Geometric Data Structures Dr. M. Gavrilova. Lecture Plan Voronoi diagrams Trees and grid variants.

Geometric Data Structures

Dr. M. Gavrilova

Lecture Plan Voronoi diagrams Trees and grid variants

Part 1

Voronoi diagram Delaunay triangulation

Voronoi diagrams VD: Thiessen polygons, Delaunay

triangulations, tessellations

Tessellation Tessellation is often obtained using

Delaunay triangulation.

Voronoi diagram Given a set of N sites (points) in the

plane or a 3D space Distance function d(x,P) between point x

and site P is defined according to some metric

Voronoi region Vor(P) is the set of all points which are closer to P than to any other site

Voronoi diagram is the union of all Voronoi regions

Voronoi Diagram

Voronoi diagram

Definition 1 For a finite set dES , a Voronoi diagram (VD) of S associates to eachSp a Voronoi region )(pVor such that

}}{),,(),(|{)( pqqxpxxp SddEVor d ,

where ,d denotes the Euclidean distance function [Okabe et. al. 92].

Voronoi diagram

C o n s i d e r t h e b i s e c t o r ),(),(),( xqxpxqp ddB b e t w e e n t w o s i t e s Sqp , o f a

V o r o n o i d i a g r a m i n dE . T h e b i s e c t o r d i v i d e s t h e s p a c e i n t o t w o h a l f s p a c e s . D e n o t e t h e

h a l f s p a c e t h a t c o n t a i n s t h e p o i n t p a s ),(),(),( xqxpxqp ddH . B y d e f i n i t i o n t h i s

h a l f s p a c e a l s o c o n t a i n s t h e b i s e c t o r ),( qpB .

D e f i n i t i o n 2 F o r a f i n i t e s e t dES , a V o r o n o i d i a g r a m o f S i s t h e c o l l e c t i o n o f a l l V o r o n o i

r e g i o n s SVor pp ),( s u c h t h a t

pq

qpp

S

HVor ),()( [ O k a b e e t . a l . 9 2 ] .

Voronoi diagram properties

Assumption 1. No four points from the set S are cocircular.Property 1. Voronoi vertex is the intersection of 3 Voronoi

edges and a common point of 3 Voronoi regionsProperty 2. Voronoi vertex is equidistant from 3 sites. It

lies in the center of a circle inscribed between 3 citesProperty 3. Empty circle property This inscribed circle is

empty, i.e. it does not contain any other sitesProperty 4. Nearest-neighbor property If Q is the nearest

neighbor of P then their Voronoi regions share an edge (to find a nearest neighbor it is sufficient to check only neighbors in the VD)

Property 5. Voronoi region Vor(P) is unbounded if and only if P belongs to the boundary of convex hull of S.

VD propertiesProperty 1. A Voronoi vertex v is the intersection of 3 Voronoi edges and a common point of3 Voronoi regions.

1) assume 3)deg(v .

3,,...

...,,, 321

npvd

pvdpvdpvd

n

Contradicts the non-circumcircularity assumption.

2) assume 3)deg(v . Then both edges are parts of the bisector

21,ppB . Both edges lie on the same line. Then v is not avertex.

v

1p

np3p

2p

v

1p

2p

VD properties

Property 2. Voronoi vertex is equidistant from 3 sites. It lies in the centerof a circle inscribed between 3 cites.

321 ,,, pvdpvdpvd

v

1p

3p

2p

VD properties

Property 3 (empty circle property). The inscribed circle is empty, i.e. it does notcontain any other sites.

1) assume a site q lies inside the inscribed circle C .

Then 1,, pvdrqvd C . This contradicts the fact

that 1pVorv .

2) assume q lies on the boundary of the inscribed circle.

Then 321 ,,,, pvdpvdpvdqvd . This

contradicts non-circumcircularity assumption.

v

1p

3p

2p

q

VD properties

Property 4 (Nearest-neighbor property). If q is the nearest neighbor of p then their

Voronoi regions share an edge (to find a nearest neighbor it is sufficient to check onlyneighbors in the VD)

Assume pVor and qVor don't have common

points. Segment pq crosses the boundary of pVorat a point x . Assume x belongs to the Voronoi edge

between pVor and sVor .

qxdpxdqpd ,,, ,

sxdpxdspd ,,, , sxdqxd ,, because qVorxsVorx ,

Then spdqpd ,, – contradicts the fact that q is the nearest neighbor of p .

p s

q

x

VD properties

Property 5. Voronoi region pVor is unbounded if and only if p belongs to the boundary of

the convex hull SCH .

