Degree-driven algorithm design for computing the Voronoi diagram

Degree-driven algorithm design for computing the Voronoi diagram Jack SnoeyinkFWCG08Oct 31, 2008David L. MillmanUniversity of North Carolina - Chapel HillVoronoi diagrams2Voronoi diagramsImplicit Voronoi Diagram [LPT97] 3Implicit Voronoi Diagram [LPT97]Topological componentPlanar embeddingGeometric ComponentEach vertex (vx,vy) of Voronoi diagram of S 4

Basic ProblemGiven:sites S ={s1,s2,,sn} w/ b-bit integer coords Construct: implied Voronoi V*(S) with minimum precision.

Note: precision < 5b bits precludes computing the Voronoi Diagram5Previous WorkHandling the precision requirements of geometric computation:Rely on machine precisionExact Geometric Computation [Y97]Arithmetic Filters [FV93][DP99]Adaptive Predicates [P92][S97] Topological Consistency [SI92]Degree-driven algorithmic design [LPT97]6Fortune and VanWyk FiltersPriest thesis for predicates6Cell Graph7Cell VertexCell EdgeGrid Cell VertexNon-Grid Cell VertexRandomized Incramental [SI92]8bisectorInCell9Given:Two sites s1, s2, and a grid cell G

Decide: Whether b12 passes through G

Arithmetic DegreeArithmetic degree monomial, sum of the arithmetic degree of its variablespolynomial, largest arithmetic degree of its monomials

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Using Liotta, Preperata and Tamassias def10bisectorInCell11Given:Two sites s1, s2, and a grid cell G

Decide: Whether b12 passes through G

Degree 2 and constant time

stabbingOrderingGiven: Two bisectors b12 & b34 that stab a grid cell G

Determine: The order in which the bisectors intersect the cell walls12Degree 3 and constant time

bisectorWalkGiven:Two sites s1, s2 and a direction to walk

bisectorWalk:a traversal of a subset of the cells that b12 passes though.

13Degree 2 and log(g)bisectorIntersection14Degree 3 and log(g)Given: Four sites si, i={1,2,3,4}

Find: The grid cell that contains the intersection of bisectors b12 & b34ResultsMethod for computing the implicit Voronoi diagram using predicates of max degree 3.Running time is in O(n (log n + log g)), where g is the max bisector length. First construction of the implicit Voronoi w/o computing the full Voronoi diagram.15Future WorkCan we do this in degree 2?Generalizing to other diagramsDiagrams with non-linear bisectorsIdentify the grid cell containing a bisector intersection in constant time

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Not in Grid CellIn Grid Cell

b12

g4

G

g2 g3

g1 g4

G

b12

g2 g3

g1

Not in Grid CellIn Grid Cell

b12

g4

G

g2 g3

g1 g4

G

b12

g2 g3

g1

b34

b12

w + 23w + 21 w + 22w

s1

s2

b12

w + 20

b34

w + 20 w + 21 w + 22 w + 23

s4

s3

w

s1

s2

b12