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GD 2011September 21, 2011 Stephane Durocher Debajyoti Mondal Md. Saidur Rahman Sue Whitesides Rahnuma Islam Nishat Dept. of Computer Science and Engg. Bangladesh University of Engineering and Technology Department of Computer Science University of Victoria Department of Computer Science University of Manitoba

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GD 2011September 21, 2011 a b c d e f g h i A plane graph G A point set P 1

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GD 2011September 21, 2011 a b c d e f g h i A plane graph G An embedding of G on P a b c d e f g h i 2

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GD 2011September 21, 2011 Gritzmann et al. (1991), Castañeda and Urrutia (1996) Outerplanar graphsO(n 2 ) Bose (2002)Outerplanar graphsO(n lg 3 n) Cabello (2006)Biconnected 2-outerplanar graphs NP-complete Nishat et al. (2010) Plane 3-trees Partial plane 3-trees O(n 2 ), NP-complete Moosa et al. (2011) Plane 3-trees O(n 4/3 + ɛ log n) This Presentation Reference Graph Class Time complexity 3

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GD 2011September 21, 2011 a b c d e f g h i j k l m n o A plane 3-tree G f g h i j k l m n o a b c d e A construction for G 4

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GD 2011September 21, 2011 a b c d e f g h i j k l m n o A plane 3-tree G f g h i j k l m n o a b c d e The representative vertex of G k l e A plane 3-tree A construction for G c o c g m n d A plane 3-tree 5

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GD 2011September 21, 2011 a b c d e Convex Hull f g h A plane 3-tree G A point set P 6

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GD 2011September 21, 2011 b c d e a c b a f g h A plane 3-tree G A point set P We can map the outervertices in Six different ways. 7

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GD 2011September 21, 2011 d b c d e a a c b Find a valid mapping for the representative vertex. f g h Valid mapping?? a b c d e f g h n 1 = 1 n 2 = 1 n 3 = 2 8

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GD 2011September 21, 2011 a c b d Find a valid mapping for the representative vertex. b c d e a f g h a b c d e f g h n 1 = 1 n 2 = 1 n 3 = 2 9

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GD 2011September 21, 2011 b c d e a a c b d f g h e h f g Find a valid mapping for the representative vertex recursively. a b c d e f g h 10

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GD 2011September 21, 2011 How fast can we find a valid mapping for the representative vertex, if such a mapping exists? 11

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GD 2011September 21, 2011 b c d e a c Representative vertex cannot be mapped in the shaded regions. At most min{n 1, n 2, n 3 }+1 points in the white region are candidates. Representative vertex cannot be mapped in the shaded regions. At most min{n 1, n 2, n 3 }+1 points in the white region are candidates. f g h a b c d e f g h Assume that n 1 ≤ min{n 2,n 3 }. a n 3 = 2 n 2 = 1 b n 1 = 1 n 2 = 1 n 3 = 2 12

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GD 2011September 21, 2011 a b c d e a c b f g h a b c d e f g h z Choose a random point z Δ abc We need n 3 points in this region Choose a random point z Δ abc We need n 3 points in this region How do we select the shaded regions? n 1 = 1 n 2 = 1 n 3 = 2 13

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GD 2011September 21, 2011 a b c d e a c b f g h a b c d e f g h How do we select the shaded regions? z Choose a random point z Δ abc We need n 3 points in this region Choose a random point z Δ abc We need n 3 points in this region n 1 = 1 n 2 = 1 n 3 = 2 14

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GD 2011September 21, 2011 How do we select the shaded regions? b c d e a f g h a b c d e f g h c a b n 3 = 2 n 2 = 1 n 1 = 1 n 2 = 1 n 3 = 2 Selecting the shaded regions takes O(t n log n) expected time. At most min{n 1, n 2, n 3 }+1 points in the white region are candidates. Selecting the shaded regions takes O(t n log n) expected time. At most min{n 1, n 2, n 3 }+1 points in the white region are candidates. 15

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GD 2011September 21, 2011 a c b d Find a valid mapping in f n = O(t n log n) + O(t n min{n 1, n 2, n 3 }) time. T(n) = T(n 1 )+ T(n 2 )+ T(n 3 )+ f n = O(n 4/3 + ɛ ), for any ɛ > 0, using Chazelle’s DS. Find a valid mapping in f n = O(t n log n) + O(t n min{n 1, n 2, n 3 }) time. T(n) = T(n 1 )+ T(n 2 )+ T(n 3 )+ f n = O(n 4/3 + ɛ ), for any ɛ > 0, using Chazelle’s DS. b c d e a f g h a b c d e f g h n 1 = 1 n 2 = 1 n 3 = 2

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GD 2011September 21, 2011 b c d e a c b a f g h A plane 3-tree G A point set P and a prespecified mapping for the outervertices of G 16

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GD 2011September 21, 2011 Instance: A set of 3m nonzero positive integers S = {a 1, a 2,...,a 3m } and an integer B > 0, where a 1 +a a 3m = mB and B/4 0, where a 1 +a a 3m = mB and B/4

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GD 2011September 21, 2011 a1a1 a2a2 a3ma3m B B Y b c Z X a b c a 18 G P

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GD 2011September 21, 2011 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} Example: BB B B 19 x

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GD 2011September 21, 2011 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} Example: BB B B a1 a1 a2 a2 a|S| a|S| A fan A divider 20 x

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GD 2011September 21, 2011 S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 }, B = 32 S 1 ={10, 10, 12}, S 2 ={ 9, 11, 12}, S 3 ={ 9, 9,14}, S 4 ={ 10, 11,11} Example: 20 x {10, 10, 12} { 9, 11, 12} { 9, 9,14} { 10, 11,11}

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GD 2011September 21, 2011 a1a1 a2a2 a3ma3m B B Y b c Z X a b c a 21 G P

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GD 2011September 21, 2011 B B b c b c a A fan A divider a A spine vertex A spine vertex 22 G P

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GD 2011September 21, 2011 B B b c b c a A fan A divider a A spine vertex A spine vertex 23 G P

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b GD 2011September 21, 2011 B B c b c a Edge Crossings? Edge Crossings? a 24 A spine vertex A spine vertex A fan A divider G P

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GD 2011September 21,

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GD 2011September 21, 2011 Dept. of Computer Science and Engg. Bangladesh University of Engineering and Technology Department of Computer Science University of Victoria Department of Computer Science University of Manitoba

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