Deletion in Abstract Voronoi diagrams in Expected Linear Time Kolja - PowerPoint PPT Presentation

Universit` a della Our results Svizzera italiana Introduce Voronoi-like diagrams – relaxed version of a Voronoi diagram (easier to compute) A simple, randomized incremental algorithm for updating abstract Voronoi diagrams after deletion of one site in expected linear time . 11

Universit` a della Our results Svizzera italiana Introduce Voronoi-like diagrams – relaxed version of a Voronoi diagram (easier to compute) A simple, randomized incremental algorithm for updating abstract Voronoi diagrams after deletion of one site in expected linear time . – adapt to the farthest abstract Voronoi diagram , after the sequence of its faces at infinity is known. 11

Universit` a della Overview Svizzera italiana • Define abstract Voronoi diagrams (AVDs). • Define Voronoi-like diagrams . • Properties of Voronoi-like diagrams. • Define an insertion operation on Voronoi-like diagrams. • Sketch a randomized incremental algorithm. 12

Universit` a della Abstract Voronoi diagrams Svizzera italiana Rolf Klein. Concrete and Abstract Voronoi Diagrams . 1989. 13

Universit` a della Abstract Voronoi diagrams Svizzera italiana S abstract sites, n = | S | . J ( p, q ) bisector p q Rolf Klein. Concrete and Abstract Voronoi Diagrams . 1989. 13

Universit` a della Abstract Voronoi diagrams Svizzera italiana S abstract sites, n = | S | . J ( p, q ) dominance region of p D ( p, q ) p q Rolf Klein. Concrete and Abstract Voronoi Diagrams . 1989. 13

Universit` a della Abstract Voronoi diagrams Svizzera italiana S abstract sites, n = | S | . J ( p, q ) p q D ( q, p ) dominance region of q Rolf Klein. Concrete and Abstract Voronoi Diagrams . 1989. 13

Universit` a della Abstract Voronoi diagrams Svizzera italiana S abstract sites, n = | S | . J ( p, q ) p q Given a set of bisectors J := { J ( p, q ) : p � = q ∈ S } . Rolf Klein. Concrete and Abstract Voronoi Diagrams . 1989. 13

Universit` a della Abstract Voronoi diagrams Svizzera italiana r p VR ( p ) p q VR ( p ) = � q ∈ S \{ p } D ( p, q ) Voronoi region: Rolf Klein. Concrete and Abstract Voronoi Diagrams . 1989. 13

Universit` a della Abstract Voronoi diagrams Svizzera italiana V ( S ) = R 2 \ � Voronoi diagram: p ∈ S VR ( p, S ) Rolf Klein. Concrete and Abstract Voronoi Diagrams . 1989. 13

Universit` a della Admissible bisector system Svizzera italiana Given J := { J ( p, q ) : p � = q ∈ S } . For every S ′ ⊆ S : (A1) Voronoi regions are non-empty and connected. (A2) Voronoi regions cover the plane. (A3) Bisectors are unbounded Jordan curves. (A4) Transversal and finite # intersections. Rolf Klein. Concrete and Abstract Voronoi Diagrams . 1989. 14

Universit` a della Admissible bisector system Svizzera italiana • For simplicity we always assume a big circle Γ , containing all intersections. We restrict all computations in the interior of Γ . • VR ( s ) can be bounded, unbounded, and have several openings to infinity ( Γ -arcs). Γ Γ Γ VR( s ) VR( s ) VR( s ) Rolf Klein. Concrete and Abstract Voronoi Diagrams . 1989. 15

Universit` a della Site deletion Svizzera italiana Problem : Compute V ( S \ s ) ∩ VR ( s ) (within VR ( s ) ). VR ( s ) 16

Universit` a della Site deletion Svizzera italiana Problem : Compute V ( S \ s ) ∩ VR ( s ) (within VR ( s ) ). VR ( s ) Lemma: V ( S \ s ) ∩ VR ( s ) is a forest with one face per Voronoi edge of ∂ VR ( s ) . 16

Universit` a della Voronoi regions of arcs Svizzera italiana Idea : Treat the boundary arcs (Voronoi edges) of VR ( s ) as sites . 17

Universit` a della Voronoi regions of arcs Svizzera italiana Idea : Treat the boundary arcs (Voronoi edges) of VR ( s ) as sites . • Denote these arcs by S . S 17

