Computing $$L_1$$ Shortest Paths Among Polygonal Obstacles in the Plane

Chen, Danny Z.; Wang, Haitao

doi:10.1007/s00453-018-00540-x

Cited by 6 publications

(11 citation statements)

References 33 publications

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“…The 𝐿 1 shortest path problem where the path length is measured in the 𝐿 1 metric also attracted much attention, e.g., [1,9,12,13,33,35]. For polygons with holes, Mitchell [33,35] gave an algorithm that can build a shortest path map for a source point in 𝑂 (𝑛 log 𝑛) time and 𝑂 (𝑛) space; for small ℎ, Chen and Wang [9] proposed an algorithm of 𝑂 (𝑛 +ℎ log ℎ) time and 𝑂 (𝑛) space, after the free space is triangulated. For simple polygons, Bae and Wang [1] built a data structure in 𝑂 (𝑛) time that can answer each two-point shortest path query in 𝑂 (log 𝑛) time.…”

Section: Previous Workmentioning

confidence: 99%

A new algorithm for Euclidean shortest paths in the plane

Wang

2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

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Given a set of pairwise disjoint polygonal obstacles in the plane, finding an obstacle-avoiding Euclidean shortest path between two points is a classical problem in computational geometry and has been studied extensively. Previously, Hershberger and Suri [SIAM J. Comput. 1999] gave an algorithm of 𝑂 (𝑛 log 𝑛) time and 𝑂 (𝑛 log 𝑛) space, where 𝑛 is the total number of vertices of all obstacles. Recently, by modifying Hershberger and Suri's algorithm, Wang [SODA 2021] reduced the space to 𝑂 (𝑛) while the runtime of the algorithm is still 𝑂 (𝑛 log 𝑛). In this paper, we present a new algorithm of 𝑂 (𝑛 + ℎ log ℎ) time and 𝑂 (𝑛) space, provided that a triangulation of the free space is given, where ℎ is the number of obstacles. Our algorithm builds a shortest path map for a source point 𝑠, so that given any query point 𝑡, the shortest path length from 𝑠 to 𝑡 can be computed in 𝑂 (log 𝑛) time and a shortest 𝑠-𝑡 path can be produced in additional time linear in the number of edges of the path. CCS CONCEPTS• Theory of computation → Computational geometry; Design and analysis of algorithms.

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Section: Previous Workmentioning

confidence: 99%

A new algorithm for Euclidean shortest paths in the plane

Wang

2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

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“…The high-level scheme of the algorithm is similar in spirit to that for the L 1 problem [10]. The algorithm has three main steps.…”

Section: Expanding Spm (M) Into All Canalsmentioning

confidence: 99%

A New Algorithm for Euclidean Shortest Paths in the Plane

Wang

2021

Preprint

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Given a set of pairwise disjoint polygonal obstacles in the plane, finding an obstacle-avoiding Euclidean shortest path between two points is a classical problem in computational geometry and has been studied extensively. Previously, Hershberger and Suri [SIAM J. Comput. 1999] gave an algorithm of O(n log n) time and O(n log n) space, where n is the total number of vertices of all obstacles. Recently, by modifying Hershberger and Suri's algorithm, Wang [SODA 2021] reduced the space to O(n) while the runtime of the algorithm is still O(n log n). In this paper, we present a new algorithm of O(n + h log h) time and O(n) space, provided that a triangulation of the free space is given, where h is the number of obstacles. The algorithm, which improves the previous work when h = o(n), is optimal in both time and space as Ω(n + h log h) is a lower bound on the runtime. Our algorithm builds a shortest path map for a source point s, so that given any query point t, the shortest path length from s to t can be computed in O(log n) time and a shortest s-t path can be produced in additional time linear in the number of edges of the path.

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“…Better results are also known for one-point queries in the L 1 metric [8,11,12,22,24,25], i.e., s is fixed in the input and only t is a query point. In particular, Mitchell [24,25] built a data structure of O(n) size in O(n log n) time that can answer each such query in O(log n) time.…”

Section: Related Workmentioning

confidence: 99%

“…In particular, Mitchell [24,25] built a data structure of O(n) size in O(n log n) time that can answer each such query in O(log n) time. Later Chen and Wang [8] reduced the preprocessing time to O(n + h log h) if P is already triangulated (which can be done in O(n log n) or O(n + h log 1+ǫ h) time for any ǫ > 0 [3,4]), while the query time is still O(log n).…”

Section: Related Workmentioning

confidence: 99%

A Divide-and-Conquer Algorithm for Two-Point $L_1$ Shortest Path Queries in Polygonal Domains

Wang

2019

Preprint

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Let P be a polygonal domain of h holes and n vertices. We study the problem of constructing a data structure that can compute a shortest path between s and t in P under the L1 metric for any two query points s and t. To do so, a standard approach is to first find a set of ns "gateways" for s and a set of nt "gateways" for t such that there exist a shortest s-t path containing a gateway of s and a gateway of t, and then compute a shortest s-t path using these gateways. Previous algorithms all take quadratic O(ns • nt) time to solve this problem. In this paper, we propose a divide-and-conquer technique that solves the problem in O(ns + nt log ns) time. As a consequence, we construct a data structure of O(n + (h 2 log 3 h/ log log h)) size in O(n + (h 2 log 4 h/ log log h)) time such that each query can be answered in O(log n) time.

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Computing $$L_1$$ Shortest Paths Among Polygonal Obstacles in the Plane

Cited by 6 publications

References 33 publications

A new algorithm for Euclidean shortest paths in the plane

A new algorithm for Euclidean shortest paths in the plane

A New Algorithm for Euclidean Shortest Paths in the Plane

A Divide-and-Conquer Algorithm for Two-Point $L_1$ Shortest Path Queries in Polygonal Domains

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