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Minimum-Perimeter Enclosing k-gons Joe Mitchell and Valentin Polishchuk

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Given: Convex n-gon P Find: k-gon Q P Q P Q perimeter of Q → min Problem 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]

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Given: Convex n-gon P Find: k-gon Q P Q P Q area of Q → min Previous Work Solved in: [Aggarwal, Chang and Yap’85] [Chang and Yap’86] [Aggarwal, Klawe, Moran, Shor, Wilber‘86] P Q

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Given: Convex n-gon P Find: 3-gon Q P Q P Q perimeter of Q → min Previous Work P Q Solved in: [Bhattacharya and Mukhopadhyay’02] [Medvedeva and Mukhopadhyay’03]

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FPTAS Based on [MPA’92] shortest k-link path in simple polygon Our Results P Q Small print: Vertices of Q given by roots of high-degree polynomials Exact solution not possible O ¡ n 3 k 3 l og ( N k ² 1 = k ) ¢

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To “Rock” h ab = a h rocks on a h rocks on a P Q h a b

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To Be “Flush” with Pe P Q e f e f is flush with Q f

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Everybody in Polygon Optimization Uses Flushness! [Boyce, Dobkin, Drysdale and Guibas’85] [Chang’86, DePano’87] [Aggarwal and Park’88] [Mitchell, Piatko and Arkin’92] [Piatko’93] [Bhattacharya and Mukhopadhyay’02] [Aggarwal, Chang and Yap’85] [Chang and Yap’86] [Aggarwal, Klawe, Moran, Shor and Wilber‘86] And so will we…

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The Flushness Lemma: Lemma: Q is flush with P in opt Lemma: Q is flush with P in opt P Q

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The Algorithm For each edge e of P Guess: e is flush with Q Transform to simple P’ Find shortest (k+1)-link s-t path shortest (k+1)-link s-t path Complete into Q e Chose min over e P e Q P’ s t QeQe [MPA’92] e

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Running Time n ¢ SP ( k ; n ) [MPA’92] Can do better than Looking into details of [MPA ’92] : or n ¢ SP ( k ; n ) to find shortest k-link path in a simple n-gon O ¡ n 3 k 3 l og ( N k ² 1 = k ) ¢ O ( n 3 k = ² ) SP ( k ; n ) = O ¡ n 3 k 3 l og ( N k ² 1 = k ) ¢ SP ( k ; n ) = O ( k 3 n 3 = ² )

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“Rotating calipers” [Toussaint’83] Local optimality: Local optimality: Each edge of Q either flush with (an edge of) P either flush with (an edge of) P or rocks on a vertex of P or rocks on a vertex of P perimeter of Q = If not flush, rotate Q by θ perimeter of Q(θ) = P k i = 1 a i s i n ® i 2 cos ( ¯ i ¡ µ ) ¡ ( ° i + µ ) 2 Proof of the Flushness Lemma keeping ‘s fixed const concave fcn of θ Attains min at θ min or θ max ← when Q is flush with P P k i = 1 a i s i n® i ( s i n ¯ i + s i n ° i ) = P k i = 1 a i s i n ® i 2 cos ¯ i ¡ ° i 2

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A Fast Approximation Algorithm

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Interleaving k-gons

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Locally Optimal k-gons Interleave Otherwise – slide an edge until supported by P P

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Perimeters of Interleaving k-gons r 1 +r 2 ≤ b/sin( ) p(R) ≤ p(B)/sin( min ) b r1r1 r2r2 R B

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The Approximation Algorithm Wrap equiangular k-gon Q k min (k-2) k min (k-2) k p(Q k ) ≤ p(OPT)/cos( k) Linear-time 1/cos( k) - approximation 1/cos( k) - approximation P QkQk

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Extensions Minimum-perimeter enclosing envelope k-gon with given angle sequence A Flush with P by the Flushness Lemma O(nk log k) time [Mount and Silverman’94]

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Extensions Restricted envelope Given P out, P in P out, sequence A of k angles sequence A of k angles Find P in Q P out Q has angle sequence A Q has angle sequence A perimeter of Q → min Application: Classification Build low-complexity separator Build low-complexity separator P in Q P out

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Either flush or bash If flush guess the flush edge guess the flush edgewrap with P in with P out P in Q P out by the Flushness Lemma

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ii a2a2 a1a1 If bash For {p j,p l } – vertices of P in For e – edge of P out For q i – vertex of Q Guess q i-1 q i rocks on p j q i-1 q i rocks on p j q i q i+1 rocks on p l q i q i+1 rocks on p l q i is bash with e q i is bash with e the bash point is a 1 or a 2 because i is fixed P in P out e qiqi pjpj plpl O(n in 2 n out k) guesses

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Know the direction of an edge of Q O(min{ n in, k log n in }) to wrap If flush O(n in min{ n in, k log n in }) If bash O(n in 2 n out min{ n in, k log n in }) Restricted Enclosures P in Q P out O( n in 2 n out min{ n in, k log n in } )

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Conclusion Algorithms for minimum-perimeter enclosing k-gon Linear-time 1/cos( k) – approximation 1/cos( k) – approximationExtensionsenvelopes restricted envelopes restricted envelopes Open: 3D min-surface-area polytope? or O ¡ n 3 k 3 l og ( N k ² 1 = k ) ¢ O ( n 3 k = ² )

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Polygon Inclusion/Enclosure Given polygon P Given polygon P Find polygon Q Find polygon Q Q P – inclusion problem Q P – inclusion problem P Q – enclosure problem P 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 Objectives Objectives areaarea perimeterperimeter

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