Universit` a della Voronoi regions of arcs Svizzera italiana Idea : Treat the boundary arcs (Voronoi edges) of VR ( s ) as sites . • Denote these arcs by S . • Voronoi diagram of S is V ( S ) = V ( S \ s ) ∩ VR ( s ) . S 17

Universit` a della Voronoi regions of arcs Svizzera italiana Idea : Treat the boundary arcs (Voronoi edges) of VR ( s ) as sites . • Denote these arcs by S . • Voronoi diagram of S is V ( S ) = V ( S \ s ) ∩ VR ( s ) . • For an arc α ∈ S , assign VR ( α ) = face of V ( S ) incident to α . α S 17

Universit` a della Voronoi regions of arcs Svizzera italiana Idea : Treat the boundary arcs (Voronoi edges) of VR ( s ) as sites . • Denote these arcs by S . • Voronoi diagram of S is V ( S ) = V ( S \ s ) ∩ VR ( s ) . • For an arc α ∈ S , assign VR ( α ) = face of V ( S ) incident to α . Site s α can have Θ( n ) faces within VR ( s ) . Treat each face independently (different arc). VR ( s α ) s α = site defining α α S 17

Universit` a della Svizzera italiana Wish : Voronoi diagram of a subset of arcs S ′ ⊆ S . But that does not exist. S ′ ⊆ S 18

Universit` a della Svizzera italiana Wish : Voronoi diagram of a subset of arcs S ′ ⊆ S . But that does not exist. Instead we define a Voronoi-like diagram for a subset of arcs S ′ ⊆ S . S ′ ⊆ S 18

Universit` a della Svizzera italiana Wish : Voronoi diagram of a subset of arcs S ′ ⊆ S . But that does not exist. Instead we define a Voronoi-like diagram for a subset of arcs S ′ ⊆ S . Next: Definitions... S ′ ⊆ S 18

Universit` a della p -monotone paths Svizzera italiana Let p ∈ S be a site. Let J p be the arrangement of all p -related bisectors. p t p r J p p q 19

Universit` a della p -monotone paths Svizzera italiana Let p ∈ S be a site. Let J p be the arrangement of all p -related bisectors. A path in the arrangement J p is p -monotone , if any two adjacent edges α, β coincide locally with the Voronoi edges of VR ( p, { p, s α , s β } ) . VR ( p ) α β p p s α s β p -monotone path p t p r J p p q VR( p ) α p t p p p s α s β q β P 19

Universit` a della p -monotone paths Svizzera italiana Let p ∈ S be a site. Let J p be the arrangement of all p -related bisectors. A path in the arrangement J p is p -monotone , if any two adjacent edges α, β coincide locally with the Voronoi edges of VR ( p, { p, s α , s β } ) . A path in J p is the p -envelope , if it is the boundary of VR ( p ) p -envelope p t p t p r J p p q 19

Universit` a della p -monotone paths Svizzera italiana Let p ∈ S be a site. Let J p be the arrangement of all p -related bisectors. A path in the arrangement J p is p -monotone , if any two adjacent edges α, β coincide locally with the Voronoi edges of VR ( p, { p, s α , s β } ) . VR ( p ) α β p p s α s β p -monotone path p t p r J p p q p t p q P 19

Universit` a della Boundary curve Svizzera italiana Let S ′ ⊆ S = boundary arcs (Voronoi edges) along ∂ VR ( s ) . S ′ ⊆ S 20

Universit` a della Boundary curve Svizzera italiana Let S ′ ⊆ S = boundary arcs (Voronoi edges) along ∂ VR ( s ) . Consider the arrangement of all s -related bisectors of arcs in S ′ . S ′ 20

Universit` a della Boundary curve Svizzera italiana Let S ′ ⊆ S = boundary arcs (Voronoi edges) along ∂ VR ( s ) . A boundary curve P for S ′ is an s -monotone path in the arrangement of s -related bisectors that contains every arc in S ′ . S ′ P 20

Universit` a della Boundary curve Svizzera italiana Let S ′ ⊆ S = boundary arcs (Voronoi edges) along ∂ VR ( s ) . A boundary curve P for S ′ is an s -monotone path in the arrangement of s -related bisectors that contains every arc in S ′ . S ′ can have different boundary curves. S ′ P 20

Universit` a della Boundary curve Svizzera italiana original arc (contains an arc of S ′ ) S ′ Γ -arc auxiliary arc (does not contain an arc of S ′ ) 20

Universit` a della Boundary curve Svizzera italiana S ′ domain D P P 20

Universit` a della Voronoi-like diagram Svizzera italiana Definition: Given a boundary curve P , the Voronoi-like diagram V l ( P ) is a subdivision of the domain D P such that: P 21

