273 research outputs found

    Minimum Perimeter-Sum Partitions in the Plane

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    Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P_1 and P_2 such that the sum of the perimeters of CH(P_1) and CH(P_2) is minimized, where CH(P_i) denotes the convex hull of P_i. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n^2) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in O(n log^4 n) time and a (1+e)-approximation algorithm running in O(n + 1/e^2 log^4(1/e)) time

    On Embeddability of Buses in Point Sets

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    Set membership of points in the plane can be visualized by connecting corresponding points via graphical features, like paths, trees, polygons, ellipses. In this paper we study the \emph{bus embeddability problem} (BEP): given a set of colored points we ask whether there exists a planar realization with one horizontal straight-line segment per color, called bus, such that all points with the same color are connected with vertical line segments to their bus. We present an ILP and an FPT algorithm for the general problem. For restricted versions of this problem, such as when the relative order of buses is predefined, or when a bus must be placed above all its points, we provide efficient algorithms. We show that another restricted version of the problem can be solved using 2-stack pushall sorting. On the negative side we prove the NP-completeness of a special case of BEP.Comment: 19 pages, 9 figures, conference version at GD 201

    Small grid embeddings of 3-polytopes

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    We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by O(27.55n)=O(188n)O(2^{7.55n})=O(188^{n}). If the graph contains a triangle we can bound the integer coordinates by O(24.82n)O(2^{4.82n}). If the graph contains a quadrilateral we can bound the integer coordinates by O(25.46n)O(2^{5.46n}). The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face

    Almost optimal asynchronous rendezvous in infinite multidimensional grids

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    Two anonymous mobile agents (robots) moving in an asynchronous manner have to meet in an infinite grid of dimension 未> 0, starting from two arbitrary positions at distance at most d. Since the problem is clearly infeasible in such general setting, we assume that the grid is embedded in a 未-dimensional Euclidean space and that each agent knows the Cartesian coordinates of its own initial position (but not the one of the other agent). We design an algorithm permitting the agents to meet after traversing a trajectory of length O(d 未 polylog d). This bound for the case of 2d-grids subsumes the main result of [12]. The algorithm is almost optimal, since the 鈩(d 未) lower bound is straightforward. Further, we apply our rendezvous method to the following network design problem. The ports of the 未-dimensional grid have to be set such that two anonymous agents starting at distance at most d from each other will always meet, moving in an asynchronous manner, after traversing a O(d 未 polylog d) length trajectory. We can also apply our method to a version of the geometric rendezvous problem. Two anonymous agents move asynchronously in the 未-dimensional Euclidean space. The agents have the radii of visibility of r1 and r2, respectively. Each agent knows only its own initial position and its own radius of visibility. The agents meet when one agent is visible to the other one. We propose an algorithm designing the trajectory of each agent, so that they always meet after traveling a total distance of O( ( d)), where r = min(r1, r2) and for r 鈮 1. r)未polylog ( d r