### the order-k Voronoi diagram

```On Farthest, Higher-order, and
Hausdorff Voronoi diagrams
Overview
Università della Svizzera italiana -- USI
(University of Lugano)
1
Project Overview
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Order-k Voronoi diagrams of line segments
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
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Farthest (k=n-1) Voronoi diagrams of line segments


In 2,3,d dimensions
Hausdorff Voronoi diagram – min-max type


Completely ignored in CG literature (surprisingly)
Application: line segments forming polygons and planar graphs
Captures defects destroying shapes / clusters of points
L simplifications (motivated by VLSI applications)

LVoronoi diagram of line segments in CGAL
2
Talk Overview

Summarize our work on
 Order-k
line-segment Voronoi diagram
 Farthest line-segment Voronoi diagram in 2D, 3D
 L software

Elena Khramtcova will talk on the Hausdorff Voronoi
diagram

Maksym Zavershyskyi on recent algorithms for order-k
diagrams
3
Order-k Voronoi diagram

Given a set of sites, the order-k Voronoi diagram is a
partitioning of the plane into regions such that all points
within one region have the same k nearest sites.
Order-1
Order-2
4
Order-k line-segment Voronoi diagram
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
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Line segments: surprisingly different properties from points
Disconnected regions
A single order-k region may disconnect into (n) disjoint
faces
5
Order-k line-segment Voronoi diagram



Line segments: surprisingly different properties from points
Disconnected regions
A single order-k region may disconnect into (n) disjoint
faces
Ω(n-k) bounded faces. For 1<k<n/2, Ω(n-k) = Ω(n)
6
Order-k line-segment Voronoi diagram



Line segments: surprisingly different properties from points
Disconnected regions
A single order-k region may disconnect into (n) disjoint
faces
k
n-k
Ω(k) unbounded faces. For n/2<k n, Ω(k) = Ω(n)
7
Overall structural complexity: O(k(n-k))

Same as points, where Voronoi regions are connected
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Disjoint line segments
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Intersecting line segments, for k>n/2
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For k>n/2, complexity is independent of #intersections
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O(k(n-k) +I) for k< n/2; I: # intersections
Line segments forming a PSLG
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Important for applications -- polygonal objects
[joint work with M. Zavershynskyi]
8
Order-k VD of PSLG
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Planar straight-line graph (PSLG). Non-disjoint sites
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Inherently degenerate
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Areas where multiple segments are equidistant -- their
number is independent of k
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Standard definition assumes |H|=k -- not sufficient


Vk (H , S )  x | s  H , t  S \ H , d x, s  d x, t 
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Must allow > k elements in an order-k set
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Challenge: Augment definition of order-k region
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Order-k disk Dk(x)
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Order-k disk Dk(x): Disk of minimum radius that
intersects at least k line-segments (centered at x)
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Proper Dpk(x): touches exactly one elementary site p
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The set of segments intersected by such a disk is an
order-k subset
p
k = 5,6,7
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Order-k subset
Order-k subset H
1. |H|=k (Type-1)
2.
