move on the surface of a 3-dimensional polyhedral body; how many pursuers are required to capture the evader? We show that 4 pursuers always suffice (upper bound), and that 3 are sometimes necessary (lower bound), for any polyhedral surface with genus zero. Generalizing this bound to surfaces of genus g, we prove the sufficiency of (4g + 4) pursuers. Finally, we show that 4 pursuers also suffice under the "weighted region" constraints, where the movement costs through different regions of the (genus zero) surface have (different) multiplicative weights. Next we consider a more general problem with a less restrictive sensing model. The pursuers' sensors are visibility based, only providing the location of the evader if it is in direct line of sight. We begin my making only the minimalist assumption that pursuers and the evader have the same maximum speed. When the environment is a simply-connected (hole-free) polygon of n vertices, we show that Θ(n^1/2 ) pursuers are both necessary and sufficient in the worst-case. When the environment is a polygon with holes, we prove a lower bound of Ω(n^2/3 ) and an upper bound of O(n^5/6 ) pursuers, where n includes the vertices of the hole boundaries. However, we show that with realistic constraints on the polygonal environment these bounds can be drastically improved. Namely, if the players' movement speed is small compared to the features of the environment, we give an algorithm with a worst case upper bound of O(log n) pursuers for simply-connected n-gons and O(√h + log n) for polygons with h holes. The final problem we consider takes a small step toward addressing the fact that location sensing is noisy and imprecise in practice. Suppose a tracking agent wants to follow a moving target in the two-dimensional plane. We investigate what is the tracker's best strategy to follow the target and at what rate does the distance between the tracker and target grow under worst-case localization noise. We adopt a simple but realistic model of relative error in sensing noise: the localization error is proportional to the true distance between the tracker and the target. Under this model we are able to give tight upper and lower bounds for the worst-case tracking performance, both with or without obstacles in the Euclidean plane

Visibility-Based Pursuit-Evasion In The Plane

As technological advances further increase the amount of memory and computing power available to mobile robots, we are seeing an unprecedented explosion in the utilization of deployable robots for various tasks. The speed at which robots begin to enter various domains is largely dependent on the availability of robust and efficient algorithms that are capable of solving the complex planning problems inherent to the given domain. One such domain which is experiencing unprecedented growth in recent years requires a robot to detect and/or track a mobile agent or group of agents. In these scenarios, there are typically two players with diametrically opposed goals. For matters of security, we have a guard and an intruder. The guard’s goal is to ensure that if an intruder enters the premises they are caught in a timely manner. Analogously, the intruder wishes to evade detection for as long as possible. Search and rescue operations are often framed as a two-player game between rescuers and survivors. Though the survivors are unlikely to behave antagonistically, an agnostic model is useful for the rescuers to guarantee that the survivors are found, regardless of their movements. Both of these tasks, are at their core, pursuit-evasion problems. There are many variants of the pursuit-evasion problem, the common theme amongst them is that one group of agents, the “pursuers”, attempts to track members of another group, the “evaders”. Geometric formulations of the pursuit-evasion problem require a pursuer(s) to systematically search an environment to locate one or more evaders ensuring that all evaders will be captured by the pursuer(s) in a finite time. The visibility-based pursuit-evasion problem is a geometric variant of the pursuit-evasion problem that defines a visibility-region which corresponds to the region of the environment that the pursuer(s) can actively perceive. If an evader lies within this visibility region then it is captured (detected). This thesis contains four novel contributions that solve various visibility-based pursuit-evasion problems. The first contribution is an algorithm that computes the optimal (minimal path length) pursuer trajectory for a single pursuer. The second contribution is an algorithm that generates a joint motion strategy for multiple pursuers. Motivated by the result of the second contribution, the third result is a sampling-based algorithm for the multiple pursuer scenario. The fourth contribution is a complete algorithm that computes a trajectory for a pursuer that has a very limited sensor footprint

Capture Bounds for Visibility-Based Pursuit Evasion

We investigate the following problem in the visibility-based discrete-time model of pursuit evasion in the plane: how many pursuers are needed to capture an evader in a polygonal environment with obstacles under the minimalist assumption that pursuers and the evader have the same maximum speed? When the environment is a simply-connected (hole-free) polygon of n vertices, we show that Θ(n 1/2) pursuers are both necessary and sufficient in the worst-case. When the environment is a polygon with holes, we prove a lower bound of Ω(n 2/3) and an upper bound of O(n 5/6) for the number of pursuers that are needed in the worstcase, where n is the total number of vertices including the hole boundaries. More precisely, if the polygon contains h holes, our upper bound is O(n 1/2 h 1/4), for h ≤ n 2/3, and O(n 1/3 h 1/2) otherwise. These bounds show that capture with minimal assumptions requires significantly more pursuers than what is possible either for visibility detection where pursuerswin if one of themcan see the evader[Guibas et al. 1999], or for capture when players’ movement speed is small compared to“features”of the environment [Klein an

Straight Line Movement in Morphing and Pursuit Evasion

Piece-wise linear structures are widely used to define problems and to represent simplified solutions in computational geometry. A piece-wise linear structure consists of straight-line or linear pieces connected together in a continuous geometric environment like 2D or 3D Euclidean spaces. In this thesis two different problems both with the approach of finding piece-wise linear solutions in 2D space are defined and studied: straight-line pursuit evasion and straight-line morphing. Straight-line pursuit evasion is a geometric version of the famous cops and robbers game that is defined in this thesis for the first time. The game is played in a simply connected region in 2D. It is a full information game where the players take turns. The cop’s goal is to catch the robber. In a turn, each player may move any distance along a straight line as long as the line segment connecting their current location to the new location is not blocked by the region’s boundary. We first prove that the cop can always win the game when the players move on the visibility graph of a simple polygon. We prove this by showing that the visibility graph of a simple polygon is “dismantlable” (the known class of cop-win graphs). Polygon visibility graphs are also shown to be 2-dismantlable. Two other settings of the game are also studied in this thesis: when the players are free to move on the infinitely many points inside a simple polygon, and inside a splinegon. In both cases we show that the cop can always win the game. For the case of polygons, the proposed cop strategy gives an asymptotically tight linear bound on the number of steps the cop needs to catch the robber. For the case of splinegons, the cop may need a quadratic number of steps with the proposed strategy, while our best lower bound is linear. Straight-line morphing is a type of morphing first defined in this thesis that provides a nice and smooth transformation between straight-line graph drawings in 2D. In straight- line morphing, each vertex of the graph moves forward along the line segment connecting its initial position to its final position. The vertex trajectories in straight-line morphing are very simple, but because the speed of each vertex may vary, straight-line morphs are more general than the commonly used “linear morphs” where each vertex moves at uniform speed. We explore the problem of whether an initial planar straight-line drawing of a graph can be morphed to a final straight-line drawing of the graph using a straight-line morph that preserves planarity at all times. We prove that this problem is NP-hard even for the special case where the graph drawing consists of disjoint segments. We then look at some restricted versions of the straight-line morphing: when only one vertex moves at a time, when the vertices move one by one to their final positions uninterruptedly, and when the edges morph one by one to their final configurations in the case of disjoint segments. Some of the variations are shown to be still NP-complete while some others are solvable in polynomial time. We conjecture that the class of planar straight-line morphs is as powerful as the class of planar piece-wise linear straight-line morphs. We also explore a simpler problem where for each edge the quadrilateral formed by its initial and final positions together with the trajectories of its two vertices is convex. There is a necessary condition for this case that we conjecture is also sufficient for paths and cycles