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Proceedings

Scandinavian Thins over a Fat Top:on the Smallest One-Size-Fits-All Box

Esther Arkin Applied Mathematics and Statistics and Computer Science, Stony BrookUniversity, USA

Alon Efrat Computer Science, The University of Arizona, USA

George Hart

Irina Kostitsyna Computer Science, Stony Brook University, USA

Alexander Kroeller Institute of Operating Systems and Computer Networks, BraunschweigInstitute of Technology, Germany

Joseph Mitchell Applied Mathematics and Statistics and Computer Science, Stony BrookUniversity, USA

Valentin Polishchuk Computer Science, University of Helsinki, Finland and Science andTechnology, Linkoping University, Sweden

Scandinavian Thins on Top of Cake:on the Smallest One-Size-Fits-All Box

ProgramDecoratingChristmas tree

Listening toJingle Bells

Merry Xmas

Thinking aboutsomething FUN cookiesthinsGingerbread

http://en.wikipedia.org/wiki/Ginger_biscuit:

http://en.wikipedia.org/wiki/Gingerbread:

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Alc. by vol.:3-20 %

Mulledwine

Consumption: adults

Consumption: kidsConstraint: Equal rectanglesObjective: Minimize area

Problem statement Given:polygons P1…PK

Find:min-area rectangle Rs.t. Pk can fit into R

aftertranslation and rotation

is not

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• asdg-

asdga “Stacking” minimizearea

Packing• simultaneous fit

and• no overlap

Hard problem

We (“stacking”)• One at a time

or• Overlap allowed

Efficient algorithms

Fitting 1 polygon into rectangle

Rotating calipers [Toussaint, IEEE MELECON'83]

A( ) = l( )w( )

2 pairs of calipersMonotone

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Flushness

Min-area enclosing rectangle:choose best flush edge

Linear time

pt on arc

l

w

Elementary arc

l

w(l,w)

fits into l w box (l,w) above FF graphLower envelope

Fit Function

(l,w)

l

w

fits into l w box (l,w) above FF graphs(l,w) above max FF graphs

Alg MinRectangle(P1…PK):For each k = 1…K

Build Fit Function for PkCompute FF* = max FFsFind (l,w) on FF* to min lw a breakpoint of FF*

lw = const

DS sequence of order s on {1…n}: s+2 alternations x…y…x…y…x…y

no xx

Max length:

s(n)

Runtime: Davenport–Schinzel sequences

n functionsany 2 intersect at most s times

Lower envelope:

s+2(n) complexity

s+1(n) log n time to build[Hershberger, Information Processing Letters ‘89]

DS sequences and Lower envelopes

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l

w

Runtime

Pk

m = |Pk| functionsany 2 intersect at most 4 times

Lower envelope:

6(m) complexity

5(m) log m time

Fit Function:

6(m) complexity

5(m) log m time

l

w

Runtime FF* = max FFs6(n) functions

any 2 intersect at most 4 times

6 6(n)) complexity

5 6(n)) log n time

Alg MinRectangle(P1…PK):For each k = 1…K

Build Fit Function for PkCompute FF* = max FFsFind (l,w) on FF* to min lw

Fit Function:

6(m) complexity

MinRectangle runs in5 6(n)) log n time

l

w

maximize areas.t. cookies sit tight in the box

Rip-off ANNA’s problem

all tight boxes

1 polygon: flushness of min-area rectangle

A( ) = l( )w( )

Monotone

Multiple polygons:min-area rectangle not necessarily flush

Min enclosing rectangle of rectangles Min enclosing rectangle of rectanglesflush with widest enclosed rectangle

Proof: in the paper

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Min enclosing rectangle of rectanglesflush with widest enclosed rectangle

Linear time

Polygons Rectangular enclosure Convex enclosure

Convex5 6(n)) log n

PTASes (3D too)n/ log(1/ ) , n + K/ 3/2 log(1/ )

NP-hardPTAS for min-perimeter

min-area: open

Rectangles O(n) Open

l

w

(l,w)P1…PK fit into l w box?

[Martin, Stephenson, Theory and practice of geometric modeling’89]

[Agarwal, Har-Peled, Varadarajan, Journal of the ACM’04]

linear time

2

1

points

Min surface area Breads

Capsule hotel Polygons Rectangular enclosure Convex enclosure

Convex5 6(n)) log n

PTASesn/ log(1/ ) , n + K/ 3/2 log(1/ )

NP-hardPTAS for min-perimeter

min-area: open

Rectangles O(n) Open

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Polygons Rectangular enclosure Convex enclosure

Convex5 6(n)) log n

PTASesn/ log(1/ ) , n + K/ 3/2 log(1/ )

NP-hardPTAS for min-perimeter

min-area: open

Rectangles O(n) Open

[Chazelle, Advances in Computing Research’83]

2

1

points

sides

OPT

B per(OPT) (1+ ) • per (B)

For each b-gon Bb =

P1…PK fit into B ?

Polygons Rectangular enclosure Convex enclosure

Convex5 6(n)) log n

PTASesn/ log(1/ ) , n + K/ 3/2 log(1/ )

NP-hardPTAS for min-perimeter

min-area: open

Rectangles O(n) Open

2

1

points

sidesB

per(OPT) (1+ ) • per (B)

0 area(OPT) ?? • area (B)

OPT

?Flipsallowed?

b =