Shortest paths in intersection graphs of unit disks

Cabello, Sergio; Jejčič, Miha

doi:10.1016/j.comgeo.2014.12.003

Cited by 30 publications

(61 citation statements)

References 22 publications

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“…FMT *’s strategy is reminiscent of the approach used for the computation of shortest paths over acyclic graphs (Sniedovich, 2006). Indeed, the idea of leveraging graph structure to compute shortest paths over disk graphs is not new and was recently investigated in Roditty and Segal (2011)— under the name of bounded leg shortest path problem — and in Cabello and Jejčič (2014). Both works, however, do not use “direct” dynamic programming arguments, but rather combine Dijkstra ’s algorithm with the concept of bichromatic closest pairs (Chan and Efrat, 2001).…”

Section: The Fast Marching Tree Algorithm (Fmt*)mentioning

confidence: 99%

Fast marching tree: A fast marching sampling-based method for optimal motion planning in many dimensions

Janson

Szczerbicki

Clark

et al. 2015

The International Journal of Robotics Research

454

384

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In this paper we present a novel probabilistic sampling-based motion planning algorithm called the Fast Marching Tree algorithm (FMT*). The algorithm is specifically aimed at solving complex motion planning problems in high-dimensional configuration spaces. This algorithm is proven to be asymptotically optimal and is shown to converge to an optimal solution faster than its state-of-the-art counterparts, chiefly PRM* and RRT*. The FMT* algorithm performs a “lazy” dynamic programming recursion on a predetermined number of probabilistically-drawn samples to grow a tree of paths, which moves steadily outward in cost-to-arrive space. As such, this algorithm combines features of both single-query algorithms (chiefly RRT) and multiple-query algorithms (chiefly PRM), and is reminiscent of the Fast Marching Method for the solution of Eikonal equations. As a departure from previous analysis approaches that are based on the notion of almost sure convergence, the FMT* algorithm is analyzed under the notion of convergence in probability: the extra mathematical flexibility of this approach allows for convergence rate bounds—the first in the field of optimal sampling-based motion planning. Specifically, for a certain selection of tuning parameters and configuration spaces, we obtain a convergence rate bound of order O(n−1/d+ρ), where n is the number of sampled points, d is the dimension of the configuration space, and ρ is an arbitrarily small constant. We go on to demonstrate asymptotic optimality for a number of variations on FMT*, namely when the configuration space is sampled non-uniformly, when the cost is not arc length, and when connections are made based on the number of nearest neighbors instead of a fixed connection radius. Numerical experiments over a range of dimensions and obstacle configurations confirm our the-oretical and heuristic arguments by showing that FMT*, for a given execution time, returns substantially better solutions than either PRM* or RRT*, especially in high-dimensional configuration spaces and in scenarios where collision-checking is expensive.

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Section: The Fast Marching Tree Algorithm (Fmt*)mentioning

confidence: 99%

Fast marching tree: A fast marching sampling-based method for optimal motion planning in many dimensions

Janson

Szczerbicki

Clark

et al. 2015

The International Journal of Robotics Research

454

384

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“…We describe here the algorithm of Cabello and Jejčič [3] to compute a shortest path tree in G from a given root point r ∈ P . As it is usually done for shortest path algorithms, we use tables dist[·] and π[·] indexed by the points of P to record, for each point p ∈ P , the distance d G (s, p) and the ancestor of p in a shortest (s, p)-path.…”

Section: Description Of Algorithms 21 Shortest-path Tree In Unit-dismentioning

confidence: 99%

“…The pseudocode of the algorithm, which we call UnweightedShortestPath, is in Figure 2. We explain the intuition, taken almost verbatim from [3]. We start by computing the Delaunay triangulation DT (P ) of P .…”

Section: Description Of Algorithms 21 Shortest-path Tree In Unit-dismentioning

confidence: 99%

“…Cabello and Jejčič [3] show that the algorithm correctly computes the shortest-path tree from r. If for nearest neighbors we use a data structure that, for n points, has construction time T c (n) and query time T q (n), and the Delaunay triangulation is computed in T DT (n) time, then the algorithm takes O(T DT (n) + T c (n) + nT q (n)) time. Standards tools in Computational Geometry imply that T DT (n) = O(n log n), T c (n) = O(n log n) and T q (n) = O(log n).…”

Section: Description Of Algorithms 21 Shortest-path Tree In Unit-dismentioning

confidence: 99%

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Two optimization problems for unit disks

Cabello

Milinković

2018

Computational Geometry

Self Cite

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We present an implementation of a recent algorithm to compute shortest-path trees in unit disk graphs in O(n log n) worst-case time, where n is the number of disks.In the minimum-separation problem, we are given n unit disks and two points s and t, not contained in any of the disks, and we want to compute the minimum number of disks one needs to retain so that any curve connecting s to t intersects some of the retained disks. We present a new algorithm solving this problem in O(n 2 log 3 n) worst-case time and its implementation.

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“…In 'heat flow' methods, geodesic distances can be found in terms of the small time asymptotics of a heat kernel on the space. For instance, see [10], [11], [12], [13], [33].…”

Section: Introductionmentioning

confidence: 99%

Approximating geodesics via random points

Davis

Sethuraman²

2019

Ann. Appl. Probab.

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Given a 'cost' functional F on paths γ in a domain D ⊂ R d , in the form F (γ) = 1 0 f (γ(t),γ(t))dt, it is of interest to approximate its minimum cost and geodesic paths. Let X 1 , . . . , Xn be points drawn independently from D according to a distribution with a density. Form a random geometric graph on the points where X i and X j are connected when 0 < |X i − X j | < , and the length scale = n vanishes at a suitable rate.For a general class of functionals F , associated to Finsler and other distances on D, using a probabilistic form of Gamma convergence, we show that the minimum costs and geodesic paths, with respect to types of approximating discrete 'cost' functionals, built from the random geometric graph, converge almost surely in various senses to those corresponding to the continuum cost F , as the number of sample points diverges. In particular, the geodesic path convergence shown appears to be among the first results of its kind.2010 Mathematics Subject Classification. 60D05, 58E10, 62-07, 49J55, 49J45, 53C22, 05C82.

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Shortest paths in intersection graphs of unit disks

Cited by 30 publications

References 22 publications

Fast marching tree: A fast marching sampling-based method for optimal motion planning in many dimensions

Fast marching tree: A fast marching sampling-based method for optimal motion planning in many dimensions

Two optimization problems for unit disks

Approximating geodesics via random points

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