|H| > k (Type-2) and there is a proper order-k
disk Dpk(x) that intersects H
p -- representative of H
H
p
H is order-k subset for k = 5,6,7
Order-k Voronoi region of non-disjoint sites
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
Define in terms of order-k subsets -- not cardinality-k
H Type 1 – standard definition of a Voronoi region
Vk (H, S) = { x | "s Î H, "t Î S \ H, d ( x, s) < d ( x, t )}
Vk (H, S) = { x | Sk (x) = H } type 1
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H = Hp Type 2 (representative p)
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Sk(x) the set of line segments intersected by Dk(x)
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Type-2 Voronoi regions
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Non-standard configurations – pose no problems
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Degree 4, 5, 6 Voronoi vertices
Type-2 regions expand (as k increases) until they become Type-1
k=1
k=2
k=3,..,6
k=7
Algorithms -- order-k VD of line segments
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Standard iterative construction (all cases)
O(k2nlogn)
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Plane sweep (all cases) [P. & Zavershyskyi 2013]
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Maintain the level-k in an arrangement of segment wave-curves
Maintain the level <= k
Time O(k2nlogn)
Randomized techniques (disjoint sites) appropriate for
larger k (Maksym’s talk)
14
Order-k AVDs
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Generalized complexity results to order-k Abstract
Voronoi Diagrams

Represent a much wider class of order-k diagrams of disjoint sites

Disjoint line segments: a concrete instance of order-k AVDs
(PSLG or intersecting line segments not such an instance)
[Joint work with Bonn]
[Bohler, Cheilaris, Klein, Liu, Papadopoulou, Zavershynskyi, ICALP 2013]
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Gaussian Map for VD
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Structure that encodes the unbounded regions of a VD

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Maps VD at infinity on the sphere of directions
Signature of VD at infinity
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Useful for VDs that consist of unbounded regions
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Example: the farthest line-segment Voronoi diagram
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Farthest line-segment Voronoi diagram

[Aurenhammer, Drysdale, Krasser, IPL 06]
Properties surprisingly different from points

Not related to convex hull
 Disconnected Voronoi regions
 A single segment may have (n) disconnected faces!

Tree structure, size: O(n)
 Constructed in O(nlog n) time
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Farthest line-segment Voronoi diagram
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Faces of the FVD still characterized by the farthest-hull
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Convex-hull-like structure
Closed polygonal curve represented by its Gaussian map (Gmap)
[P. & Dey 2012]
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Farthest hull -- Supporting line

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A segment s (or segment endpoint p) has a nonempty
farthest Voronoi region if it admits a supporting line l
Line l is supporting if it defines an open half-plane H(l)
that intersects all segments except s
H(l)
H(l)
l
s
s
Hull vertex
Hull segment
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Supporting segment

Two points p1,p2 define a supporting segment if the line l
through p1,p2 defines an open half-plane H(l) that
intersects all segments except segments incident to p1,p2.
p1
p2
H(l)
Unbounded bisector in FVD
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Farthest line-segment hull (f-hull)
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
The sequence of hull segments and supporting
segments ordered in the angular order of their unit
normals yields the farthest hull and its Gaussian map on
the unit circle.
The farthest hull is a closed polygonal curve.
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Gaussian map: simple circular list

The f-hull may look complicated but its Gmap is a simple
circular list (like a convex hull)


a vertex may appear multiple times on Gmap (multiple arcs)
a segment may appear twice (two opposite unit vectors)
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Connection with point-line duality
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A point can be mapped into a line (in dual space) and vice
versa
A line segment can be mapped into a wedge below (or
above) two intersecting lines
[Aurenhammer, Drysdale, Krasser, IPL 06]
lines passing above AB
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Faces FVD = Upper envelope of wedges

Faces of FVD, unbounded in direction 0 to  = upper
envelope of wedges
[Aurenhammer, Drysdale, Krasser, IPL 06]
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Upper (lower) Gmap = Upper (lower) envelope of wedges
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Gmap arc = envelope edge
Gmap unit vector = envelope vertex
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Computing the FVD
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Compute Gmap, O(nlogh) time
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h: # Voronoi faces
D&C, quick hull, incremental, Chan’s output-sensitive, Jarvis
march (no Graham scan)
[P. & Dey 12]
Compute FVD from circular list of unbounded bisectors,
O(h log h) time
[Aurenhammer, Drysdale, Krasser, IPL 06]
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Conjecture: FVD can also be computed in O(h) time, given
the Gmap
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FVD -- line segments (and lines) in 3D
