Voronoi Diagram for Services Neighboring a Highway

1. Voronoi Diagram for Services Neighboring a Highway Information Processing Letters 86 (2003) 283-288 M. Abellanas

2. Outline • Introduction • Time distance • Time Voronoi Diagram • The v∞ Straight Line Transportation Model

3. Introduction • Voronoi Diagram • The partitioning of a plane with n points into n convex polygon such that each polygon contains exactly one point and every point in a given polygon is closer to its central point than to any other. A Voronoi diagram is sometimes also known as a Dirichlet tessellation.

4. Voronoi Diagram

5. Introduction • When several suppliers provide similar services, which service can each customer reach faster? • Distance  Time distance • Assumptions on the nature of the movements in the plane have to be made precise.

6. Straight Line Transportation Model • The model we consider in this paper is the following: • There is a big high way crossing some area which we will describe as a line in the plane

7. Straight Line Transportation Model • Travelers can enter the high way at any point and travel in it at speed v in both directions • Out of the high way travelers can move freely, and the traveling speed in any direction is v0 <<v

8. Some Assumptions • The transportation line L lies on the x-axis • L+(L-): The halfplanes containing the points with non-negative (non-positive) y-coordinates. • Let v0 = 1

9. Time Distance • Giving two points p, qR2, the time distance between p and q is the time required to follow any of the shortest time paths between them. • How to find the shortest time path? • Where to enter the high way? Where to exit the high way?

10. Where to Enter the High Way p sinα = v0/v = 1/v α q pl pr L α

11. Terminology • d(p,q): the Euclidean distance between p and q • α(0,π/2), sinα=1/v • For p  L+, we denote by pr (right foot-point for p) and pl the intersection of L with the lines through p whose slopes are -1/tanα and 1/tanα

12. Time Distance q p L pr ql α ps Sr(p)

13. Network Function • net: R2×R2 → R 0 • Let q be to the right of p, pL+. If qL+(L-), let t be the intersection point of L with the segment qps (pq), where ps is the symmetric point of p with respect to L

14. Network Function if t is to the right of pr otherwise

15. Terminology • Given p L+, let Sl(p) (Sr(p)) be the halfline with endpoint ps and slope tanα (-tanα) to the left (right) of ps

16. Network Function q p If q is in the shadowed region, then net(p,q)=d(q,Sr(p)) L pr ql α ps Sr(p)

17. Lemma 1 • Let p L+ and let q be an arbitrary point in the plane such that t is to the right of pr if q L+ if q  L-

18. Lemma 2 • Time distance • dt: R2×R2 → R 0 • For any two points p, qR2

19. Theorem 3 • The function dt: R2×R2 → R 0 is a distance function

20. Theorem 4 • Let p L+ and let q be an arbitrary point in the plane if q L+ if q  L-

21. Time Voronoi Diagram • Unit ball Unit ball for a point far away from L (p1), a point close to L (p2) and a point on L (p3) L

22. Terminology Sl+(p) Sl+(q) Sr+(p) Sr+(q) p, qs L ps, q Sr-(p) Sr-(q) Sl-(p) Sl-(q)

23. Terminology • Sa (above): the set of all Sl-(p) and Sr-(p) and allp S, Sb (below) is similar but exchange + and - • VR(x, X): the Euclidean Voronoi Region of a point xX with respect to the set X • TVR(p, S): the Time Voronoi Region for a point p  S with respect to the set S

24. Theorem 5 • The Time Voronoi Region, TVR(p, S), for a point p S with respect to the set S is

25. Time Voronoi Region

26. Lemma 6 • Only the Euclidean Voronoi Region of the line segments on the upper envelope of Sa can intersect L+

27. Lemma 7 • The upper envelope of the halflines in Sa can be computed in O(nlogn) time • fi: combine Sl-(pi) and Sr-(pi) • Upper envelope = , x R • Every function can contribute with at most one piece • Devide and conquer?

28. Theorem 8 • The Time Voronoi diagram of a point set under the Straight Line Transportation Model can be computed in time O(nlogn) via a direct specific algorithm

29. The v∞ Straight Line Transportation Model • Let S be a set of points in the plane where no two points share the samey-coordinate • The first point reaching L, will dominate the whole transportation line • Network function: net(p, q)=|py|+|qy|

30. The v∞ Straight Line Transportation Model • Sr(p) and Sl(p) are parallel to L • If p S is the closest point in S to the transportation line, then p is the only point in S such that TVR(p, S)∩L is not empty

31. Theorem 9 • The Time Voronoi Diagram of a point set under the v∞ Straight Line Transportation Model can be computed in time O(nlogn) via an adaptation of the sweepline algorithm for the standard Euclidean Voronoi Diagram