On Bounded Leg Shortest Paths Problems

Roditty, Liam; Segal, Michael

doi:10.1007/s00453-009-9322-3

Cited by 14 publications

(22 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

453

365

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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

453

365

View full text Add to dashboard Cite

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“…These are defined as follows: Let S ⊂ R 2 be a finite set of point sites, each with an associated weight w p > 0, p ∈ S. The disk intersection graph for S, D(S), has the sites in S as vertices, and there is an edge between two sites pq in S if and only if |pq| ≤ w p + w q , i.e., if the disk around p with radius w p intersects the disk around q with radius w q . If all weights are 1, we call D(S) the unit disk graph for S. Disk intersection graphs are a popular model for geometrically defined graphs and enjoy an increasing interest in the research community, in particular due to applications in wireless sensor networks [10,14,25,40]. The following table gives an overview of our results.…”

Section: Problemmentioning

confidence: 99%

“…Dynamic connectivity in disk intersection graphs n 20/21 update n 1/7 query [14] Ψ 2 log 9 nλ s+2 (log n) update log n/ log log n query BFS tree in a disk intersection graph n 1+ε [40] n log 9 nλ s+2 (log n)…”

Section: Old Boundmentioning

confidence: 99%

Dynamic Planar Voronoi Diagrams for General Distance Functions and their Algorithmic Applications

Kaplan¹,

Mulzer²,

Roditty³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We describe a new data structure for dynamic nearest neighbor queries in the plane with respect to a general family of distance functions that includes Lp-norms and additively weighted Euclidean distances, and for general (convex, pairwise disjoint) sites that have constant description complexity (line segments, disks, etc.). Our data structure has a polylogarithmic update and query time, improving an earlier data structure of Agarwal, Efrat and Sharir that required O(n ε ) time for an update and O(log n) time for a query [1]. Our data structure has numerous applications, and in all of them it gives faster algorithms, typically reducing an O(n ε ) factor in the bounds to polylogarithmic. To further demonstrate its effectiveness, we give here two new applications: an efficient construction of a spanner in a disk intersection graph, and a data structure for efficient connectivity queries in a dynamic disk graph.To obtain this data structure, we combine and extend various techniques and obtain several side results that are of independent interest. Our data structure depends on the existence and an efficient construction of "vertical" shallow cuttings in arrangements of bivariate algebraic functions. We prove that an appropriate level in an arrangement of a random sample of a suitable size provides such a cutting. To compute it efficiently, we develop a randomized incremental construction algorithm for finding the lowest k levels in an arrangement of bivariate algebraic functions (we mostly consider here collections of functions whose lower envelope * A full version is available on the arXiv [20] has linear complexity, as is the case in the dynamic nearestneighbor context). To analyze this algorithm, we improve a longstanding bound on the combinatorial complexity of the vertical decomposition of these levels. Finally, to obtain our structure, we plug our vertical shallow cutting construction into Chan's algorithm for efficiently maintaining the lower envelope of a dynamic set of planes in R 3 . While doing this, we also revisit Chan's technique and present a variant that uses a single binary counter, with a simpler analysis and an improved amortized deletion time.

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“…We would like to thank Timothy Chan, Alon Efrat, and David Eppstein for several useful comments. In particular, we are indebted to Timothy Chan for pointing out the work of Roditty and Segal [15] and to Alon Efrat for explaining the alternative algorithm for the unweighted case discussed in the introduction.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…Exact computation of shortest paths in unit disks is considered by Roditty and Segal [15], under the name of bounded leg shortest path problem. They show that, for the weighted case, a shortest path tree can be computed in O(n 4/3+ε ) time.…”

Section: Introductionmentioning

confidence: 99%