Time-Based Voronoi Diagram Institute of Information Science Academia Sinica, Taipei, Taiwan D. T. Lee Institute of Information Science Academia Sinica, Taipei, Taiwan Jointly with C. S. Liao, W. B. Wang, IIS.

2/ 57 Outline Introduction Preliminaries Good intersection condition General condition Conclusion

3/ 57 Outline General condition Two highways case Parallel two highways An extreme case Multiple highways case Conclusion

4/ 57 Multiple Highways Model Input: A set S of points, S={p 1, …, p n } in the plane and k highways L 1, …, L k, modeled as lines. Travelers can enter the highways at any point and move along L i at speed v i in both directions. Off the highways travelers can move freely in any direction at speed v 0 << v 1,…, v k. Output: A Voronoi diagram for the input based on traveling time, i.e. Time-based Voronoi Diagram

5/ 57 Time Distance Given two points p, q in the plane, the shortest time path sp t (p, q) is a path that takes the shortest time traveling between p to q. The time distance d t (p, q) between p and q is the time required to follow any shortest time paths between p and q.

6/ 57 One Highway Problem Abellanas, Hurtado, Sacristan, Icking, Ma, Klein, Langetepe, Palop IPL, 2003 Assumption L 1 lies on the x-axis. sine  = v 0 /v 1 = 1/v 1 L 1 + : the half-plane above L 1 L 1 - : the half-plane below L 1

7/ 57 Where to Enter the Highway α p q sine  = v 0 /v 1 = 1/v 1 α prpr plpl L1L1

8/ 57 Time Distance L1L1 α p q pL1pL1 prpr qlql

9/ 57 Terminology : the symmetric point of p reflecting by L 1. Given a site p, let be the half-ray with endpoint p and of slope tan  (-tan  ) respectively.

10/ 57 L1L1 p q

11/ 57 L1L1 p q

12/ 57 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

13/ 57 Definition P a, P b denote the sets of objects used in defining the time-based Voronoi diagram above L 1 and below L 1.

14/ 57 Vor() & Vor t () Vor(x, X): the Euclidean Voronoi Region of a site or a line x  X with respect to the set X. Vor t (x, X): the time-based Voronoi Region of a site or a line x  X with respect to the set X.

15/ 57 Theorem [Abellanas, et al.] For p  L + For p  L -

16/ 57 Envelope & Objects Involved The envelope of the objects below L 1 The Voronoi diagram above L 1

17/ 57 Envelope L1L1

18/ 57 Two Highways Problem O is the intersection of L 1 and L 2  is the angle between L 1 and L 2 is the union of and for L 1 is similarly defined for L 2 Four “quadrants” Q 0, Q 1, Q 2, Q 3

19/ 57 L1L1 L2L2 O Q3Q3 Q1Q1 Q2Q2 Q0Q0

20/ 57 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.

21/ 57 Two Highways q p L1L1 L2L2

22/ 57 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.

23/ 57 Two Highways A B DC q p L1L1 L2L2

24/ 57 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)

25/ 57 Two Highways A B DC q p  L1 L2L2

26/ 57 Two Highways q1q1 p  L1 L2L2 q2q2 q3q3 L1L1 L2L2

27/ 57 Good Condition Let p, q be any two points on the plane. If the shortest time path from p to q is unique, and it walks along both highways, then the path must walk through the intersection of two highways.

28/ 57 Good Condition for Highway Intersection Highways L 1, L 2 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.

29/ 57 How to Choose a Highway P b 1 : bisector of h 1 and h 2 L1L1 L2L2 O h1h1 L1L1 h2h2 L2L2 sin  L1 =1/v 1 sin  L2 =1/v 2

30/ 57 Determine the Time Distance L1L1 L2L2 O Q P h1h1

31/ 57 O-Domination Site p O is the O-domination site if O is in the Voronoi region of O-domination site p O

32/ 57  -Distance-Line-from-O L1L1 L2L2 O  Q3Q3 Q2Q2 Q1Q1 Q0Q0

33/ 57 O-Domination Line The  -distance-line-from-O,, is called O-domination line in Q i, where   = d t (O, p O ).

34/ 57 Trivial Site Any site which is not the O-domination site is a trivial site

35/ 57 Some Properties For a point q  Vor t (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  Vor t (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.

36/ 57 Some Properties (cont’d) For a point q  Vor t (p, S), and p is a trivial site, then the q to p path never enters both highways. For a trivial site p in Q i, Vor t (p, S)  Q (i+2) mod 4 =  We need not consider trivial sites in Q i when we compute the Voronoi diagram in Q (i+2) mod 4

37/ 57 Trivial Site If there is a start point in the plane, and the path to the nearest (take the shortest time) site is through O, then the destination site must be O-domination site. For a trivial site, we don’t care the path through both highways. Its time-based Voronoi region cannot overstride to opposite quadrant.

38/ 57 Notations Let L i+ be the line that borders quadrant Q i and Q (i+1) mod 4, and L i- borders quadrant Q i and Q (i-1) mod 4 QiQi Q (i+1) mod 4 Q (i-1) mod 4 Li-Li- Li+Li+

39/ 57 Good Condition Case The time-based Voronoi diagram in Q i, is determined by the set of objects P i :

40/ 57 Envelope & Objects Involved Li+Li+ Li-Li- O

41/ 57 Time-Based Voronoi Diagram The time-based Voronoi diagram in a quadrant Q i is The time-based Voronoi diagram is It is our general form.

42/ 57 Algorithm Find the O-domination site p O and let  =d t (O, p O ) Compute the O-domination line for Q i, i=0,1,2,3 Compute the set P i of objects used for constructing the Voronoi diagram in each quadrant Q i for i=0,1,2,3. i.e, the envelope surrounding Q i, and all the sites in Q i Compute the ordinary Voronoi diagram in Q i. i.e, Vor(P i )  Q i For all sites p, collect all regions associated with, and p

43/ 57 Theorem The Voronoi diagram for a set S of n sites in the presence of two highways L 1 and L 2 in the plane that satisfy the good intersection condition, can be computed in O(n log n) time.

44/ 57 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.

45/ 57 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.

46/ 57 Propagation

47/ 57 Time Complexity n sites, k highways To determine all intersection-domination sites with propagation costs O(kn + k 3 log k) time To compute all time-based Voronoi region costs O(n log n) time The total time is O(kn + k 3 log k + n log n)

48/ 57 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.

49/ 57 Two-Highway Model in General (cont’d) The time-based Voronoi diagram in Q i, is determined by the set of objects P i :

50/ 57 Time-Based Voronoi Diagram The time-based Voronoi diagram in a quadrant Q i 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.

51/ 57 Special Cases Two parallel highways

52/ 57 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

53/ 57 Two Parallel Highways Problem p q qL1qL1 qL2qL2 L1L1 L2L2

54/ 57  L2   L1 +  L 1 nullifies L 2 No shortest time path along both highways We solve the problem as in two parallel highways case.

55/ 57  L2   L1 +  p O L1L1 L2L2

56/ 57 General Multiple Highways Case Hard to determine the shortest time path Hard to determine the intersection domination sites Propagation doesn’t work OPEN?

57/ 57 Conclusion n sites, k highways If good intersection condition holds, we can solve the problem in O(k 3 log k + kn + n log n) time If good intersection condition doesn’t hold, we can solve two highways problem in O(n log n) time.