Minimum Rectilinear Polygons for Given Angle Sequences W. Evans 1 , - PowerPoint PPT Presentation

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea:                  S                

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea:                  S                

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea:                  S                

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea:                  S                

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea:                  S                

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea:                  S                

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea:                  S                

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea:                  S                

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea:                  S                

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea:

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea: Distance always 2 · 3 m

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea: Distance always 2 · 3 m Blocks:

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea: Distance always 2 · 3 m Blocks: 4 · 6 m · s j 6 m · ( m + 1)

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea: Distance always 2 · 3 m Blocks: 4 · 6 m · s j 6 m · ( m + 1)

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea: Distance always 2 · 3 m Blocks: 4 · 6 m · s j 6 m · ( m + 1) Walls:

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea: Distance always 2 · 3 m Blocks: 4 · 6 m · s j 6 m · ( m + 1) Walls: ≈ m 4 B

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea: Distance always 2 · 3 m Blocks: 4 · 6 m · s j 6 m · ( m + 1) Walls: ≈ m 4 B

Reduction (Sketch) Given an instance S = ( s 1 , . . . , s 3 m ) of 3-Partition, compute in poly( m ) time K > 0 and an instance σ of MinBBLRSequence s.t. area( σ ) < K ⇔ S is yes-instance Idea: Distance always 2 · 3 m Blocks: 4 · 6 m · s j 6 m · ( m + 1) Walls: ≈ m 4 B reduction ∈ poly( m , B )

Minimizing the Area of xy -monotone Polygons

Polygon canonical if . . .

Polygon canonical if . . . extreme edges

Polygon canonical if . . . Bounding Box extreme edges Bounding Box

Polygon canonical if . . . Bounding Box extreme edges . . . connected to a blue stair! Bounding Box

Example: Canonical Polygon connected Def.

Example: Canonical Polygon connected Def.

Example: Canonical Polygon connected Def.

Example: Canonical Polygon connected Def. not connected!

Example: Canonical Polygon However . . . connected Def.

Example: Canonical Polygon However . . . connected Def. connected! . . . is canonical.

connected Lemma : There is an optimum Def. canonical polygon!

connected Lemma : There is an optimum Def. canonical polygon! How to find it?

Computing Optimum Canonical Polygon Given angle sequence . . . TL TR BL BR

Computing Optimum Canonical Polygon Given angle sequence . . . . . . O (1) candidates for OPT. TR TL TR TL BR BL BR BL

Computing Optimum Canonical Polygon Given angle sequence . . . . . . O (1) candidates for OPT. TL TR TL TR TL or TR BR BL BL BR BL BR

Computing Optimum Canonical Polygon Given angle sequence . . . . . . O (1) candidates for OPT. TL TR TL TR TL or TR BR BL BL BR BL BR

Computing Optimum Canonical Polygon Given angle sequence . . . . . . O (1) candidates for OPT. TL TR TL TR TL or TR BR BL BL BR BL BR

Computing Optimum Canonical Polygon Given angle sequence . . . . . . O (1) candidates for OPT. TL TR TL TR TL or TR BR BL BL BR BL BR

Computing Optimum Canonical Polygon Given angle sequence . . . . . . O (1) candidates for OPT. TL TR TL TR TL or TR BR BL BL BR BL BR

Computing Area of a Candidate Given a candidate . . .

Computing Area of a Candidate Given a candidate . . . ≤ 2 stairs

Computing Area of a Candidate Given a candidate . . . Easy to solve! ≤ 2 stairs

Computing Area of a Candidate Given a candidate . . . O ( n ) time ≤ 2 stairs OPT 1 OPT 2 OPT 3

Computing Area of a Candidate Given a candidate . . . Put together! O ( n ) time ≤ 2 stairs OPT 1 OPT 2 OPT 3

Computing Area of a Candidate Given a candidate . . . O ( n ) time ≤ 2 stairs OPT 1 OPT 2 OPT 3

Computing Area of a Candidate Given a candidate . . . O ( n ) time ≤ 2 stairs OPT 1 OPT 2 OPT 3

Computing Area of a Candidate Given a candidate . . . O ( n ) time ≤ 2 stairs OPT 1 OPT 2 OPT 3

Computing Area of a Candidate Given a candidate . . . O ( n ) time ≤ 2 stairs OPT 1 OPT 2 OPT 3

Algorithm Algorithm : For every candidate × O (1) compute OPT candidate compute OPT candidate O ( n ) OPT = min OPT candidate O ( n )

Minimizing Bounding Box of x -monotone Polygons

Canonical x -monotone Polygons Bounding Box (BB)

Canonical x -monotone Polygons

Canonical x -monotone Polygons

Canonical x -monotone Polygons Canonical

Canonical x -monotone Polygons Red part fixed!

Canonical x -monotone Polygons Red part fixed! Only unknowns: a) Height of BB b) Width of extreme edges

Canonical x -monotone Polygons Red part fixed! Only unknowns: a) Height of BB Algorithm : Guess height of BB b) Width of extreme edges Use DP to find widths

DP Algorithm T [ p , e ] h

DP Algorithm T [ p , e ] p

DP Algorithm T [ p , e ] e p

DP Algorithm T [ p , e ] = minimum w s.t. left part ∈ BB( w × h ) e left part h p w