Euclidean shortest paths in the presence of rectilinear barriers

Lee, Der-Tsai; Preparata, Franco P.

doi:10.1002/net.3230140304

Cited by 353 publications

(184 citation statements)

References 11 publications

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“…Computing one of them individually can be done in O(n) time [17]. However, this time complexity can be reduced by utilizing similarities between shortest paths that start and end close to each other.…”

Section: The Computation In Stepmentioning

confidence: 99%

Computational Science and Its Applications – ICCSA 2004

2004

Lecture Notes in Computer Science

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Abstract. Given a simple n-sided polygon in the plane with a boundary partitioned into subchains some of which are convex and colored, we consider the following problem: Which is the shortest route (closed path) contained in the polygon that passes through a given point on the boundary and intersects at least one vertex in each of the colored subchains? We present an optimal algorithm that solves this problem in O(n) time. Previously it was known how to solve the problem optimally when each colored subchain contains one vertex only. Moreover, we show that a solution computed by the algorithm is at most a factor 2+c c times longer than the overall shortest route that intersects the subchains (not just at vertices) if the minimal distance between vertices of different subchains is at least c times the maximal length of an edge of a subchain. Without such a bound its length can be arbitrarily longer. Furthermore, it is known that algorithms for computing such overall shortest routes suffer from numerical problems. Our algorithm is not subject to such problems.

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Section: The Computation In Stepmentioning

confidence: 99%

Computational Science and Its Applications – ICCSA 2004

2004

Lecture Notes in Computer Science

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“…and g~ as Evaluating these at x = b yields Therefore, we see that we can always choose a direction to move in so as to make J{~ negative, which in turn implies \Vhich implies that we can al ways move qo in sorne direction so as not to decrease the lengths of SI and S2. In fact, we can rnove to the nearest vertex in said direction, since once we move a distance 6.x, the path lengths are no longer equal, and The algorithm for the construction requires linear plus triangulation time and is an extension of an algorithm due to Lee and Preparata [10].…”

Section: A Qo)mentioning

confidence: 99%

Computing the external geodesic diameter of a simple polygon

Samuel

Toussaint

1990

Computing

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Given a simple polygon P of n vertices, we present an algorithm that finds the pair of points on the boundary of P that maximize the externat shortest path between them. This path is defined as the externat geodesic diameter of P. The algorithm takes O(n 2 ) time and requires D(n) space. Surprisingly, this problem is quite different from that of computing the internai geodeslc diameter of P. WhiIe the internai diameter le; determined by a pair of verticls of P, this is not the case for the external diameter.Finally, we show how this algo~ithm can be extended to solve the ail external geodesicfurthest neighbours problem.-- R~suméCe travail présente un a1gorithme qui trouve une paire de points sur la frontière d'un polygone simple P de n coins, qui maximise le trajectoire externe le plus court entre les deux points. Ce trajectoire est définit par le diamètre géodésique externe de P. L'algorithrrle prend un temps O(n 2 ) et requiert un champ O(n). Fait surprennant, ce ploblème n'est pas semblable au prlJblème de calculer le diamètre géodésique interne. Bien que le diamètre interne est delerminé par une paire de coins de P, ce n'est pas le cas pour le diamètre externe. En fin, ce travail demontre comment cet algorithme peut être étendu pour résoudre le problème de tous les voisins à la plus grande distance géodésique externe.

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“…A free path will cross several unconstrained edges resulting in a "channel" of connected simplexes formed of all traversed triangles. A path solution through this channel is determined with a "funnel" algorithm developed by Lee and Preparata, and Chazelle [13,14] as cited by Hershberger [15]. The funnel algorithm has been demonstrated under multiple applications, including path finding for autonomous agents [16], querying visible points in large data sets to define shortest paths [17], shortest paths for tethered robots [18], and robots in extreme terrain [19].…”

Section: Introductionmentioning

confidence: 99%

Simplex Solutions for Optimal Control Flight Paths in Urban Environments

Zollars¹,

Rg²,

Dj³

2017

J Aeronaut Aerospace Eng

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This paper identifies feasible fight paths for Small Unmanned Aircraft Systems in a highly constrained environment. Optimal control software has long been used for vehicle path planning and has proven most successful when an adequate initial guess is presented flight to an optimal control solver. Leveragingfast geometric planning techniques, a large search space is discretized into a set of simplexes where a Dubins path solution is generated and contained in a polygonal search corridor free of path constraints. Direct optimal control methods are then used to determine the optimal flight path through the newly defined search corridor. Two scenarios are evaluated. The first is limited to heading rate control only, requiring the air vehicle to maintain constant speed. The second allows for velocity control which permits slower speeds, reducing the vehicles minimum turn radius and increasing the search domain. Results illustrate the benefits gained when including speed control to path planning algorithms by comparing trajectory and convergence times, resulting in a reliable, hybrid solution method to the SUAS constrained optimal control problem.

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Euclidean shortest paths in the presence of rectilinear barriers

Cited by 353 publications

References 11 publications

Computational Science and Its Applications – ICCSA 2004

Computational Science and Its Applications – ICCSA 2004

Computing the external geodesic diameter of a simple polygon

Simplex Solutions for Optimal Control Flight Paths in Urban Environments

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