1) assume SCHp . Select a ray pVorr . r will

intersect the boundary of SCH . Assume r intersects a

segment 21ss . It is easy to show that if rx , x

then either pxdsxd ,, 1 or

pxdsxd ,, 2 .

Contradicts the fact that pVorx

2) if SCHp then it is possible to draw a line l through p such that

all sites of S lie on the same side of l . Select a ray lr originating atp . It is easy to show that any rx is closer to p than to any other site.

Then pVorr , i.e. pVor is unbounded.

p xr

s2

s1

p

x

r

l

Delaunay triangulation

Definition 3. A Delaunay triangulation (DT) is the straight-line dual of the Voronoi diagram obtained by joining all pairs of sites whose Voronoi regions share a common Voronoi edge [Delaunay 34].

Follows from the definition: If two Voronoi regions Vor(P) and Vor(Q) share an edge, then

sites P and Q are connected by an edge in the Delaunay triangulation

If a Voronoi vertex belongs to Vor(P), Vor(Q) and Vor(R), then DT contains a triangle (P,Q,R)

Delaunay triangulation

DT propertiesAssumption 2. No three points from the set S lie

on the same straight line.

Theorem. The straight-line dual of the Voronoi diagram is a triangulation of S [Preparata and Shamos 85].

Property 5. The circumcircle of any Delaunay triangle does not contain any points of S in its interior [Lawson 77].

Property 6. If each triangle of a triangulation of the convex hull of S satisfies the empty circle property, then this triangulation is the Delaunay triangulation of [Lawson 77].

VD and DT Property 7 For the Voronoi diagram of n points on the plane and its dual

Delaunay triangulation the following relationships are true when 3n :

nvDT ,

63 nee VDDT ,

52 nvf VDDT ,

where VDv and DTv represent the number of vertices in VD and DT,

correspondingly, VDe and DTe represent the number of edges in VD and DT

and DTf is the number of triangles in DT

DT propertiesThe edge of the quadrilateral satisfies the local min-max

criterion if the following equation holds:

A triangulation satisfies the global min-max criterion if every internal edge of a convex quadrilateral in the triangulation satisfies the local min-max criterion.

Property 8. The Delaunay triangulation satisfies the global min-max criterion [Lawson 77].

Property 9. If a triangulation of the convex hull of satisfies the global min-max criterion then it is the Delaunay triangulation of [Lawson 77].

)(min)(min ii

ii

b

c

d

a

b

c

d

a

VD and DT

Both DT and VD effectively represent the proximity information for the set of sites. They can be easily transformed into each other.

VD contains geometrical information, while DT contains topological information.

Generalized Voronoi diagram

Given a set S of n sites (spheres) in d-dimensional space

Distance function d(x,P) between a point x and a site P is defined.

Generalized Voronoi diagramA generalized Voronoi diagram (GVD)

for a set of objects in space isthe set of generalized Voronoi regions

where d(x,P) is a distance function between a point x and a site P in the

d-dimensional space.

}{\,,, PSQQdPdPVor xxx

Generalized Delaunay tessellation

A generalized Delaunay triangulation (GDT) is the dual of the generalized Voronoi diagram obtained by joining all pairs of sites whose Voronoi regions share a common Voronoi edge.

General metricsGeneralized distance functions

Power

Additively weighted

Euclidean

Manhattan

supremum

22),(),( pp rdPd pxx

pe rdPd ),(),( pxx

p

d

iii rpxPd

1),(x

p

d

iii rpxPd

1

2)(),(x

piidi

rpxPd ..1max),(x

x

P

d(x,P)

P

x Pd ,x

P

x

pr

Pxd ,

P

x

pr

Pxd ,

Power and Euclidean Voronoi diagrams

P Q

B(P,Q)

Euclidean bisectorPower bisector

Power diagram and Delaunay triangulation

P Q

B(P,Q)

Euclidean diagram and Delaunay triangulation

Manhattan and Supremum VD

Supremum bisectorsManhattan bisectors

Manhattan diagram and Delaunay triangulation Supremum diagram and Delaunay triangulation

Properties of Generalized VD and DT The vertex of generalized Voronoi diagram is a center of a

sphere inscribed between d +1 Voronoi sites (spheres). The inscribed sphere is empty, i.e. it does not contain any

other sites. One of the facets of the generalized Voronoi region Vor(P)

defines a nearest-neighbor of P. A Voronoi region Vor(P) is unlimited if and only if site P

belongs to the convex hull of S. The sphere inscribed between the sites comprising a simplex

of generalized Delaunay tessellation is an empty sphere. The power Delaunay tessellation of a set of spheres S is a

tetrahedrization.

