We would wait for the system to reach an equilibrium, which hopefully would minimize some potential function that measures the distance between our embedding and a unit disk graph. I have an idea but it's rather silly: I thought of running a physical simulation in which we place an attractive force on edges between vertices that are far apart, and a repelling force on non-edges between vertices that are close together. "Low distortion Embeddings" is something related but different: There, the goal is that all edges should have similar lengths, whereas in my case I don't care about eliminating short edges. Has this type of thing been studied? In particular, I haven't been able to find an appropriate algorithm for the first item. Hopefully, if the embedding of $G$ is close to a unit disk graph, $A$ will return a good approximation to the $NP$-hard problem. Use the algorithm $A$ to find a solution for the graph.Of course, the resulting graph is probably not going to be a unit disk graph, but hopefully it is close to such a graph in some sense. Given a graph $G$, first embed it into the plane in such a way that vertex pairs connected by an edge tend to be close together, whereas vertex pairs that aren't connected tend to be farther apart.Let $A$ be an algorithm that solves an $NP$-hard problem in polynomial time in the special case of unit disk graphs. I am interested in designing approximation algorithms by adapting such algorithms to general graphs as follows: Some $NP$-hard problems become solvable in polynomial time for unit disk graphs. 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)).A unit disk graph is defined by a collection of $n$ vertices corresponding to $n$ points on the plane, with an edge between any two vertices whose distance is at most $r$. 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. 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. 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. Two vertices u v 2 V are connected by an edge uv 2 E if and only if their Euclidean distance is less than. An unit disk graph G (V E) is dened on a nite set of points V in the plane. 1 Introduction We address the problem of nding all maximal cliques in an Unit Disk Graph (UDG) 1. We also show that an axis-parallel line intersecting O(m+n) disks exists, but each halfplane may contain up to 4n/5 disks. Keywords: Cliques, Unit Disk Graphs, Ad-Hoc Networks. that for the unit-disk graph metric of n points in the plane and for any constant c 1, there exists a c-well- separated pair decomposition with O(n log. We present simple and provably good heuristics for a number of. 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. Abstract Unit disk graphs are intersection graphs of circles of unit radius in the plane. T1 - Balanced line separators of unit disk graphs
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