Universit` a della Voronoi-like diagram Svizzera italiana Definition: Given a boundary curve P , the Voronoi-like diagram V l ( P ) is a subdivision of the domain D P such that: P 21

Universit` a della Voronoi-like diagram Svizzera italiana Definition: Given a boundary curve P , the Voronoi-like diagram V l ( P ) is a subdivision of the domain D P such that: • Each boundary arc α ∈ P has one region R ( α ) . α R ( α ) P 21

Universit` a della Voronoi-like diagram Svizzera italiana Definition: Given a boundary curve P , the Voronoi-like diagram V l ( P ) is a subdivision of the domain D P such that: • Each boundary arc α ∈ P has one region R ( α ) . R ( α ) • ∂R ( α ) is an s α -monotone path plus α . α α R ( α ) P 21

Universit` a della Properties of Voronoi-like diagrams Svizzera italiana • Voronoi-like regions are supersets of the real Voronoi regions. R ( α ) α P 22

Universit` a della Properties of Voronoi-like diagrams Svizzera italiana • Voronoi-like regions are supersets of the real Voronoi regions. R ( α ) α VR ( α ) P 22

Universit` a della Properties of Voronoi-like diagrams Svizzera italiana • Voronoi-like regions are supersets of the real Voronoi regions. • For all arcs S , V l ( S ) equals the real diagram V ( S )= V ( S \ s ) ∩ VR ( s ) . S 22

Universit` a della Properties of Voronoi-like diagrams Svizzera italiana • Voronoi-like regions are supersets of the real Voronoi regions. • For all arcs S , V l ( S ) equals the real diagram V ( S )= V ( S \ s ) ∩ VR ( s ) . • Missing arc lemma: Suppose an α -related bisector appears within R ( α ) . Then there is an arc β “missing” from P . R ( α ) α P 22

Universit` a della Properties of Voronoi-like diagrams Svizzera italiana • Voronoi-like regions are supersets of the real Voronoi regions. • For all arcs S , V l ( S ) equals the real diagram V ( S )= V ( S \ s ) ∩ VR ( s ) . • Missing arc lemma: Suppose an α -related bisector appears within R ( α ) . Then there is an arc β “missing” from P . β J ( s, s β ) R ( α ) α P 22

Universit` a della Uniqueness of Voronoi-like diagrams Svizzera italiana Theorem: The Voronoi-like diagram V l ( P ) of a boundary curve P is unique . V l ( P ) P 23

Universit` a della No monotonicity property Svizzera italiana Voronoi-like regions do not have the standard monotonicity property of real Voronoi regions: Voronoi diagram: S ′ ⊆ S ⇒ VR ( p, S ) ⊆ VR ( p, S ′ ) Voronoi-like diagram: S ′ ⊆ S �⇒ R ( α, S ) ⊆ R ( α, S ′ ) 24

Universit` a della No monotonicity property Svizzera italiana Voronoi-like regions do not have the standard monotonicity property of real Voronoi regions: Voronoi diagram: S ′ ⊆ S ⇒ VR ( p, S ) ⊆ VR ( p, S ′ ) Voronoi-like diagram: S ′ ⊆ S �⇒ R ( α, S ) ⊆ R ( α, S ′ ) In proofs, use missing-arc lemma instead 24

Universit` a della Arc insertion Svizzera italiana 25

Universit` a della Arc insertion Svizzera italiana Problem : Given a boundary curve P for S ′ ⊂ S and its Voronoi-like diagram V l ( P ) , insert arc β ∗ ∈ S \ S ′ , prolong β ∗ ⊆ β , compute R ( β ) , and update the diagram to V l ( P ) ⊕ β . V l ( P ) P 25

Universit` a della Arc insertion Svizzera italiana Problem : Given a boundary curve P for S ′ ⊂ S and its Voronoi-like diagram V l ( P ) , insert arc β ∗ ∈ S \ S ′ , prolong β ∗ ⊆ β , compute R ( β ) , and update the diagram to V l ( P ) ⊕ β . V l ( P ) β ∗ P 25

Universit` a della Arc insertion Svizzera italiana Problem : Given a boundary curve P for S ′ ⊂ S and its Voronoi-like diagram V l ( P ) , insert arc β ∗ ∈ S \ S ′ , prolong β ∗ ⊆ β , compute R ( β ) , and update the diagram to V l ( P ) ⊕ β . V l ( P ) β P ⊕ β 25