Algorithmic Strategies Incremental Divide-and-conquer Sweep-line/plane Dimension reduction Geometric decomposition

Algorithmic Strategies

Algorithm development techniques on example of Voronoi Diagrams

Incremental construction Divide and conquer Sweep-plane Geometric transformations Dynamic data structures

Incremental construction

INCIRCLE condition

power

Euclidean

Supremum

Polynomial of 4th order in the plane

Polynomial of 8th order in the plane

System of linear equations and inequalities in the plane

Incremental construction

Incremental method outline Insert new site PPerform swap operations on quadrilaterals where the empty-sphere condition is not satisfied

Method complexity is O(n2) in the plane.

Sweep-plane algorithm Algorithm description

Throw pebbles in the water Intersection of waves gives the Voronoi

diagram Add time as 3rd dimension: waves

transform to pyramids. Sweep pyramids with the sweep-plane to

get the Voronoi diagram

Sweep-plane algorithm Properties

The complexity of the sweep-plane algorithm in generalized Manhattan metric is O(n log n).

Sweep-plane method is not applicable to the power diagram construction.

Sweep-plane algorithm

An example of a sweep-plane construction of a Voronoi diagram in L1 metric

Cones in L1


Sweeping the cones


Parabolas

sweep direction


Site event


two half-bisectors are createdone is growing


The half-bisector changes its direction


Site event2 half-bisectors created


The half-bisectorChanges its direction


Circle eventTriangle added to DT

2 half-bisectors deleted1 half-bisector created


Half-bisectorchanges its

direction


No moreevents


Resulting Voronoi diagram and Delaunay triangulation



Waves in Manhattan metric Waves in

power metric

Swap method: from VD to power diagram

Relationship between Voronoi bisector and power bisector: The power bisector is moved relative to the Voronoi bisector by

Bisector transformation

),(2

22

qpd

rr qp

Swap method: from VD to power diagram Algorithm overview

Inflate VD sites Transform edges of VD Perform INCIRCLE test on quadrilaterals and

perform swap operations if needed.The worst-case complexity of algorithm is

O(n2).

Swap method Voronoi to power

(a)

p

(b)

p

e e

e *

1

p

p

p

2

13

4

2

p

p

p3

4

e *

Dynamic data structures – swap applicationDynamic data structures – swap applicationApproach

construct and maintain a dynamic Delaunay triangulation for the set of moving disks (in some metric).

Collisions checks are performed only along the Delaunay edges.

Dynamic VD and DT

Problem Given a set of N moving and/or

changing (i.e. growing) sites Construct the VD (DT) for the set Maintain the VD (DT) dynamically

Dynamic VD and DTDynamic Voronoi diagram

Dynamic Delaunay triangulation

Dynamic VD and DT Dynamic DT is easier to maintain than

dynamic VD in a sense that the coordinates of VD vertices must be calculated, while coordinates of DT vertices (the sites) are already known for any moment of time

As sites move, the topological structure of DT (and VD) changes only at discrete moments of time, which are called topological events.

Topological event involves swapping of the diagonal in a quadrilateral in the DT

In the VD corresponding edge shrinks to zero and another edge starts to grow

Swap condition

INCIRCLE P P P P( , , , )1 2 3 4 0 INCIRCLE P P P P( , , , )1 2 3 4 0 INCIRCLE P P P P( , , , )1 2 3 4 0

P1

P2

P3

P4

P1

P2

P3

P4

P1

P2

P3

P4

INCIRCLE test

1. Power metric x y x y r

x y x y r

x y x y r

x y x y r

poly trajectory

1 1 12

12

12

2 2 22

22

22

3 3 32

32

32

4 4 42

42

42

4

1

1

1

1

( )

To determine the swap time we have to solve INCIRCLE P t P t P t P t( ( ), ( ), ( ), ( ))1 2 3 4 0

2. Euclidean metric

x x r r p

x x r r p

x x r r p

r x y y p

r x y y p

r x y y p

x x y y p

x x y y p

x x y y p

p x x y y r r i

poly trajectoryi i i i

1 4 1 4 1

2 4 2 4 2

3 4 3 4 3

21 4 1 4 1

2 4 2 4 2

3 4 3 4 3

21 4 1 4 1

2 4 2 4 2

3 4 3 4 3

2

42

42

42

8

1 2 3

, , ,

( )

3. Manhattan metric (L)

INCIRCLE P t P t P t P t linear trajectory( ( ), ( ), ( ), ( )) ( )1 2 3 4

Dynamic DT maintenance Preprocessing

Construct the static Delaunay triangulation for the original site distribution

For every quadrilateral in DT calculate the potential topological event. Insert all such events into the priority event queue sorted according to the time order

Iteration Take the next event from the event queue Perform the swap operation and update the data

structure Delete topological events planned for the four

disappearing quadrilaterals from the event queue. Compute the new topological events for the four new

quadrilaterals and insert them into the queue.

