1. Danny Z. Chen and Haitao Wang University of Notre Dame Indiana, USA
A Nearly Optimal Algorithm for Finding L1 Shortest 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
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

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