Time-Based Voronoi DiagramTime-Based Voronoi Diagram
D. T. LeeInstitute of Information Science AcadeInstitute of Information Science Acade
mia Sinica, Taipei, Taiwanmia Sinica, Taipei, Taiwan
[email protected]@iis.sinica.edu.tw
Jointly with C. S. Liao, W. B. Wang, IIS.
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Outline Introduction Preliminaries Good intersection condition General condition Conclusion
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Multiple Highways Model Input: A set S of points, S={p1, …, pn} in th
e plane and k highways L1, …, Lk, modeled as lines. Travelers can enter the highways at any point a
nd move along Li at speed vi in both directions. Off the highways travelers can move freely in a
ny direction at speed v0 << v1,…, vk. Output: A Voronoi diagram for the input ba
sed on traveling time, i.e. Time-based Voronoi Diagram
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Time Distance Given two points p, q in the plane, the short
est time path spt(p, q) is a path that takes the shortest time traveling between p to q.
The time distance dt(p, q) between p and q is the time required to follow any shortest time paths between p and q.
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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
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Where to Enter the Highway
α
p
q
sine = v0/v1 = 1/v1α
prplL1
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Time Distance
L1
α
p
q
pL1
pr ql
1ˆ Lp
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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.
1Lp
)ˆ( ˆ pp
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L1
p
q
1ˆ Lp1ˆ Lp
1L1Lp
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L1
p
q
p̂p̂
1L
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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
11 ˆ,ˆ, LL ppp
ppp ˆ,ˆ,
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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.
Xx
YxVorYXVor
),(),(
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Theorem [Abellanas, et al.] For p L+
For p L-
)),ˆ(),ˆ(),((
)),ˆ(),ˆ(),((),(
1
111
bbb
aLaLat
PpVorPpVorPpVorL
PpVorPpVorPpVorLSpVor
)),ˆ(),ˆ(),((
)),ˆ(),ˆ(),((),(
111
1
bLbLb
aaat
PpVorPpVorPpVorL
PpVorPpVorPpVorLSpVor
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Envelope & Objects Involved
The envelope of the objects below L1
The Voronoi diagram above L1
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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
1ˆ Lp p̂ p̂
2ˆ Lp
2010 /sin ,/sin21
vvvv LL
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L1
L2
O
Q3
Q1
Q2
Q0
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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.
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Two Highways
q
p
L1
L2
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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.
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Two Highways
A B
DC
q
p
L1
L2
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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)
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Two Highways
q1
p
L1
L2
q2
q3
L1
L2
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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.
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O-Domination Site
pO is the O-domination site if
O is in the Voronoi region of O-domination site pO
),(min),( pOdpOd tSp
Ot
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-Distance-Line-from-O
L1
L2
O
2
Q3
213 )( O
Q2
Q1
Q0
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O-Domination Line
The -distance-line-from-O, , is called
O-domination line in Qi, where = dt(O, pO).
)(Oi
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Trivial Site Any site which is not the O-domination site
is a trivial site
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Some Properties
For a point qVort(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 qVort(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.
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Some Properties (cont’d) For a point qVort(p, S), and p is a trivial si
te, 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
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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
QiQ(i+1) mod 4
Q(i-1) mod 4
Li-
Li+
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Good Condition Case
The time-based Voronoi diagram in Qi, is determined by the set of objects Pi:
4 mod)1(4 mod)1(
ˆˆ
ˆˆ)(
ii
i
ii
Qpi
Qpi
Qp
Li
Lii
i
pp
pppOP
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Envelope & Objects Involved
Li+
Li-
O
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Time-Based Voronoi Diagram The time-based Voronoi diagram in a quadr
ant Qi is The time-based Voronoi diagram is
It is our general form.
ii QPVor )(
3
0
)(
i
ii QPVor
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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 Vor
onoi 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
pp ˆ ,ˆ
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Theorem The Voronoi diagram for a set S of n sites i
n 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.
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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.
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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.
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Propagation
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Time Complexity n sites, k highways To determine all intersection-domination sit
es 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)
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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.
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Two-Highway Model in General (cont’d)
The time-based Voronoi diagram in Qi, is determined by the set of objects Pi:
)(ˆˆˆˆ
ˆˆˆˆ
4 mod)1(4 mod)2(
4 mod)1(
Opppp
ppppSP
iQp
iLi
Qpii
Qp
Lii
Qp
Li
Li
i
i
i
i
i
i
i
ii
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Time-Based Voronoi Diagram The time-based Voronoi diagram in a quadr
ant 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.
ii QPVor )(
3
0
)(
i
ii QPVor
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Special Cases Two parallel highways
12 LL
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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
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Two Parallel Highways Problem
p
q
qL1
qL2
L1
L2
21 vv
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L2 L1+ L1 nullifies L2
No shortest time path along both highways We solve the problem as in two parallel hig
hways case.
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L2 L1+
p
OL1
L2
1
ˆLO
2
ˆLO
21 vv
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General Multiple Highways Case Hard to determine the shortest time path Hard to determine the intersection dominati
on sites Propagation doesn’t work
OPEN?
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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.