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FVD: cell complex – all regions unbounded
Voronoi skeleton is connected
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The region of one segment may induce Θ(n2) unbounded
features
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Total number of unbounded features in FVD is Θ(n2)
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FVD of lines in 3D, Θ(n2) unbounded features
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As NN-VD of lines in 3D, diagram complexity open, O(n3+ε)
[Recent joint work with Gill Barequet]
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FVD -- line segments (and lines) in d-D
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Results seem to generalize in d-dimensions by induction
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The total number of unbounded features of FVD is Θ(nd-1)
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Tool used: the Gaussian Map, Duality
[Recent joint work with Gill Barequet]
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Supporting plane
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A plane P is supporting
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
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P passes through 3 endpoints of (three or two) segments of ;
One closed halfspace +, fully contains these segments
The other open halfspace, −, intersects all other segments
3-leg
tricycle
29
Gaussian map – line segments
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Planar map on the sphere of directions as implied by
the unbounded features of FVD
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
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Vertices: Correspond to supporting planes (either 3-leg or tricycle)
Edges: Correspond to rotations of supporting planes
Two types of rotations
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Type I: rotations about a segment
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

Not part of FVD
Connect tricycle vertices
Type II: about the line defined by two
endpoints of different segments (bisectors)
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Vertices / Edges

3-leg vertices reveal unbounded trisectors in the FVD
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Type II edges correspond to unbounded bisectors of
FVD
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Tricycle vertices reveal hull segments (like in 2D)

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They are breakpoints of Type II edges (bisectors)
Type I arcs are not part of the regular Gmap



Can be included in an augmented Gmap
Reveal hull segments (like in 2D)
Represent two parallel planes separating the open portion and
endpoints (in an augmented FVD)
31
Faces of Gaussian map
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Segment faces

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Correspond to Voronoi cells of entire segments
Contain a Type I rotation arc
One segment can have Θ(n) faces around its Type-I great circle
Endpoint faces


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Correspond to Voronoi cells of segment endpoints
Bounded by Type-II edges only
One segment endpoint can induce Θ(n2) vertices on the Gmap
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One endpoint may induce Θ(n2) vertices



Put n-1 long segments connecting almost antipodal points on a
sphere and 1 short segment near the center of the sphere
Any pair of long segments together with the short segment define a
distinct supporting plane.
The region of the short segment induces Θ(n2) vertices on Gmap
2D example of
[Aurenhammer, Drysdale, Krasser, IPL 06]
34
Gaussian map segments in 3-d
(lower bound)
V
36
Complexity of Gmap is Θ(n2)

The worst case complexity of the Gmap is Θ(n2)

Lower bound
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n line segments connecting antipodal points on a sphere
n great circles of supporting planes. Intersection points are all
features of the Gmap.
Upper bound



Based on duality transformation of segments into 3D wedges
Upper Gmap = upper envelope of 3D wedges (roofs)
Use the fact that the upper envelope of 2D wedges is linear
[Edelsbrunner, Maurer, Preparata, Rosenberg, Welzl, Wood, BIT 82]
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Complexity of Gmap is O(n2)
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
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Complexity of the upper envelope of 3D wedges (roofs)
Count the complexity of envelopes on each slope. There are O(n)
slopes
On a given slope we essentially have the envelope of 2D wedges
(almost), which is O(n)

On slope of wedge w: Intersection with another wedge is a 2D wedge (full
or trimmed by apex a)
 Another wedge w’ may partially or fully contain a (w’ takes over w). But
then features are counted on the slope of w’
38
Algorithms

Compute the Gmap: D&C O(n2logn) time


O(n2) seems possible
Compute FVD, given the Gmap: O(k log n) time, k:
complexity of FVD



Compute vertices in decreasing order of weight (radius)
Property: Voronoi skeleton forms a DAG using the order of
vertices in increasing weight (radius of implied sphere)
39
Gmap of lines in 3D – simpler
one type of faces
V
40
L segment Voronoi diagrams
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Interested in the L metric because of simplicity
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Straight-line skeleton (no parabolic parts) -- valuable proximity
Voronoi vertices on rational coordinates (if input is rational)
Lower algorithmic degree (faster computations)
Ideal for VLSI applications -- majority rectilinear edges
Often simpler (more efficient) algorithms

L order-k VD of points in optimal O(m log n) time