A Sweepline Algorithm for Higher Order Voronoi Diagrams

Maksym Zavershynskyi
joint work with
Evanthia Papadopoulou
Università della Svizzera Italiana
Lugano, Switzerland
Supported in part by the Swiss National Science Foundation (SNF) grant 200021-127137.
Also by SNF grant 20GG21-134355 within the collaborative research project EuroGIGA/VORONOI of the European Science Foundation.
A Sweepline Algorithm for
Higher OrderVoronoi Diagrams
Tuesday, July 16, 2013

Nearest NeighborVoronoi Diagram
The nearest neighborVoronoi diagram is the
partitioning of the plane into regions, such that all
points within a region have the same closest site.

Higher OrderVoronoi Diagram
The order-kVoronoi diagram is the partitioning of
the plane into regions, such that all points within an
order-k region have the same k nearest sites.
V2({s1, s2}, S)
s1 s2

Related Work
• Structural complexity of the order-kVoronoi
diagram:
• for points [Lee’82]
• line segments [Papadopoulou&Zavershynskyi’12]
O(k(n − k))
O(k(n − k))

Related Work
• Iterative construction algorithm:
• Time
• Space
• Can be used to construct all order-iVoronoi
diagrams, for
O

k2
n log n

O

k2
(n − k)

i = 1, . . . , k

Construction Algorithms
Construction Time Reference
ChazelleEdelsbrunner’87
Clarkson’87
Aurenhammer’90
Mulmuley’91
Boissonnat et al.’93
Agarwal et al.’98
Chan’98
Ramos’99
O

n2
+ b log2
n

O

n1+
k

O

nk2
+ n log n

O

nk3
+ n log n

O

n log3
n + nk log n

O (n log n + nk log n)
O

n log n + nk2c log∗
k

O

nk2
log n


Construction Time Reference
ChazelleEdelsbrunner’87
Clarkson’87
Aurenhammer’90
Mulmuley’91
Boissonnat et al.’93
Agarwal et al.’98
Chan’98
Ramos’99
O

n2
+ b log2
n

O

n1+
k

O

nk2
+ n log n

O

nk3
+ n log n

O

n log3
n + nk log n

O (n log n + nk log n)
O

n log n + nk2c log∗
k

O

nk2
log n

Expected
For Points!

• Based on Fortune’s algorithm
• For points and line segments
• Deterministic
• Simple
• Easy to implement
Sweepline approach?

The Algorithm

The Idea
• Consider horizontal line l
• Wave-curve w(s) is the locus of points equidistant
from l to s

• Let be the set of line segments that intersect the
upper halfplane.
• Consider an arrangement of wave-curves

The Idea
S
w(s), s ∈ S

The Idea
• Consider k-level Ak
• Lower envelope is 1-level
Ak

The Idea
• Consider k-level Ak
• Lower envelope is 1-level
Breakpoint
Wave

The Idea
• While the horizontal line moves down, the
breakpoints of Ak and Ak+1 levels move along the
edges of the order-kVoronoi diagram.

The Events
• Simulate the discrete event points that change the
topological structure of k-level while line l moves
down.
• We store the adjacency relations!
Site Events Circle Events

Site Events
• Occur when the line hits a new line segment.
• We insert the corresponding wave-curve and
update all levels.

Site Events
a1, am1. . . ,
b1, bm2. . . ,
c1, cm3
. . . ,
. . . ,
d1, dm4
A1 :
A2 :
A3 :
A4 :
A1
A2
A3
A4
. . . ,
. . . ,
. . . ,
. . . ,
ai,
bj,
cr,
ds,
Adjacency relations:
x

Site Events
a1, am1. . . ,
b1, bm2. . . ,
c1, cm3
. . . ,
. . . ,
d1, dm4
A1 :
A2 :
A3 :
A4 :
A1
A2
A3
. . . ,
. . . ,
. . . ,
. . . ,
ai,
bj,
cr,
ai, , ai,
bj, bj,
cr, cr,
ds, ds,
x, x,
x, x,
x, x,
x
A4
A5 : x, xds,

Circle Events
• Occur when 3 wave-curves intersect at
a common point
a
b
c

Circle Events
a
b
c
Ai :
Ai+1 :
Ai+2 :
a, b,
c,
c
b, a, b
c, a

Circle Events
a
b
c
Ai :
Ai+1 :
Ai+2 :
a,
b,
c,
c
b, a, b
c, a

Circle Events
• Precompute the circle events of all triples of the
consecutive waves.
• A new triple appears - create the circle event
• An old triple disappears - remove the circle event

Maintaining the ≤k-level
• The breakpoints of k-level and (k+1)-level move
along the edges of the order-kVoronoi diagram
• We need to keep track of k-level!
• A new wave may be introduced to k-level:
• By Site Events
• From (k-1)-level and (k-2)-level by Circle Events
• We need to maintain ≤k-level!

Data Structures
• Event queue:
• Queue:
Site Events, sorted by y-coordinate.
• Balanced Binary Tree:
Circle Events, sorted by y-coordinate of the
bottom-most point of the circle.
• Levels 1,...,k:
• Balanced Binary Tree for each level:
Sequence of waves that constitute the level.

