1. L1 Shortest Path Queries among Polygonal Obstacles in the Plane
Danny Z. Chen1 and Haitao Wang2
1University of Notre Dame
2Utah State University

2. Shortest paths among polygonal obstacles
Input: A set of h polygonal obstacles with a total of 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. Shortest path queries
Build a shortest path map (SPM), such that for any query point t, compute the shortest path from s to t t t t s
obstacle

4. Our problem: the L1 version
L1 shortest path: a path 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

5. Previous work (a single shortest path)
Studied by Clarkson, Lee, Kapoor, Widmayer, Wu, Wong, Vaidya, Chen, Klenk, Tu, Inkulu, etc.
O(nlogn) time and O(n) space, Mitchell 92’
O(T+n+hlogh) time and O(n) space, Chen and Wang 11’
T = O(n+hlog1+εh) : the time for triangulating the free space, Bar-Yehuda and Chazelle, 94’

6. Previous work and our new result (shortest path queries)
Mitchell 92’
Time for building an SPM: O(nlogn)
Space: O(n)
Query time: O(k+logn)
k: the number of edges of the path
Our new result (in this talk)
Time for building an SPM: O(T+n+hlogh)
O(n+hlog h) time, if a triangulation is given

7. More… Our algorithm runs in Θ(T) time Why?
building an SPM is equivalent to triangulating the free space
Why?
Lower bound: T=Ω(n+hlogh)
Given a triangulation,
build an SPM in O(n+hlogh) time
Given an SPM
obtain a triangulation in O(n) time

8. Our approach (previous work)
Reduce the problem to the convex case where all obstacles are convex
Solve the convex case
O(n+hlogh) time and O(n) space
an SPM is also built
Open: Building an SPM for the general (non-convex) problem
the key: solving a sub-problem optimally

9. The key sub-problem a shortest path from s to t s c r t P
The green region is
the Voronoi region
of r, denoted by VR(r) d

10. The key sub-problem Each site has a weight : the length of
the L1 shortest path from s to it c
Goal: Compute the L1 geodesic
Voronoi diagram for the sites that
influence P only through cd, or
compute the Voronoi region for
each site P
N: the number of vertices of P
M: the number of red sites d

11. Difficulties How to handle the interactions of the bisectors? c
Goal: building the SPM
in O(n+hlogh) time
Goal for solving the
sub-problem: O(M+N) time P d

12. The standard approaches
O((N+M)log(N+M))  overall O(nlogn) time
Continuous Dijkstra paradigm
Sweeping
Divide-and-conquer
Our approach: incremental
O(N+M)

13. Bisectors of two weighted sites
middle segment
A bisector: a middle segment, and two half-lines, vertical or horizontal

14. Our algorithm Order the sites r1, r2, … rm by
the sub-segments of cd contained
in their cells from c to d r1 c r2 r3 P r4 d

15. Obtaining the rays r1 c r2 Middle segments must appear in the Voronoi
diagram r3 P r4 r5 d

16. Preprocessing c sites for the yellow region r2 p q
sites for the brown region r3 P d

17. Processing the rays incrementally
VR(r1) r1 c
Vertical rays are stored
in a stack r2 r3
VR(r3) P r4 r5 d

18. Processing the rays incrementally
VR(r1) r1 c r2 r3
VR(r3) r4 P r5 d

19. Processing the rays incrementally
VR(r1) r1 c
Vertical rays are stored
in a stack r2 r3 r4 P r5 d

20. Processing the rays incrementally
VR(r1) r1 c r2 r3 r4
VR(r4) P r5 d

21. Processing the rays incrementally
VR(r1) r1 c r2 r3 r4
VR(r4) P
VR(r3) r5 d

22. Processing the rays incrementally
VR(r1) r1 c r2 r3
VR(r2) r4 P r5
VR(r3) d

23. Implementation
Data structures: linked lists, a stack, vertical and horizontal visibility decompositions of P
horizontal visibility decomposition
vertical visibility decomposition

24. Conclusions Build an L1 shortest path map in Θ(T+n+hlogh) time
Build a special L1 weighted geodesic Voronoi diagram
Linear time

25. Thank You