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 (in SIAM Journal on Computing , 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 (in SODA’21) reduced the space to O(n) while the runtime of the algorithm is still O(n log n ). In this article, 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 is better than the previous work when h is relatively small. 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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Shortest Paths in the Plane with Obstacle Violations
We study the problem of finding shortest paths in the plane among h convex obstacles, where the path is allowed to pass through (violate) up to k obstacles, for 𝑘≤ℎ. Equivalently, the problem is to find shortest paths that become obstacle-free if k obstacles are removed from the input. Given a fixed source point s, we show how to construct a map, called a shortest k-path map, so that all destinations in the same region of the map have the same combinatorial shortest path passing through at most k obstacles. We prove a tight bound of 𝛩(𝑘𝑛) on the size of this map, and show that it can be computed in 𝑂(𝑘2𝑛log𝑛) time, where n is the total number of obstacle vertices.
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- Award ID(s):
- 1814172
- PAR ID:
- 10168685
- Date Published:
- Journal Name:
- Algorithmica
- Volume:
- 82
- ISSN:
- 0178-4617
- Page Range / eLocation ID:
- 1813-1832
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/s00453-020-00673-y