Algorithm complexity Time

Constructing initial static DT - O(N log N) Each swap - O(1) Insertion into the priority queue - O(log N) Deletion from the queue can be performed in O(1) Number of topological events depends on the trajectories of

moving sites. It can vary from from O(1) up to O(n3) and even to infinity (in the case of periodic trajectories) When sites do not move, but just grow, we conjecture that

the number of topological events will not be greater than O(N) in general case.

Space DT occupies O(N) Queue requires O(N) because only one event can be

scheduled for each edge in the DT

Geometric Data Structures:

Part 2: Grid files and tree variants

Part 2

Grid file Tree Variants

Geometric data structures

Voronoi diagramsRegular grid (mesh)Grid fileQuad treek-d treeInterval tree

Trees BST – search trees, O(n) AVL, IPR – balanced O(log n) B-trees – for indexing and

searching in data bases: Grow from the leaf level More compact – faster search

B+, B* - used for indexing, store data in leaves, nodes are more full

Operations on spatial trees Spatial queries Point location Stabbing query (which

intervals/polygons contain the point)

Window query – which objects (polygons, points) are intersecting the given window (polygon)

Spatial queries (2D) Point query – find an object

containing a point (find Voronoi region containing a point)

Window query – find an object overlapping a rectangle

Spatial join – join parts of objects satisfying some relationship (intersection, adjacency, containment)

Interval trees Geometric, 1-dimensional tree Interval is defined by (x1,x2) Split at the middle (5), again at the middle (3,7),

again at the middle (2,8) All intervals intersecting a middle point are stored

at the corresponding root.

1 2 3 4 5 6 7 8 9

(4,6) (4,8)

(6,9)

(7.5,8.5)

(2,4)

Interval trees Finding intervals – by finding x1, x2

against the nodes Find interval containing specific value –

from the root Sort intervals within each node of the

tree according to their coordinates Cost of the “stabbing query”– finding all

intervals containing the specified value is O(log n + k), where k is the number of reported intervals.

SAM (Spatial Access Method) Constructs the minimal bounding box (mbb) Check validity (predicate) on mbb Refinement step verifies if actual objects

satisfy the predicate.

The grid Fixed grid: Stored as a 2D array, each entry

contains a link to a list of points (object) stored in a grid.

a,b

Page overflow

Too many points in one grid cell:Split the cell!

Rectangle indexing with grids Rectangles may share different

grid cells Duplicates are stored Grid cells are of fixed size

Grid file vs. grid In a grid file, the index is dynamically

increased in size when overflow happens.

The space is split by a vertical or a horizontal line, and then further subdivided when overflow happens!

Index is dynamically growing Boundaries of cells of different sizes are

stores, thus point and stabbing queries are easy

The quadtree Instead of using an array as an

index, use tree!

Quadtree decomposition – cells are indexed by using quaternary B-tree.

All cells are squares, not polygons. Search in a tree is faster!

Grid file Example of a grid file

Linear quadtree B+ index – actual references to

rectangles are stored in the leaves, saving more space+ access time

Label nodes according to Z or “pi” order

Linear quadtree Level of detail increases as the

number of quadtree decompositions increases!

Decompositions have indexes of a form:00,01,02,03,10,11,12,13, 2,300301 ,302 ,303 ,31 ,32 ,33 Stores as Bplus tree

Finer Grid

R-tree Each object is decomposed and

stored as a set of rectangles Object decomposition: larger areas of

a grid are treated as one element Raster decomposition: each smaller

element is stored separately

R-trees

R-tree Objects are grouped together according to

topological properties not a grid. More flexibility.

R * tree- Optimizes Node overlapping Areas covered by the node

R+ tree – B+ tree, bounding rectangles do not intersect

K-d tree Used for point location, k –number

of the attributes to perform the search

Geometric interpretation – to perform search in 2D space – 2-d tree

Search components (x,y) interchange

K-d tree

a

c

b

ed

d

b

f

f

c a e

Conclusions

Numerous applications exist based on Voronoi diagram methodology and tree/grid based data structures.

Geometric Data Structures Dr. M. Gavrilova. Lecture Plan Voronoi diagrams Trees and grid variants.

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Transcript of Geometric Data Structures Dr. M. Gavrilova. Lecture Plan Voronoi diagrams Trees and grid variants.