Shortest Paths in the Plane with Obstacle Violations

Hershberger, John; Kumar, Neeraj; Suri, Subhash

doi:10.1007/s00453-020-00673-y

Cited by 4 publications

(3 citation statements)

References 29 publications

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“…They also enter into the definition of key structural properties such as the closeness, the efficiency, or the betweenness centrality [11,12]. The definition of a shortest path between two nodes can be generalized to include, for example, constraints or obstacles (see, e.g., [13][14][15][16]) or to reach several targets from one or several sources (see, e.g., [17,18]). A great deal of work in the applied discrete mathematics, theoretical computer science, and statistical physics communities has dealt with the solutions of such problems, and the computational complexity of the algorithms involved to find them.…”

Section: Introductionmentioning

confidence: 99%

Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks

Gutiérrez

Pérez‐Espigares

2021

Phys. Rev. E

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Numerous problems of both theoretical and practical interest are related to finding shortest (or otherwise optimal) paths in networks, frequently in the presence of some obstacles or constraints. A somewhat related class of problems focuses on finding optimal distributions of weights which, for a given connection topology, maximize some kind of flow or minimize a given cost function. We show that both sets of problems can be approached through an analysis of the large-deviation functions of random walks. Specifically, a study of ensembles of trajectories allows us to find optimal paths, or design optimal weighted networks, by means of an auxiliary stochastic process (the generalized Doob transform). The paths are not limited to shortest paths, and the weights must not necessarily optimize a given function. Paths and weights can in fact be tailored to a given statistics of a time-integrated observable, which may be an activity or current, or local functions marking the passing of the random walker through a given node or link. We illustrate this idea with an exploration of optimal paths in the presence of obstacles, and networks that optimize flows under constraints on local observables.

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Section: Introductionmentioning

confidence: 99%

Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks

Gutiérrez

Pérez‐Espigares

2021

Phys. Rev. E

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show abstract

“…The definition of a shortest path between two nodes can be generalized to include, for example, constraints or obstacles (see e.g. [13][14][15][16]) or to reach several targets from one or several sources (see e.g. [17,18]).…”

Section: Introductionmentioning

confidence: 99%

Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks

Gutiérrez,

Pérez-Espigares

2020

Preprint

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Numerous problems of both theoretical and practical interest are related to finding shortest (or otherwise optimal) paths in networks, frequently in the presence of some obstacles or constraints. A somewhat related class of problems focus on finding optimal distributions of weights which, for a given connection topology, maximize some kind of flow or minimize a given cost function. We show that both sets of problems can be approached through an analysis of the large-deviation functions of random walks. Specifically, a study of ensembles of trajectories allows us to find optimal paths, or design optimal weighted networks, by means of an auxiliary stochastic process (the generalized Doob transform). In contrast to standard approaches, the paths are not limited to shortest paths nor the weights need to optimize a given function. Paths and weights can in fact be tailored to a given statistics of a time-integrated observable, which may be an activity or current, or local functions marking the passing of the random walker through a given node or link. We illustrate this idea with an exploration of optimal paths in the presence of obstacles and optimal networks for flows in the presence of forbidden regions or upper bounds in the local activity.

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“…Later [7], he proposed an O(n 4 ) time algorithm for a variant of this problem in which the obstacles are allowed to occupy the same area of the plane (i.e., intersecting obstacles). A recently introduced model [9] considers another variation of this problem, where the path is allowed to pass through k obstacles. They present an O(k 2 n log n) time algorithm, where n is the total number of obstacle vertices.…”

Section: Introductionmentioning

confidence: 99%

Rectilinear Shortest Paths Among Transient Obstacles

Maheshwari

Nouri

Sack

2018

Combinatorial Optimization and Applications

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This paper presents an optimal Θ(n log n) algorithm for determining time-minimal rectiliear paths among n transient rectilinear obstacles. An obstacle is transient if it exists in the scene only for a specific time interval, i.e., it appears and then disappears at specific times. Given a point robot moving with bounded speed among transient rectilinear obstacles and a pair of points s, d, we determine a time-minimal, obstacle-avoiding path from s to d. The main challenge in solving this problem arises as the robot may be required to wait for an obstacle to disappear, before it can continue moving toward the destination. Our algorithm builds on the continuous Dijkstra paradigm, which simulates propagating a wavefront from the source point. We also solve a query version of this problem. For this, we build a planar subdivision with respect to a fixed source point, so that minimum arrival time to any query point can be reported in O(log n) time, using point location for the query point in this subdivision.

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Shortest Paths in the Plane with Obstacle Violations

Cited by 4 publications

References 29 publications

Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks

Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks

Generalized optimal paths and weight distributions revealed through the large deviations of random walks on networks

Rectilinear Shortest Paths Among Transient Obstacles

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