Time and Space Analysis

Maximum Size of the ≤k-level?
• The complexity of the ≤k-level in arrangement
of Jordan curves [SharirAgarwal’95, Clarkson’87]
• Complexity of the lower envelope [Fortune’87]
• Therefore the complexity of the ≤k-level is:
g≤k(n) = O

k2
g1(n/k)

g1(n) = O(n)
g≤k(n) = O(kn)

Maximum Size of the Event Queue?
• Event Queue:
• Site Events
• Circle Events
Each Circle Event corresponds to a triple of
consecutive waves in ≤k-level
O(n)
O(kn)

Time Complexity?
• Site Event
• For each i-level, i = 1,...,k
• Find insertion point
• Update adjacency relations at the point
• For all Site Events:
O(log |Ai|) ≤ O(log (i(n − i))) ≤ O(log n)
O(nk)O(log n) = O(nk log n)

Number of Circle Events
• Every processed Circle Event corresponds to
some order-iVoronoi vertex, i = 1,...,k
• Therefore total number of processed Circle
Events is the total complexity of order-i
Voronoi diagrams, i = 1,...,k
k
i=1
i(n − i) = O

k2
n


Time Complexity
• Circle Event
• Remove Circle Event from the queue
• Update adjacency relations relations
on 3 levels:
• For all Circle Events:
O

log O

k2
n

= O(log n)
3O(log n) = O(log n)
O(k2
n)O(log n) = O(k2
n log n)

Results
• Time complexity
• Space complexity
O(k2
n log n)
O(kn)

Line Segments Forming
a Planar Straight-Line Graph

Planar Straight-Line Graph
• Weak general position assumption:
“No more than 3 elementary sites touch the
same circle”

Bisectors containing 2-dimensional portion
d the maxi-
A1, . . . , Ak.
wing bound
(2)
of levels
exity of the
envelope of
Voronoi di-
e maximum
s1 s2
b(s1, s2)
b(s1, s2)
Fig. 3: Bisector containing 2-dimensional portion

Bisectors intersecting non-transversely
A1, . . . , Ak.
owing bound
(2)
of levels
exity of the
envelope of
Voronoi di-
e maximum
nk).
rcle-event in
ent waves at
er of circle-
A1, . . . , Ak,
ed to run in
ertion of the
mber of site-
n). When a
insert it in
on every list
s1 s2
b(s1, s2)
Fig. 3: Bisector containing 2-dimensional portion
s1
s2
s3
s4
b(s1, s4)
b(s2, s4)
b(s3, s4)b(s1, s4)
b(s2, s4), b(s3, s4)
Fig. 4: Bisectors intersecting non-transversely

We want to keep the information on the shared
endpoints without altering the structure of the order-k
Voronoi diagram of disjoint line segments.

• - a minimal disk centered at point x that
intersects at least k line segments, where p is
an elementary site that touches the disk.
Dp
k(x)
x

• - a minimal disk centered at point x that
intersects at least k line segments, where p is
an elementary site that touches the disk.
Dp
k(x)
x
p
Dp
3(x)
Sk(x)

Generalized Deﬁnition
Deﬁnition: A set is called an order-k subset if
type-1:
type-2: and there exists a proper order-k
disk , where p is an endpoint common
at least two segments, that intersects exactly
all line segments in .
p - the representative of
H, H ⊆ S
|H| = k
|H| k
Dp
k(x)
Hp
H

The order-k Voronoi region deﬁned:
type-1 - in the ordinary way.
type-2 :
Vk(Hp, S) = {x | ∃Dp
k(x) ∩ Sk(x) = Hp}

Order-1
1
2
3
45
6
7
8
V (a)
V (d)
V (f) V (g)
V (b)
V (5) V (4)
V (3)
V (2)
V (1)
V (6)
V (7)
V (8)
V (c)V (e)
a
b
c
d
e
f g

Order-2
1
2
3
456
7
8
V (1, 2)
V (6, 5) V (3, 4)
V (3, 8)
V (2, 8)
V (4, 5)
V (7, 5)
V (7, 8)
V (c, 3)
V (c, 8)
V (e, 5)
V (2, 7)
V (d)
V (g)
V (1, 7)
V (6, 7)
a b
c
d
e
f g

Order-3
1
2
3
456
7
8
V (2, 3, 8)
V (3, 4, 8)
V (3, 4, 5)
V (4, 5, 6)V (d, 5)
V (5, 6, 7)
V (1, 6, 7)
V (d, 2)
V (g, 1)
V (1, 2, 8)
V (1, 2, 7)
V (2, 7, 8)
V (1, 7, 8)
V (3, 7, 8)
V (4, 5, 8)
V (4, 5, 7)
V (5, 7, 8)
V (6, 7, 8)
V (c, 7, 8)
V (c, 3, 8)
a b
c
d
e
f g
V (e, 4, 5)

How we deﬁne the k-level?
x
a, b
c
d
e
π(x) = π−
(x), π0
(x), π+
(x)
π(x) = {a}, {b, c, d}, {e}

How we deﬁne the k-level?
x
a, b
c
d
e
π(x) = π−
(x), π0
(x), π+
(x)
Deﬁnition: k-level is a set of points x such that
and|π−
(x)| k |π−
(x) ∪ π0
(x)| ≥ k
A3

The Idea
1
2
3
456
7
8
a
b
c
d
e
f g
A1
A2
A3
l
x
Fig. 8: Constructing order-3 Voronoi diagram via sweepline
technique. Point x belongs to the order-k Voronoi region
Maintain the ≤k-level!

SUMMARY
We have developed an algorithm that constructs
the order-kVoronoi diagram of line segments in
time and space.
• Simple
• Easy to implement
• Sweepline
• Can output all order-iVoronoi diagrams for i = 1,...,k
• Generalized to the case of line segments forming
a planar straight-line graph.
O(k2
n log n) O(kn)

A Sweepline Algorithm for Higher Order Voronoi Diagrams

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