Two-Point L1 Shortest Path Queries in the Plane

Chen, Danny Z.; Inkulu, R.; Wang, Haitao

doi:10.1145/2582112.2582125

Cited by 14 publications

(58 citation statements)

References 45 publications

(213 reference statements)

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“…For two-point L 1 shortest path queries, Chen et al [8] constructed a data structure of size O(n 2 log n) in O(n 2 log 2 n) time that can answer each query in O(log 2 n) time. Recently, Chen et al [7] reduced the query time to O(log n) by building a data structure of size O(n + h 2 · log h · 4 √ log h ) in O(n + h 2 · log 2 h · 4 √ log h ) time. To find a minimum-link s-t path between two points s and t in an arbitrary polygonal domain P, Mitchell [23] gave an O(Eα(n) log 2 n) time algorithm, where α(n) is the inverse of Ackermann's function and E is the size of the visibility graph of P and E = Θ(n 2 ) in the worst case.…”

Section: Other Related Workmentioning

confidence: 99%

“…The main idea is to build an enhanced graph G E (B) of larger size on the backbone points of B, so that we only need a set of O( √ log h) gateways for each of s and t, which reduces the query time by a factor of log h. The details are given below. The enhanced graph G E (B) is still built on B with respect to the reduced domain P r introduced in [7] for more details. We first discuss the minimum-link shortest paths.…”

Section: Reducing the Query Timesmentioning

confidence: 99%

“…This can be We assume that both s and t are in junction rectangles. We first compute their gateways sets V g (s) and V g (t), which can be done again in O(log h) time (with O(n + h log 3/2 h2 √ log h ) time and O(n + h log 1/2 h2 √ log h ) space preprocessing) [7]. Since the sizes of both gateway sets are bounded by O( √ log h), we only need to consider O(log h) paths to extend to connect s and t. Therefore, the query time becomes O(log h).…”

Section: Reducing the Query Timesmentioning

confidence: 99%

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Bicriteria Rectilinear Shortest Paths Among Rectilinear Obstacles in the Plane

Wang

2019

Discrete Comput Geom

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Given a rectilinear domain P of h pairwise-disjoint rectilinear obstacles with a total of n vertices in the plane, we study the problem of computing bicriteria rectilinear shortest paths between two points s and t in P. Three types of bicriteria rectilinear paths are considered: minimum-link shortest paths, shortest minimum-link paths, and minimum-cost paths where the cost of a path is a non-decreasing function of both the number of edges and the length of the path. The one-point and two-point path queries are also considered. Algorithms for these problems have been given previously. Our contributions are threefold. First, we find a critical error in all previous algorithms. Second, we correct the error in a not-so-trivial way. Third, we further improve the algorithms so that they are even faster than the previous (incorrect) algorithms when h is relatively small. For example, for the minimum-link shortest paths, we obtain the following results. Our algorithm computes a minimum-link shortest s-t path in O(n + h log 3/2 h) time. For the one-point queries, we build a data structure of size O(n + h log h) in O(n + h log 3/2 h) time for a source point s, such that given any query point t, a minimum-link shortest s-t path can be computed in O(log n) time. For the two-point queries, with O(n + h 2 log 2 h) time and space preprocessing, a minimum-link shortest s-t path can be computed in O(log n + log 2 h) time for any two query points s and t; alternatively, with O(n + h 2 · log 2 h · 4 √ log h ) time and O(n + h 2 · log h · 4 √ log h ) space preprocessing, we can answer each two-point query in O(log n) time. Note that h 2 · log 2 h · 4 √ log h = O(h 2+ǫ ) for any ǫ > 0. These results are particularly interesting when h is relatively small. For example, if h = O(n 1/2−ǫ ) for any ǫ > 0, then all above results match the best results for the problems in simple rectilinear polygons, which are optimal. The complexities for the other two types of paths are slightly worse, but still linearly depend on n (in addition to g(h) for some functions g(h) of h).

