129 research outputs found

    Unions of Onions: Preprocessing Imprecise Points for Fast Onion Decomposition

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    Let D\mathcal{D} be a set of nn pairwise disjoint unit disks in the plane. We describe how to build a data structure for D\mathcal{D} so that for any point set PP containing exactly one point from each disk, we can quickly find the onion decomposition (convex layers) of PP. Our data structure can be built in O(nlogn)O(n \log n) time and has linear size. Given PP, we can find its onion decomposition in O(nlogk)O(n \log k) time, where kk is the number of layers. We also provide a matching lower bound. Our solution is based on a recursive space decomposition, combined with a fast algorithm to compute the union of two disjoint onionComment: 10 pages, 5 figures; a preliminary version appeared at WADS 201

    Minimum Cuts in Geometric Intersection Graphs

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    Let D\mathcal{D} be a set of nn disks in the plane. The disk graph GDG_\mathcal{D} for D\mathcal{D} is the undirected graph with vertex set D\mathcal{D} in which two disks are joined by an edge if and only if they intersect. The directed transmission graph GDG^{\rightarrow}_\mathcal{D} for D\mathcal{D} is the directed graph with vertex set D\mathcal{D} in which there is an edge from a disk D1DD_1 \in \mathcal{D} to a disk D2DD_2 \in \mathcal{D} if and only if D1D_1 contains the center of D2D_2. Given D\mathcal{D} and two non-intersecting disks s,tDs, t \in \mathcal{D}, we show that a minimum ss-tt vertex cut in GDG_\mathcal{D} or in GDG^{\rightarrow}_\mathcal{D} can be found in O(n3/2polylogn)O(n^{3/2}\text{polylog} n) expected time. To obtain our result, we combine an algorithm for the maximum flow problem in general graphs with dynamic geometric data structures to manipulate the disks. As an application, we consider the barrier resilience problem in a rectangular domain. In this problem, we have a vertical strip SS bounded by two vertical lines, LL_\ell and LrL_r, and a collection D\mathcal{D} of disks. Let aa be a point in SS above all disks of D\mathcal{D}, and let bb a point in SS below all disks of D\mathcal{D}. The task is to find a curve from aa to bb that lies in SS and that intersects as few disks of D\mathcal{D} as possible. Using our improved algorithm for minimum cuts in disk graphs, we can solve the barrier resilience problem in O(n3/2polylogn)O(n^{3/2}\text{polylog} n) expected time.Comment: 11 pages, 4 figure
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