Width-Optimal Visibility Representations of Plane Graphs Speaker: Chun-Cheng Lin Coauthors: Jia-Hao Fan Hsueh-I Lu Hsu-Chun Yen National Taiwan University, Taipei, Taiwan (Presented at the 18th International Symposium on Algorithms and Computation (ISAAC 2007)) 1/19
Outline Introduction 1 Preliminaries 2 Our Width-Optimal Drawing Algorithm 3 Analysis 4 Conclusion 5 2/19
Introduction Visibility Representation (a.k.a., Visibility Drawing) Visibility Representation Node segment Edge segment Measuring the drawing area in a grid 3 3 2 2 8 8 7 4 5 4 9 5 6 9 7 6 1 1 3/19
Introduction Visibility Representation (a.k.a., Visibility Drawing) Visibility Representation Node segment Edge segment Measuring the drawing area in a grid 3 3 2 2 8 8 7 4 5 4 9 5 6 9 7 6 1 1 3/19
Introduction Visibility Representation (a.k.a., Visibility Drawing) Visibility Representation Node segment Edge segment Measuring the drawing area in a grid 3 3 2 2 8 8 7 4 5 4 9 5 6 9 7 6 1 1 3/19
Introduction Visibility Representation (a.k.a., Visibility Drawing) Visibility Representation Node segment Edge segment Measuring the drawing area in a grid 3 3 2 2 8 8 7 4 5 4 9 5 6 9 7 6 1 1 3/19
Introduction Compactness of Visibility Representation Otten and van Wijk (1978) first known algorithm for visibility drawings; no bound for the compactness of the output. worst-case upper bound required height required width n − 1 2 n − 5 (Rosenstiehl & Tarjan, 1986; (Rosenstiehl & Tarjan,1986; Tamassia & Tollis, 1986) Tamassia & Tollis, 1986; ⌊ 15 n 16 ⌋ (Zhang & He, 2003) Nummenmaa, 1992) ⌊ 5 n ⌊ 22 n − 42 6 ⌋ ⌋ (Zhang & He, 2005) (Lin, Lu, and Sun, 2004) 15 ⌊ 4 n − 1 ⌊ 13 n − 24 ⌋ ⌋ (Zhang & He, 2006) (Zhang & He, 2005) 5 3 + 2 ⌈√ n ⌉ 9 2 n 4 n � 3 + 2 ⌈ n / 2 ⌉ (He & Zhang, 2006) (He & Zhang, 2006) lower bound The size of the required area is at least ⌊ 2 n 3 ⌋× ( ⌊ 4 n 3 ⌋− 3 ) (Zhang & He, 2005) 4/19
Introduction Compactness of Visibility Representation Otten and van Wijk (1978) first known algorithm for visibility drawings; no bound for the compactness of the output. worst-case upper bound required height required width n − 1 2 n − 5 (Rosenstiehl & Tarjan, 1986; (Rosenstiehl & Tarjan,1986; Tamassia & Tollis, 1986) Tamassia & Tollis, 1986; ⌊ 15 n 16 ⌋ (Zhang & He, 2003) Nummenmaa, 1992) ⌊ 5 n ⌊ 22 n − 42 6 ⌋ ⌋ (Zhang & He, 2005) (Lin, Lu, and Sun, 2004) 15 ⌊ 4 n − 1 ⌊ 13 n − 24 ⌋ ⌋ (Zhang & He, 2006) (Zhang & He, 2005) 5 3 + 2 ⌈√ n ⌉ 9 2 n 4 n � 3 + 2 ⌈ n / 2 ⌉ (He & Zhang, 2006) (He & Zhang, 2006) lower bound The size of the required area is at least ⌊ 2 n 3 ⌋× ( ⌊ 4 n 3 ⌋− 3 ) (Zhang & He, 2005) 4/19
Introduction Compactness of Visibility Representation Otten and van Wijk (1978) first known algorithm for visibility drawings; no bound for the compactness of the output. worst-case upper bound required height required width n − 1 2 n − 5 (Rosenstiehl & Tarjan, 1986; (Rosenstiehl & Tarjan,1986; Tamassia & Tollis, 1986) Tamassia & Tollis, 1986; ⌊ 15 n 16 ⌋ (Zhang & He, 2003) Nummenmaa, 1992) ⌊ 5 n ⌊ 22 n − 42 6 ⌋ ⌋ (Zhang & He, 2005) (Lin, Lu, and Sun, 2004) 15 ⌊ 4 n − 1 ⌊ 13 n − 24 ⌋ ⌋ (Zhang & He, 2006) (Zhang & He, 2005) 5 3 + 2 ⌈√ n ⌉ 9 2 n 4 n � 3 + 2 ⌈ n / 2 ⌉ (He & Zhang, 2006) (He & Zhang, 2006) lower bound The size of the required area is at least ⌊ 2 n 3 ⌋× ( ⌊ 4 n 3 ⌋− 3 ) (Zhang & He, 2005) Lin, Lu, and Sun (2004) conjectured ... no wider than 4 n 3 + O ( 1 ) . 4/19
Introduction Compactness of Visibility Representation Otten and van Wijk (1978) first known algorithm for visibility drawings; no bound for the compactness of the output. worst-case upper bound required height required width n − 1 2 n − 5 (Rosenstiehl & Tarjan, 1986; (Rosenstiehl & Tarjan,1986; Tamassia & Tollis, 1986) Tamassia & Tollis, 1986; ⌊ 15 n 16 ⌋ (Zhang & He, 2003) Nummenmaa, 1992) ⌊ 5 n ⌊ 22 n − 42 6 ⌋ ⌋ (Zhang & He, 2005) (Lin, Lu, and Sun, 2004) 15 ⌊ 4 n − 1 ⌊ 13 n − 24 ⌋ ⌋ (Zhang & He, 2006) (Zhang & He, 2005) 5 3 + 2 ⌈√ n ⌉ 9 2 n 4 n � 3 + 2 ⌈ n / 2 ⌉ (He & Zhang, 2006) (He & Zhang, 2006) lower bound The size of the required area is at least ⌊ 2 n 3 ⌋× ( ⌊ 4 n 3 ⌋− 3 ) (Zhang & He, 2005) Lin, Lu, and Sun (2004) conjectured ... no wider than 4 n 3 + O ( 1 ) . 4/19
