minimum-perimeter enclosing k-gons joe mitchell and valentin polishchuk

Minimum-Perimeter Minimum-Perimeter Enclosing k-gonsEnclosing k-gons

Joe Mitchell and Valentin PolishchukJoe Mitchell and Valentin Polishchuk

Given: Convex n-gon PGiven: Convex n-gon P

Find: Find: kk-gon -gon QQ

P P QQ

perimeterperimeter of of QQ → min→ min

Problem FormulationProblem Formulation

P

Q Stated as open problem:[Boyce, Dobkin, Drysdale, Guibas’85][Chang’86][DePano’87][Aggarwal and Park’88] [Mitchell, Piatko and Arkin’92][Piatko’93][Bhattacharya and Mukhopadhyay’02]

Given: Convex n-gon PGiven: Convex n-gon P

Find: k-gon QFind: k-gon Q

P P Q Q

areaarea of Q of Q → min→ min

Previous WorkPrevious Work

Solved in:[Aggarwal, Chang and Yap’85][Chang and Yap’86][Aggarwal, Klawe, Moran, Shor, Wilber‘86]

P

Q

Given: Convex n-gon PGiven: Convex n-gon P

Find: Find: 33-gon -gon QQ

P P QQ

perimeterperimeter of of QQ → min→ min

Previous WorkPrevious Work

P

Q

Solved in:[Bhattacharya and Mukhopadhyay’02][Medvedeva and Mukhopadhyay’03]

FPTASFPTASBased on Based on [MPA’92][MPA’92]

shortest shortest kk-link path in simple polygon-link path in simple polygon

Our ResultsOur Results

P

QSmall print:

Vertices of Q given by roots of high-degree polynomials

Exact solution not possible

O¡n3k3 log( N k

²1=k )¢

To “Rock”To “Rock”

hh ab = a ab = a

h h rocks onrocks on a a

P

Qha

b

To Be “Flush” withTo Be “Flush” with

PP

ee

P

Q

ef

e e ff

is flush with QQ

ff

Everybody in Polygon Everybody in Polygon Optimization Uses Optimization Uses

Flushness!Flushness![Boyce, Dobkin, Drysdale and Guibas’85][Boyce, Dobkin, Drysdale and Guibas’85][Chang’86, DePano’87][Chang’86, DePano’87][Aggarwal and Park’88][Aggarwal and Park’88][Mitchell, Piatko and Arkin’92][Mitchell, Piatko and Arkin’92][Piatko’93][Piatko’93][Bhattacharya and Mukhopadhyay’02][Bhattacharya and Mukhopadhyay’02][Aggarwal, Chang and Yap’85][Aggarwal, Chang and Yap’85][Chang and Yap’86][Chang and Yap’86][Aggarwal, Klawe, Moran, Shor and Wilber‘86][Aggarwal, Klawe, Moran, Shor and Wilber‘86]

And so will we…

The Flushness Lemma:The Flushness Lemma:

Lemma:Lemma: QQ is flush with P in opt is flush with P in opt

P

Q

The AlgorithmThe Algorithm

For each edge e of PFor each edge e of PGuess: e is flush with QGuess: e is flush with Q

Transform to simple P’Transform to simple P’

FindFind shortest (k+1)-link s-t shortest (k+1)-link s-t pathpath

Complete into QComplete into Qee

Chose min over e Chose min over e

P

eQ

P’

s t

Qe

[MPA’92]

e

Running TimeRunning Time

n ¢SP (k;n)

[MPA’92]

Can do better thanCan do better than

Looking into details of [MPA ’92] :Looking into details of [MPA ’92] :

or

n ¢SP (k;n)to find shortest k-link path in a simple n-gon

O¡n3k3 log( N k

²1=k )¢

O(n3k=²)

SP (k;n) = O¡n3k3 log( N k

²1=k )¢

SP (k;n) = O(k3n3=²)

““RRotating calipers” [Toussaint’83]otating calipers” [Toussaint’83]

