A Nearly Optimal Algorithm for Finding L 1 Shortest Paths among Polygonal Obstacles in the Plane

1. A Nearly Optimal Algorithm for Finding L1Shortest Paths among Polygonal Obstacles in the Plane Danny Z. Chen and Haitao Wang University of Notre Dame Indiana, USA

2. Shortest paths among polygonal obstacles • Input: A set of h polygonal obstacles with totally n vertices, and two points s and t • Output: A shortest path from s to t that avoids the obstacles t free space s obstacle

3. Our problem: the L1 version • L1 shortest path problem: Find a path from s to t with minimum L1 distance • The path can be arbitrarily polygonal • But the distance is measured by L1 metric vertical distance e horizontal distance The L1 distance of e = the horizontal distance + the vertical distance

4. Previous work (the first approach) • Construct a “path preserving graph” and then run Dijkstra’s algorithm on the graph • Key: The size of the graph should be as small as possible • O(nlog2n) time and O(nlogn) space, Clarkson, Kapoor, and Vaidya, 87’ • O(nlog1.5n) time and O(nlog1.5n) space, Clarkson, Kapoor, and Vaidya, 88’ • O(nlog1.5n) time and O(nlogn) space, Chen, Klenk, and Tu, 00’ • O(n+hlog1.5h) time and O(n+hlog1.5h) space, Inkulu and Kapoor, 09’ • a corridor structure

5. Previous work (the second approach) • Continuous Dijkstra paradigm s t

6. Continuous Dijkstra paradigm for L1 metric s The wavelets are of diamond shape

7. Previous work (cont.) • Continuous Dijkstra paradigm • O(nlogn) time and O(n) space, Mitchell 92’ • Lower bound: O(n+hlogh) time and O(n) space • Mitchell’s algorithm is worst case optimal

8. Our results O(T+n+hlogh) time and O(n) space T is the time for triangulating the free space T=O(n+hlog1+εh), Bar-Yehuda and Chazelle, 94’ t s obstacle

9. Our results (cont.) • O(T+n+hlogh) time and O(n) space • T is the time for triangulating the free space • T=O(n+hlog1+εh), Bar-Yehuda and Chazelle, 94’ • Build shortest path maps to answer L1 shortest path queries • Improvements on other problems • Shortest rectilinear path • Shortest L∞ path • Shortest path of c-orientations, for a given number c • Approximation algorithms for finding an Euclidean shortest path

10. Our approach • A combination of the corridor structure and the continuous Dijkstra paradigm (modifying Mitchell’s algorithm) • Use a corridor structure to somehow reduce the problem to the convex case where all obstacles are convex • Solve the convex case in O(n+hlogh) time and O(n) space

11. The convex case • Input: a set of h convex polygonal obstacles of totally n vertices • Output: a shortest path from s to t in the free space t s

12. The core of a convex obstacle • For any convex obstacle P, define its core to be the polygon by connecting the topmost, rightmost, bottommost, leftmost vertices of P The blue polygon is the core of the yellow obstacle

13. Why do we need the cores? • A critical observation: a shortest s-t path avoiding all cores the same L1 distance a shortest s-t path avoiding all input obstacles

14. Our algorithm for the convex case • Compute the core of each convex obstacle • Apply Mitchell’s algorithm on all cores to find a shortest path p from s to t avoiding all cores • Based on the path p, find a shortest path from s to t avoiding all input obstacles

15. An example t s

16. Our algorithm for the convex case • Compute the core of each convex obstacle • O(n) time • Apply Mitchell’s algorithm on all cores to find a shortest path p from s to t avoiding all cores • O(hlogh) time since all cores contains only O(h) vertices • Based on the path p, find a shortest path from s to t avoiding all input obstacles • O(n) time

17. Correctness of the algorithm • Ears of a convex obstacle c a e b f d

18. An example t s

19. Our algorithm for the general case • The obstacles are not convex t s obstacle

20. Our algorithm for the general case (cont.) • By using a corridor structure, partition the plane into a set P’ of O(h) convex obstacles of O(n) vertices, along with O(h) corridor paths contained in obstacles • P: the input obstacle set • A shortest path from s to t avoiding P is a shortest path avoiding P’ but possibly containing some corridor paths

21. An s-t path may contain some corridor paths a corridor path t s A standard technique used in the path preserving approach (also for the Euclidean case)

22. What’s new? • Modify the continuous Dijkstra paradigm on the convex obstacle set P’ and the corridor paths

23. Some algorithm details • Define the core of a convex obstacle in P’ differently • The endpoints of corridor paths are also considered as the core vertices • The total number of vertices in all cores of P’ is still O(h) • There are O(h) corridor paths an ordinary convex obstacle the obstacle contains a corridor path

24. Some algorithm details (cont.) • Whenever a wavelet encounters an endpoint a of a corridor path p, initiate a “pseudo-wavelet” at a • the event point: the other endpoint b of p • the even distance: the distance of the corridor path b s a There are O(h) such additional events for the endpoints of the corridor paths The overall running time of the modified Mithchell’s algorithm is still O(hlogh)

25. Conclusions • Using the cores to compute a shortest path • The combination of the corridor structure and the continuous Dijkstra paradigm

26. Thank you