Time-Based Voronoi Diagram

1. Time-Based Voronoi Diagram D. T. LeeInstitute of Information Science Academia Sinica, Taipei, Taiwan dtlee@iis.sinica.edu.tw Jointly with C. S. Liao, W. B. Wang, IIS.

2. Outline • Introduction • Preliminaries • Good intersection condition • General condition • Conclusion

3. Multiple Highways Model • Input: A set S of points, S={p1, …, pn} in the plane and k highways L1, …, Lk, modeled as lines. • Travelers can enter the highways at any point and move along Li at speed vi in both directions. • Off the highways travelers can move freely in any direction at speed v0<<v1,…, vk. • Output: A Voronoi diagram for the input based on traveling time, i.e. Time-based Voronoi Diagram

4. Time Distance • Given two points p, q in the plane, the shortest time pathspt(p, q) is a path that takes the shortest time traveling between p to q. • The time distancedt(p, q) between p and q is the time required to follow any shortest time paths between p and q.

5. One Highway Problem • Abellanas, Hurtado, Sacristan, Icking, Ma, Klein, Langetepe, Palop IPL, 2003 • Assumption • L1 lies on the x-axis. • sine = v0/v1 = 1/v1 • L1+: the half-plane above L1 • L1-: the half-plane below L1

6. Where to Enter the Highway p sine  = v0/v1 = 1/v1 α q pl pr L1 α

7. Time Distance q p L1 pr ql α pL1

8. Terminology • : the symmetric point of p reflecting by L1. • Given a site p, let be the half-ray with endpoint p and of slope tan  (-tan ) respectively.

9. q p L1

10. q L1 p

11. Approach • Transform the 1-highway problem into another problem in time distance. • If q and p are on the same side, the time distance between q to p must be one of the Euclidean distances from q to • Otherwise, the time distance between q to p must be one of the Euclidean distances from q to

12. Vor() & Vort() • Vor(x, X): the Euclidean Voronoi Region of a site or a line xX with respect to the set X. • Vort(x, X): the time-based Voronoi Region of a site or a line xX with respect to the set X.

13. Theorem [Abellanas, et al.] • For p L+ • For p L-

14. Envelope & Objects Involved The Voronoi diagram above L1 The envelope of the objects below L1

15. Two Highways Problem • O is the intersection of L1 and L2 •  is the angle between L1 and L2 • is the union of and for L1 • is similarly defined for L2 • Four “quadrants” Q0, Q1, Q2, Q3

16. L2 L1 O Q1 Q0 Q2 Q3

17. Two Highways • Lemma 3.1 • SupposeL1 + L2 = , for two points p, q on different highways. • The shortest time paths are not unique. • One of the shortest time paths from p to q is to walk along one highway then change to the other at the intersection.

18. Two Highways q L2 L1 p

19. Two Highways • Lemma 3.2 • SupposeL1 + L2 < , for two points p, q on different highways. • The shortest time path from p to q is to walk along one highway then change to the other at the intersection.

20. Two Highways q B A L2 L1 C D p

21. Two Highways • Lemma 3.3 • SupposeL1 + L2 > , for two points p, q on different highways. • The shortest time path from p to q is to walk along at most one highway. (shortcut)

22. q3 q2 q1 Two Highways L2 L2 L1 p L1

23. Good Condition for Highway Intersection • Highways L1, L2 are said to satisfy good intersection condition if and only if L1 + L2  . • Any shortest time path connecting two points on different highways that satisfy good intersection condition contains no shortcut.

24. O-Domination Site • pO is the O-domination site if • O is in the Voronoi region of O-domination site pO

25. L2 Q0 Q1  L1 O Q2 Q3 -Distance-Line-from-O

26. O-Domination Line • The -distance-line-from-O, , is called O-domination line in Qi,where = dt(O, pO).

27. Trivial Site • Any site which is not the O-domination site is a trivial site

28. Some Properties • For a point qVort(p, S), if the shortest time path from q to p passes through O, then the site p is the O-domination site. • For a point qVort(p, S), if the shortest time path from q to p enters both highways, the path must pass through O provided that the two highways satisfy good intersection condition.

29. Some Properties (cont’d) • For a point qVort(p, S), and p is a trivial site, then the q to p path never enters both highways. • For a trivial site p in Qi,Vort(p, S)  Q(i+2) mod 4 =  • We need not consider trivial sites in Qi when we compute the Voronoi diagram in Q(i+2) mod 4

30. Li+ Q(i+1) mod 4 Qi Li- Q(i-1) mod 4 Notations • Let Li+ be the line that borders quadrant Qi and Q(i+1) mod 4, and Li- borders quadrant Qi and Q(i-1) mod 4

31. Good Condition Case • The time-based Voronoi diagram in Qi, is determined by the set of objects Pi:

32. Envelope & Objects Involved Li- Li+ O

33. Time-Based Voronoi Diagram • The time-based Voronoi diagram in a quadrant Qi is • The time-based Voronoi diagram is • It is our general form.

34. Algorithm • Find the O-domination site pO and let=dt(O, pO) • Compute the O-domination line for Qi, i=0,1,2,3 • Compute the set Pi of objects used for constructing the Voronoi diagram in each quadrant Qi for i=0,1,2,3. i.e, the envelope surrounding Qi, and all the sites in Qi • Compute the ordinary Voronoi diagram in Qi. i.e, Vor(Pi) Qi • For all sites p, collect all regions associated with ,and p

35. Theorem • The Voronoi diagram for a set S of n sites in the presence of two highways L1 and L2 in the plane that satisfy the good intersection condition, can be computed in O(n log n) time.

36. Multiple Highways Problem • Idea • If good intersection condition holds, the problem is not hard. • Find domination site for each intersection. • In each cell of the arrangement, only the sites in the cell and neighboring cells determine the time-based Voronoi diagram in the cell.

37. How to Find Domination Sites? • Insert highways one at a time in order of non-descending speeds. • Rewrite and update intersection domination sites. • Propagation subroutine.

38. Propagation

39. Time Complexity • n sites, k highways • To determine all intersection-domination sites with propagation costs O(kn + k3 log k) time • To compute all time-based Voronoi region costs O(n log n) time • The total time is O(kn + k3 log k + n log n)

40. Two-Highway Model in General • No good condition now. • Lemma 5.1 • Let p, q be any two points on the plane. If the number of shortest time path from p to q is finite, and the shortest time path walks along both highways, then the path must pass through the intersection of two highways.

41. Two-Highway Model in General (cont’d) • The time-based Voronoi diagram in Qi, is determined by the set of objects Pi:

42. Time-Based Voronoi Diagram • The time-based Voronoi diagram in a quadrant Qi is • The time-based Voronoi diagram is • The time-based Voronoi diagram for n points in the presence of two highways can be computed in O(n log n) time.

43. Special Cases • Two parallel highways

44. Two Parallel highways Problem • Idea • No origin-domination site • No shortest time path along both highways • Compute the envelope associated with a proper set of hats

45. qL1 qL2 Two Parallel Highways Problem p q L1 L2

46. L2  L1+ • L1 nullifies L2 • No shortest time path along both highways • We solve the problem as in two parallel highways case.

47. L2  L1+ p L2 O L1

48. General Multiple Highways Case • Hard to determine the shortest time path • Hard to determine the intersection domination sites • Propagation doesn’t work • OPEN?

49. Conclusion • n sites, k highways • If good intersection condition holds, we can solve the problem inO(k3 log k + kn + n log n) time • If good intersection condition doesn’t hold, we can solve two highways problem inO(n log n) time.