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Section: Other Related Workmentioning

confidence: 99%

Section: Reducing the Query Timesmentioning

confidence: 99%

Section: Reducing the Query Timesmentioning

confidence: 99%

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Bicriteria Rectilinear Shortest Paths Among Rectilinear Obstacles in the Plane

Wang

2019

Discrete Comput Geom

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“…The corridor structure and its extended version for polygonal domains have been used for solving shortest path and visibility problems [Chen et al 2014;Chen and Wang 2011b, 2012, 2013a, 2013b, 2013cInkulu and Kapoor 2009;Kapoor and Maheshwari 1988;Kapoor et al 1997]. For our splinegonal domain, we generalize the approach in Kapoor et al [1997] for the polygonal domain case.…”

Section: The Corridor Structurementioning

confidence: 99%

Computing Shortest Paths among Curved Obstacles in the Plane

Chen

Wang

2015

ACM Trans. Algorithms

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A fundamental problem in computational geometry is to compute an obstacle-avoiding Euclidean shortest path between two points in the plane. The case of this problem on polygonal obstacles is well studied. In this article, we consider the problem version on curved obstacles, which are commonly modeled as splinegons. A splinegon can be viewed as replacing each edge of a polygon by a convex curved edge (polygons are special splinegons), and the combinatorial complexity of each curved edge is assumed to be O(1). Given in the plane two points s and t and a set S of h pairwise disjoint splinegons with a total of n vertices, after a bounded degree decomposition of S is obtained, we compute a shortest s-to-t path avoiding the splinegonswhere k is a parameter sensitive to the geometric structures of the input and is upper bounded by O(h 2 ). The bounded degree decomposition of S, which is similar to the triangulation of the polygonal domains, can be computed in O(n log n) time or O(n + h log 1+ h) time for any > 0. In particular, when all splinegons are convex, the decomposition can be computed in O(n + h log h) time and k is linear to the number of common tangents in the free space (called "free common tangents") among the splinegons. Our techniques also improve several previous results: (1) For the polygon case (i.e., when all splinegons are polygons), the shortest path problem was previously solved in O(n log n) time, or in O(n+ h 2 log n) time. Thus, our algorithm improves the O(n+ h 2 log n) time result, and is faster than the O(n log n) time solution for sufficiently small h, for example, h = o( n log n). (2) Our techniques produce an optimal output-sensitive algorithm for a basic visibility problem of computing all free common tangents among h pairwise disjoint convex splinegons with a total of n vertices. Our algorithm runs in O(n+h log h+k) time and O(n) working space, where k is the number of all free common tangents. Note that k = O(h 2 ). Even for the special case where all splinegons are convex polygons, the previously best algorithm for this visibility problem takes O(n + h 2 log n) time. (3) We improve the previous work for computing the shortest path between two points among convex pseudodisks of O(1) complexity each.In addition, a by-product of our techniques is an optimal O(n + h log h) time and O(n) space algorithm for computing the Voronoi diagram of a set of h pairwise disjoint convex splinegons with a total of n vertices.

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“…Figure 5: Illustration of the proof of Lemma 22: C 1 (resp., C 2 ) is the (red) chain connecting u and a (resp., b). To prove the lemma, we use some results given in [7]. First of all, we use the fact that x 1 is below x 2 .…”

Section: Computing the Center In Linear Timementioning

confidence: 99%

Computing the L 1 Geodesic Diameter and Center of a Simple Polygon in Linear Time

Bae

Korman

Okamoto

et al. 2014

LATIN 2014: Theoretical Informatics

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In this paper, we show that the L1 geodesic diameter and center of a simple polygon can be computed in linear time. For the purpose, we focus on revealing basic geometric properties of the L1 geodesic balls, that is, the metric balls with respect to the L1 geodesic distance. More specifically, in this paper we show that any family of L1 geodesic balls in any simple polygon has Helly number two, and the L1 geodesic center consists of midpoints of shortest paths between diametral pairs. These properties are crucial for our linear-time algorithms, and do not hold for the Euclidean case.

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Two-Point L1 Shortest Path Queries in the Plane

Cited by 14 publications

References 45 publications

Bicriteria Rectilinear Shortest Paths Among Rectilinear Obstacles in the Plane

Bicriteria Rectilinear Shortest Paths Among Rectilinear Obstacles in the Plane

Computing Shortest Paths among Curved Obstacles in the Plane

Computing the L 1 Geodesic Diameter and Center of a Simple Polygon in Linear Time

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