3/19/2023 0 Comments A unit disk graph![]() They can communicate only if they are within mutual transmission range. Nodes are located in the Euclidean plane and are assumed to have identical (unit) transmission radii. One prominent application of unit disk graphs can be found in the eld of wireless networking, where a unit disk graph represents an idealized multi-hop radio network. Unit disk graphs have proven to be useful in modeling various physical real world problems. Equivalently, each node is identi ed with a disk of unit radius r = 1 in the plane, and is connected to all nodes within (or on the edge of) its corresponding disk. INTRODUCTION In a unit disk graph, there is an edge between two nodes u and v if and only if the Euclidean distance between u and v is at most 1. Unit Disk Graph Approximation Fabian Kuhn Computer Engineering and Networks Laboratory ETH Zurich 8092 Zurich, Switzerland Thomas Moscibroda Computer Engineering and Networks Laboratory ETH Zurich 8092 Zurich, Switzerland Roger Wattenhofer Computer Engineering and Networks Laboratory ETH Zurich 8092 Zurich, Switzerland ABSTRACT 1. Clearly, the unit disk graph model neatly captures this behavior and it is not surprising that it has ![]() Kuhn, Fabian Moscibroda, Thomas Wattenhofer, Roger Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size O(m)).Unit disk graph approximation Unit disk graph approximation Proofs are constructive and suggest simple algorithms that run in linear time. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in n exists when we look at disks of arbitrary radii, even when m=0. the context of unit disk graphs, which are the two dimensional version of unit ball graphs, there is no established result on the complexity and approximation status for some of them in unit ball graphs. We also show that an axis-parallel line intersecting O(m+n) disks exists, but each halfplane may contain up to 4n/5 disks. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size O(m)).ĪB - We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of n unit disks in the plane there exists a line ℓ such that ℓ intersects at most O((m+n)logn) disks and each of the halfplanes determined by ℓ contains at most 2n/3 unit disks from the set, where m is the number of intersecting pairs of disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in n exists when we look at disks of arbitrary radii, even when m=0. N2 - We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of n unit disks in the plane there exists a line ℓ such that ℓ intersects at most O((m+n)logn) disks and each of the halfplanes determined by ℓ contains at most 2n/3 unit disks from the set, where m is the number of intersecting pairs of disks. T1 - Balanced line separators of unit disk graphs
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