On the relation between graph distance and Euclidean distance in random geometric graphs

Suppose that there is a family of n random points X v for v ∈ V , independently and uniformly distributed in the square S n = [− √ n/2, √ n/2] 2 .We do not see these points, but learn about them in one of the following two ways. Suppose first that we are given the corresponding random geometric graph G ∈ G (n, r), where distinct vertices u and v are adjacent when the Euclidean distance d E (X u , X v ) is at most r. Assume that the threshold distance r satisfies n 3/14 ≪ r ≪ n 1/2 . We shall see that the following holds with high probability. Given the graph G (without any geometric information), in polynomial time we can approximately reconstruct the hidden embedding, in the sense that, 'up to symmetries', for each vertex v we find a point within distance about r of X v ; that is, we find an embedding with 'displacement' at most about r. Now suppose that, instead of being given the graph G, we are given, for each vertex v, the ordering of the other vertices by increasing Euclidean distance from v. Then, with high probability, in polynomial time we can find an embedding with the much smaller displacement error O( √ log n).

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“…Let t > 0 and let σ be a symmetry. Arguing as for (9), and noting also that d * (Ψ, Φ) = d * (σ −1 • Ψ, Φ), we have…”

Section: Proof Of Theorem 12mentioning

confidence: 98%

Learning random points from geometric graphs or orderings

Dı́az

McDiarmid

Mitsche

2020

Random Struct Algorithms

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“…The results there are stated in the model of a square of side length √ n, but they can be easily translated to our setting. (In fact, in [7], a more precise result was shown, but for our purpose the following version is enough.) Theorem 3.…”

Section: Proof Of Theorem 12mentioning

confidence: 95%

“…For the upper bound, we will use the result from [7]. The results there are stated in the model of a square of side length √ n, but they can be easily translated to our setting.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Burning Graphs: A Probabilistic Perspective

Mitsche¹,

Prałat

Roshanbin

2017

Graphs and Combinatorics

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Abstract. In this paper, we study a graph parameter that was recently introduced, the burning number, focusing on a few probabilistic aspects of the problem. The original burning number is revisited and analyzed for binomial random graphs G(n, p), random geometric graphs, and the Cartesian product of paths. Moreover, new variants of the burning number are introduced in which a burning sequence of vertices is selected according to some probabilistic rules. We analyze these new graph parameters for paths.

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“…An important parameter that arises in this context is the number of transmitters a message has to traverse in order to establish a broadcast between two given transmitters. Several works have established that this parameter is proportional to the Euclidean distance between the two nodes (see, e.g., [11,15,18,20,44,46]).…”

Section: Random Geometric Graphsmentioning

confidence: 99%

New perspective on sampling-based motion planning via random geometric graphs

Solovey¹,

Salzman²,

Halperin³

Robotics: Science and Systems XII

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Abstract-Roadmaps constructed by many sampling-based motion planners coincide, in the absence of obstacles, with standard models of random geometric graphs (RGGs). Those models have been studied for several decades and by now a rich body of literature exists analyzing various properties and types of RGGs. In their seminal work on optimal motion planning Karaman and Frazzoli [31] conjectured that a sampling-based planner has a certain property if the underlying RGG has this property as well. In this paper we settle this conjecture and leverage it for the development of a general framework for the analysis of sampling-based planners. Our framework, which we call localization-tessellation, allows for easy transfer of arguments on RGGs from the free unit-hypercube to spaces punctured by obstacles, which are geometrically and topologically much more complex. We demonstrate its power by providing alternative and (arguably) simple proofs for probabilistic completeness and asymptotic (near-)optimality of probabilistic roadmaps (PRMs). Furthermore, we introduce two variants of PRMs, analyze them using our framework, and discuss the implications of the analysis.

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On the relation between graph distance and Euclidean distance in random geometric graphs

Cited by 24 publications

References 9 publications

Learning random points from geometric graphs or orderings

Learning random points from geometric graphs or orderings

Burning Graphs: A Probabilistic Perspective

New perspective on sampling-based motion planning via random geometric graphs

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