Introduction Our Main Result Theorem Given an n -node plane triangulation G , a visibility drawing of G with its width bounded by ⌊ 4 n 3 ⌋− 2 can be obtained in time O ( n ) . Our bound is the optimal because our bound differs the previously known lower bound 4 n 3 − 3 (Zhang and He, 2005) only by a unit. Answering in the affirmative a conjecture of [Lin, Lu, Sun, 2004] about whether any visibility drawing no wider than 4 n 3 + O ( 1 ) can be obtained in polynomial time. Rather than conventionally using canonical ordering, st -numbering, or Schnyder’s realizer as the initial input, our algorithm applies a new kind of ordering, called constructive ordering , of G to constructing the visibility drawing. 5/19
Introduction Our Main Result Theorem Given an n -node plane triangulation G , a visibility drawing of G with its width bounded by ⌊ 4 n 3 ⌋− 2 can be obtained in time O ( n ) . Our bound is the optimal because our bound differs the previously known lower bound 4 n 3 − 3 (Zhang and He, 2005) only by a unit. Answering in the affirmative a conjecture of [Lin, Lu, Sun, 2004] about whether any visibility drawing no wider than 4 n 3 + O ( 1 ) can be obtained in polynomial time. Rather than conventionally using canonical ordering, st -numbering, or Schnyder’s realizer as the initial input, our algorithm applies a new kind of ordering, called constructive ordering , of G to constructing the visibility drawing. 5/19
Introduction Our Main Result Theorem Given an n -node plane triangulation G , a visibility drawing of G with its width bounded by ⌊ 4 n 3 ⌋− 2 can be obtained in time O ( n ) . Our bound is the optimal because our bound differs the previously known lower bound 4 n 3 − 3 (Zhang and He, 2005) only by a unit. Answering in the affirmative a conjecture of [Lin, Lu, Sun, 2004] about whether any visibility drawing no wider than 4 n 3 + O ( 1 ) can be obtained in polynomial time. Rather than conventionally using canonical ordering, st -numbering, or Schnyder’s realizer as the initial input, our algorithm applies a new kind of ordering, called constructive ordering , of G to constructing the visibility drawing. 5/19
Introduction Our Main Result Theorem Given an n -node plane triangulation G , a visibility drawing of G with its width bounded by ⌊ 4 n 3 ⌋− 2 can be obtained in time O ( n ) . Our bound is the optimal because our bound differs the previously known lower bound 4 n 3 − 3 (Zhang and He, 2005) only by a unit. Answering in the affirmative a conjecture of [Lin, Lu, Sun, 2004] about whether any visibility drawing no wider than 4 n 3 + O ( 1 ) can be obtained in polynomial time. Rather than conventionally using canonical ordering, st -numbering, or Schnyder’s realizer as the initial input, our algorithm applies a new kind of ordering, called constructive ordering , of G to constructing the visibility drawing. 5/19
Preliminaries Coalescing and Splitting Operations v s v r v s v r v r v r v r v r v q v q v s v s 1 2 3 F k ,2 v k +1 v k +1 F k ,1 v t v q v t v q F k ,2 F k ,1 v k +1 F k ,3 F k ,1 v q v q v p = u 2 v p = u v p = u v p = u v p = u 1 v p = u 3 in G k +1 in G k in G k +1 in G k in G k +1 in G k (a) deg ( v k +1 ) = 3 in G k +1 (b) deg ( v k +1 ) = 4 in G k +1 (c) deg ( v k +1 ) = 5 in G k +1 6/19
Preliminaries Coalescing and Splitting Operations v s v r v s v r 3 F k ,2 v k +1 v t v q v t v q F k ,3 F k ,1 v p = u v p = u 3 in G k +1 in G k (c) deg ( v k +1 ) = 5 in G k +1 When deg ( v k + 1 ) = 5 , α 3 ( v k + 1 , u ) = coalescing two nodes v k + 1 and u β 3 ( v k + 1 , F k , 1 , F k , 2 , F k , 3 ) = splitting node v k + 1 at faces F k , 1 , F k , 2 , F k , 3 6/19
Preliminaries Coalescing and Splitting Operations v s v r v s v r v r v r v r v r v q v q v s v s 1 2 3 F k ,2 v k +1 v k +1 F k ,1 v t v q v t v q F k ,2 F k ,1 v k +1 F k ,3 F k ,1 v q v q v p = u 2 v p = u v p = u v p = u v p = u 1 v p = u 3 in G k +1 in G k in G k +1 in G k in G k +1 in G k (a) deg ( v k +1 ) = 3 in G k +1 (b) deg ( v k +1 ) = 4 in G k +1 (c) deg ( v k +1 ) = 5 in G k +1 When deg ( v k + 1 ) = 5 , α 3 ( v k + 1 , u ) = coalescing two nodes v k + 1 and u β 3 ( v k + 1 , F k , 1 , F k , 2 , F k , 3 ) = splitting node v k + 1 at faces F k , 1 , F k , 2 , F k , 3 6/19
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