Metadata of the chapter that will be visualized in SpringerLink Book Title Combinatorial Optimization and Applications Series Title Chapter Title Optimal Strategy for Walking in Streets with Minimum Number of Turns for a Simple Robot Optimal Strategy for

Abstract We consider the problem of walking a simple robot in an unknown street. The robot that cannot infer any geometric properties of the street traverses the environment to reach a target , starting from a point . The robot has a minimal sensing capability that can only report the discontinuities in the depth information (gaps), and location of the target point once it enters in its visibility region. Also, the robot can only move towards the gaps while moving along straight lines is cheap, but rotation is expensive for the robot. We maintain the location of some gaps in a tree data structure of constant size. The tree is dynamically updated during the movement. Using the data structure, we present an online strategy that generates a search path for the robot with optimal number of turns. Keywords (separated by '-') Computational geometry -Minimum link path -Simple robot -Street polygon -Unknown environment Abstract. We consider the problem of walking a simple robot in an unknown street. The robot that cannot infer any geometric properties of the street traverses the environment to reach a target t, starting from a point s. The robot has a minimal sensing capability that can only report the discontinuities in the depth information (gaps), and location of the target point once it enters in its visibility region. Also, the robot can only move towards the gaps while moving along straight lines is cheap, but rotation is expensive for the robot. We maintain the location of some gaps in a tree data structure of constant size. The tree is dynamically updated during the movement. Using the data structure, we present an online strategy that generates a search path for the robot with optimal number of turns

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