Local optimality:Local optimality: Each edge of Each edge of QQ either flush with either flush with (an edge of)(an edge of) P P

or rocks on a vertex of Por rocks on a vertex of P

perimeter ofperimeter of Q Q ==

If not flush, rotate If not flush, rotate QQ by by θθ

perimeter ofperimeter of Q( Q(θθ) = ) = P k

i=1ai

sin ®i2

cos (¯ i ¡ µ)¡ (° i +µ)2

Proof of the Flushness Proof of the Flushness LemmaLemma

keeping ‘s fixed

const concave fcn of θ

concave fcn of θ

Attains min at θmin or θmax

← when Q is flush with P

P ki=1

aisin ®i

(sin¯ i + sin°i ) =P k

i=1ai

sin ®i2

cos ¯ i ¡ ° i2

A Fast Approximation A Fast Approximation AlgorithmAlgorithm

Interleaving k-gonsInterleaving k-gons

Locally Optimal k-gons Locally Optimal k-gons InterleaveInterleave

Otherwise – slide an Otherwise – slide an edge edge until until supported by Psupported by P

P

Perimeters of Interleaving Perimeters of Interleaving kk--gonsgons

rr11+r+r22 ≤≤ bb/sin(/sin())

p(p(RR) ≤ ) ≤ p(p(BB)/sin()/sin(minmin))

b

r1

r2

R

B

The Approximation The Approximation AlgorithmAlgorithm

Wrap Wrap equiangularequiangular k-gon k-gon QQkk

minmin(k-2)(k-2)kk

p(p(QQkk) ≤ p() ≤ p(OPTOPT)/cos()/cos(kk))

Linear-timeLinear-time

1/cos(1/cos(kk) - ) - approximationapproximation

P

Qk

ExtensionsExtensions

Minimum-perimeter enclosing Minimum-perimeter enclosing envelopeenvelope

k-gon with given angle sequence k-gon with given angle sequence AA

Flush with PFlush with P

by the Flushness Lemmaby the Flushness Lemma

O(nk log k)O(nk log k) timetime [Mount [Mount

and Silverman’94]and Silverman’94]

ExtensionsExtensionsRestricted envelopeRestricted envelope

Given Given PPoutout , P, Pinin PPoutout ,,

sequence sequence AA of k angles of k angles

Find PFind Pinin Q Q PPoutout

Q Q has angle sequence has angle sequence AAperimeter ofperimeter of Q Q → min→ min

Application:Application: ClassificationClassification Build low-complexity separatorBuild low-complexity separator

Pin

Q

Pout

EitherEither flushflush oror bashbash

If flush If flush guess the flush edgeguess the flush edge

wrapwrap

with Pin with Pout

Pin

Q

Pout

by the Flushness Lemma

i

a2

a1

IfIf bashbash

For {pFor {pjj,p,pll} – vertices of P} – vertices of Pinin

For For ee – edge of – edge of PPoutout

For For qqii – vertex of – vertex of QQ

Guess Guess qqi-1i-1qqii rocks on p rocks on pjj

qqiiqqi+1i+1 rocks on p rocks on pll

qqii is bash with e is bash with e

the bash point is a1 or a2

because iis fixed Pin

Pout

eqi

pj

pl

O(nin2noutk) guesses

Know the direction of an edge of QKnow the direction of an edge of Q

O(min{O(min{ nninin, , kk log n log ninin}) to wrap}) to wrap

If flush If flush

O(nO(nininmin{min{ nninin, , kk log n log ninin})})

If If bashbash

O(nO(ninin22nnoutoutmin{min{ nninin, , kk log n log ninin})})

Restricted EnclosuresRestricted Enclosures

Pin

Q

Pout

O(nin2noutmin{ nin, k log nin})

ConclusionConclusion

Algorithms for Algorithms for minimum-perimeter enclosing k-gonminimum-perimeter enclosing k-gon

Linear-time Linear-time 1/cos(1/cos(kk) – approximation) – approximation

ExtensionsExtensionsenvelopesenvelopes

restricted envelopesrestricted envelopes

Open: 3D min-surface-area polytope?

orO¡n3k3 log( N k

²1=k )¢

O(n3k=²)

Polygon Inclusion/EnclosurePolygon Inclusion/Enclosure Given Given polygon Ppolygon P Find Find polygon Q polygon Q

• Q Q P – inclusion problem P – inclusion problem P P Q – enclosure problem Q – enclosure problem

Such that Such that • Q is as large as possible – inclusion problemQ is as large as possible – inclusion problem• Q is as small as possible – enclosure problemQ is as small as possible – enclosure problem

ObjectivesObjectives• areaarea• perimeterperimeter