Universit` a della Arc insertion Svizzera italiana Problem : Given a boundary curve P for S ′ ⊂ S and its Voronoi-like diagram V l ( P ) , insert arc β ∗ ∈ S \ S ′ , prolong β ∗ ⊆ β , compute R ( β ) , and update the diagram to V l ( P ) ⊕ β . V l ( P ) R ( β ) β P ⊕ β 25

Universit` a della Arc insertion Svizzera italiana Problem : Given a boundary curve P for S ′ ⊂ S and its Voronoi-like diagram V l ( P ) , insert arc β ∗ ∈ S \ S ′ , prolong β ∗ ⊆ β , compute R ( β ) , and update the diagram to V l ( P ) ⊕ β . V l ( P ) ⊕ β R ( β ) R ( β ) β P ⊕ β 25

Universit` a della Arc insertion Svizzera italiana • Compute the boundary curve P ⊕ β containing β ( β ∗ ⊆ β ). β ∗ P 26

Universit` a della Arc insertion Svizzera italiana • Compute the boundary curve P ⊕ β containing β ( β ∗ ⊆ β ). β ∗ P 26

Universit` a della Arc insertion Svizzera italiana • Compute the boundary curve P ⊕ β containing β ( β ∗ ⊆ β ). β P ⊕ β 26

Universit` a della Arc insertion Svizzera italiana • Compute the boundary curve P ⊕ β containing β ( β ∗ ⊆ β ). β P ⊕ β 26

Universit` a della Arc insertion Svizzera italiana • Compute the boundary curve P ⊕ β containing β ( β ∗ ⊆ β ). • Compute the merge curve J ( β ) ; it defines region R ( β ) . J ( β ) β P ⊕ β 26

Universit` a della Arc insertion Svizzera italiana • Compute the boundary curve P ⊕ β containing β ( β ∗ ⊆ β ). • Compute the merge curve J ( β ) ; it defines region R ( β ) . • Insert R ( β ) in V l ( P ) and derive V l ( P ) ⊕ β : J ( β ) β P ⊕ β 26

Universit` a della Arc insertion Svizzera italiana • Compute the boundary curve P ⊕ β containing β ( β ∗ ⊆ β ). • Compute the merge curve J ( β ) ; it defines region R ( β ) . • Insert R ( β ) in V l ( P ) and derive V l ( P ) ⊕ β : J ( β ) β P ⊕ β 26

Universit` a della Insertion of β splits an arc Svizzera italiana When inserting β , a face (and its arc) may split in two, creating a new auxiliary arc ( γ ′ ) that was not in S . V l ( P ) P 27

Universit` a della Insertion of β splits an arc Svizzera italiana When inserting β , a face (and its arc) may split in two, creating a new auxiliary arc ( γ ′ ) that was not in S . V l ( P ) β ∗ P 27

Universit` a della Insertion of β splits an arc Svizzera italiana When inserting β , a face (and its arc) may split in two, creating a new auxiliary arc ( γ ′ ) that was not in S . V l ( P ) β ∗ P 27

Universit` a della Insertion of β splits an arc Svizzera italiana When inserting β , a face (and its arc) may split in two, creating a new auxiliary arc ( γ ′ ) that was not in S . V l ( P ) J ( β ) β P γ ′ γ 27

Universit` a della Insertion of β splits an arc Svizzera italiana When inserting β , a face (and its arc) may split in two, creating a new auxiliary arc ( γ ′ ) that was not in S . V l ( P ) ⊕ β R ( β ) β P ⊕ β γ ′ γ 27

Universit` a della Arc insertion Svizzera italiana Theorem: The merge curve J ( β ) is an s β -monotone path. J ( β ) β P ⊕ β 28

Universit` a della Arc insertion Svizzera italiana Theorem: The merge curve J ( β ) is an s β -monotone path. Theorem: V l ( P ) ⊕ β is the Voronoi-like diagram, V l ( P ⊕ β ) . V l ( P ⊕ β ) J ( β ) β P ⊕ β 28

Universit` a della Proof sketch Svizzera italiana Theorem: The merge curve J ( β ) is an s β -monotone path. Use a bi-directional induction starting at the two endpoints of β . Show: Γ P J ( β ) β 29

Universit` a della Proof sketch Svizzera italiana Theorem: The merge curve J ( β ) is an s β -monotone path. Use a bi-directional induction starting at the two endpoints of β . Show: Γ • J ( β ) cannot hit a boundary P arc. J ( β ) β 29