The Voronoi diagram of convex objects

in the planeMenelaos Karavelas & Mariette Yvinec

Dagsthul Workshop, march 2003

A Voronoi diagram :a set of site + a distance

The distance

Axxp

AxxpAp

Ax

Ax

||,||min

||,||min),(

The set of sites

1. Pseudo-circles sets of smooth convex objects• convex objects• smooth boundaries• at most two intersection points between the

boundaries of two objects

2. Pseudo-circles sets of piecewise smooth convex objects

3. General sets of convex objectsVoronoi diagram restricted outside the complement

of the union

Pseudo-circles sets of smooth convex objects

Pseudo-circles sets of piecewise smooth convex objects

Non pseudo circles set

Previous works

• Concrete and abstract Voronoi diagramsKlein 89

• Randomized incremental construction on abstract Voronoi diagramsKlein, Mehlhorn & Meiser 93

• The Voronoi diagram of curved objects.Alt & Schwarzkopf 95

• Dynamic additively weighted Voronoi diagrams in 2DKaravelas & Yvinec 02

The case of pseudo-circles setsof smooth convex objects

• Th1 : The bisector of two sites is – either empty– or a single curve homeomorphic to ]0,1[

• Th2 : The cell of each site is simply connected

The cell of an object

• Non empty cells

• Empty cells Hidden object Ai : any maximal disk in Ai

is included in some other Aj

ji

i

AxC

Ax

)(ji

i

AxC

Ax

)(

Th1 : The bisector of two sites is – either empty– or a single curve homeomorphic to ]0,1[

bisector emptyji AA

ji AA

ji AA

),(),( ji AxAx consider the function

Th2 : Voronoi cells are simply connected

connected is

in included not

in balls maximal of centers

of axis medial

)(

)()(

)(

i

j

iii

ii

AN

A

AASAN

AAS

1)

)

to respect withshaped-star weaklyis of cell The

i

i

AN

A

(

2)

The Algorithm

Randomized incremental

The basic data structures• The 1-skeleton of the Voronoi

diagram or its dual graph• The covering graph: to keep track

of hidden sites

)K(AAMAM

A)K(A

A

ii

ii

i

,max ball

of covering a

site hidden a

that such sites of subset a

The conflict region :when inserting new site Apart of the Voronoi 1-skeletonwhere bitangent circles are• either internal and included in A• or external and intersecting A

Insertion of a new site• Find a first conflict or a covering of the new site• Find the while conflict region and repair the Voronoi

diagram• update the covering graph

Nearest neighbor query : the Voronoi hierarchy

ConstructionLevel 0 : the whole diagramLevel k : insert each sites in

level k-1 with propability

Query• At each level, the visited

sites have decreasing distances to the query point

• Expected number of sites visited : O(1/ )

+ a tree for each cell

To avoid checking all the neighbors of A to find one closer to q,

the normals through the Voronoi vertices of the cell of A are stored in a bb-tree

Time spend in each cell O( log n)nn-query time

n)O(log2

Find the first conflict or detect a hidden site

• Disjoint sites– issue a nn query for a point p of A– at least one edge of cell(nn(p)) conflicts A

• Intersecting sites– issue a nn query for any point p of ma(A)– if M(p) nn(p), at least one edge of cell(nn(p))

conflicts with A– if M(p) nn(p) prune ma(A) and iterate

Removal of object A

• Update the Voronoi diagram– insertion the neighbors of A in an annex

Voronoi diagram – copy back in the main diagram the filling of

cell(A)• Remove A from covering graph• Reinsert objects hidden by A

Voronoi diagram for pseudo-circles setExpected complexity

Objects disjoints No hidden Hidden

Insertion

Deletion n)O(log3

n)O(log2 O(n)

O(n) )O(n2

O(n)

Pseudo-circles sets of piecewise smooth objects

• add point site at the vertices• perturb the distance

)dim(),(),(' AApAp

General convex objects

Further work

• work out the predicates and implement the algorithm for ellipses

• extend to pseudo-circles set of non convex objects• Voronoi diagram for convex objects in 3d