We study the problem of computing the free space of a simple legged robot called the spider robot. The body of this robot is a single point and the legs are attached to the body. The robot is subject to two constraints: each leg has a maximal extension R #accessibility constraint# and the body of the robot must lie above the convex hull of its feet #stability constraint#. Moreover, the robot can only put its feet on some regions, called the foothold regions. The free space positions of the body of the robot such that there exists a set of accessible footholds for which the robot is stable. We present an efficient algorithm that computes log n# time and O#n ##n## space for point footholds where ##n# is an extremely slowly growing function ###n# 3 for any practical value of n#. We also presentan when the foothold regions are pairwise disjoint polygons with n edges in total. This algorithm computes in O#n # 8 #n# log n# time using # 8 #n## space ## 8 #n# is also an extremely slowly growing function#. These results are close to optimal since#, # isalower bound for the size of .

this paper we briefly survey the most recent results in the area of facility location, concentrating on versions of the problem that are likely to be unfamiliar to the transportation and management science community and we explore the interaction between facility location problems and the field of computational geometry. Such versions of the problem include the standard models of points as customers and facilities but with geodesic rather than the traditional Minkowski metrics as measures of distance, as well as more complicated models of customers and facilities such as

We present three different methods for finding solutions to the 2D translation-only containment problem: find translations for k polygons that place them inside a given polygonal container without overlap. Both the container and the polygons to be placed in it may be nonconvex. First, we provide several exact algorithms that improve results for k = 2 or k = 3. In particular, we give an algorithm for three convex polygons and a nonconvex container with running time in O(m 3 n log mn), where n is the number of vertices in the container, and m is the sum of the vertices of the k polygons. This is an improvement of a factor of n 2 over previous algorithms. Second, we give an approximation algorithm for k nonconvex polygons and a nonconvex container based on restriction and subdivision of the configuration space. Third, we develop a MIP (mixed integer programming) model for k nonconvex polygons and a nonconvex container.

We present a new algorithm to compute a motorcycle graph. It runs in O(n p n log n) time when n is the size of the input. We give a new characterization of the straight skeleton of a polygon possibly with holes.

Given n line segments in the plane, the Bentley-Ottmann sweep maintains the exact ordering of the intersections of the segments with a vertical line, as this line sweeps the plane from left to right. To accomplish this, every intersection between two segments must be processed, and the running time of the sweep can be (n 2). In this paper, it is shown how a heap on the intersections can be maintained during the sweep. This new type of sweep processes O(n log² n)intersections when sweeping over lines and O(n p n log n) intersections when sweeping over line segments. Alower bound of (n log n) is also established.

Given a polyhedral terrain T with n vertices, the two-watchtower problem for T asks to find two vertical segments, called watchtowers, of smallest common height, whose bottom endpoints (bases) lie on T, and whose top endpoints guard T, in the sense that each point on T is visible from at least one of them. There are three versions of the problem, discrete, semi-discrete, and continuous, depending on whether two, one, or none of the two bases are restricted to be among the vertices of T, respectively. In this paper we present the following results for the two-watchtower problem in R 2 and R 3: (1) We show that the discrete two-watchtowers problem in R 2 can be solved in O(n 2 log 4 n) time, significantly improving previous solutions. The algorithm works, without increasing its asymptotic running time, for the semi-continuous version, where one of the towers is allowed to be placed anywhere on T. (2) We show that the continuous two-watchtower problem in R 2 can be solved in O(n 3 α(n)log 3 n) time, again significantly improving previous results. (3) Still in R 2, we show that the continuous version of the problem of guarding a finite set P ⊂ T of m points by two watchtowers of smallest common height can be solved in O(mnlog 4 n) time.

Abstract not found

In this paper network location problems with several objectives are discussed, where every single objective is a classical median objective function. We will look at the problem of finding Pareto optimal locations and lexicographically optimal locations. It is shown that for Pareto optimal locations in undirected networks no node dominance result can be shown. Structural results as well as efficient algorithms for these multi-criteria problems are developed. In the special case of a tree network a generalization of Goldman's dominance algorithm for finding Pareto locations is presented.

We present an O(n log 2 n) time and O(n) space algorithm for computing the shortest line segment that intersects a set of n given line segments or lines in the plane. If the line segments do not intersect the algorithm may be trimmed to run in O(n log n) time. Furthermore, in combination with linear programming the algorithm will also find the shortest line segment that intersects a set of n isothetic rectangles in the plane in O(n log k) time, where k is the combinatorial complexity of the space of transversals and k 4n. These results find application in: (1) line-fitting between a set of n data ranges where it is desired to obtain the shortest line-of-fit, (2) finding the shortest line segment from which a convex n-vertex polygon is weakly externally visible, and (3) determining the shortest line-of-sight between two edges of a simple n-vertex polygon, for which O(n) time algorithms are also given. All the algorithms are based on the solution to a new fundamental geometric optim...

We provide new combinatorial bounds on the complexity of a face in an arrangement of segments in the plane. In particular, we show that the complexity of a single face in an arrangement of n line segments determined by h endpoints is O(h log h). While the previous upper bound, O(nff(n)), is tight for segments with distinct endpoints, it is far from being optimal when n = \Omega\Gamma h 2 ). Our result shows that, in a sense, the fundamental combinatorial complexity of a face arises not as a result of the number of segments, but rather as a result of the number of endpoints. 1 Introduction Let S = fs 1 ; : : : ; s n g be a finite set of n line segments in the plane. Then S induces a partition of the plane, called the arrangement A(S) of S, into O(n 2 ) faces, edges and vertices. Refer to Figure 1. Arrangements of segments play a fundamental role in computational geometry (see, e.g., [EGS]). The problem studied here is that in which the segments S are allowed to share endpoi...