Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Victoria city and Sendai city 7300km Victoria city Sendai city 1 2 Tohoku University Tohoku University was established in 1907. Spring Summer Autumu Winter GSIS, Tohoku University 3 Graduate School of Information Sciences (GSIS), Tohoku University, was established in 1993. 150 Faculties 450 students Math. Computer Science Robotics Transportation Economics Human Social Sciences Interdisciplinary School Book 4 5 Small Grid Drawings of Planar Graphs with Balanced Bipartition Xiao Zhou Takashi Hikino Takao Nishizeki Graduate School of Information Sciences, Tohoku University, Japan 6 Grid drawing In a grid drawing of a planar graph, ・ every vertex is located at a grid point, ・ every edge is drawn as a straight-line segment without any edge-intersection. Planar graph Grid drawing 2 3 2 3 4 4 1 1 5 7 5 6 7 6 7 Embedding We deal with grid drawings of a planar graph in variable embedding setting. 1 3 6 7 Planar graph Grid drawing 2 2 3 4 4 1 1 5 7 4 5 This embedding is different from a given embedding 3 2 5 6 7 6 Width and Height of grid drawing H H W W W=9, H=11 W=4, H=3 Area W×H=99 Area W×H=12 8 9 Small grid drawing Large area Small area H H W W We wish to find a small grid drawing in variable embedding setting. 10 Known results Grid drawing of Plane graph Width and Height Running time [Schnyder, 1990] [Chrobak, Kant, 1997] W=n-2, H=n-2 O(n) n : number of vertices 11 Our results Grid drawing of Plane graph Width and Height Running time [Schnyder, 1990] [Chrobak, Kant, 1997] W=n-2, H=n-2 O(n) If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. s G s G2 G1 G2 s t Planar graph G G1 G2 t Drawing Bipartition t Subgraph G1,G2 u2 s u1 G1 t Drawing of G 12 Outline of algorithm s G s G2 G1 G2 s Bipartition t t G1 (1) Planar graph G t s (2) Subgraph G1,G2 G1 u1 s G2 u2 t t (3) Maximal planar graph G1,G2 u2 s G2 u1 G1 s G2 Combining t (5) Drawing of G u2 s G1 u1 t t (4) Drawing of G1,G2 13 Our results s G G2 G1 Theorem 1 t (1) Planar graph G If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. W≤max{n1, n2}-1 u2 s G2 u1 G1 H≤max{n1, n2}-2 t (5) Drawing of G G1,G2 Width, Height n1,n2≤n/2 W,H≤n/2 n1,n2≤2n/3 W,H≤2n/3 n1,n2≤αn W,H≤αn If α<1, Balanced bipartition 14 Our results Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. G1,G2 Width, Height n1,n2≤n/2 W,H≤n/2 n1,n2≤2n/3 W,H≤2n/3 n1,n2≤αn W,H≤αn If α<1, Balanced bipartition 15 Our results Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. Planar graph Series-Parallel graph Lemma 1 Every Series-Parallel graph has a balanced bipartition (n1,n2≤2n/3). α=2/3 16 Our results Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. Planar graph Series-Parallel graph Lemma 1 Every Series-Parallel graph has a balanced bipartition (n1,n2≤2n/3). α=2/3 Theorem 2 Series-Parallel W 2 n , 2 n 1 H graph 3 3 17 Outline of algorithm s G s G2 G1 G2 s Bipartition t t G1 (1) Planar graph G t s (2) Subgraph G1,G2 G1 u1 s G2 u2 t t (3) Maximal planar graph G1,G2 u2 s G2 u1 G1 s G2 Combining t (5) Drawing of G u2 s G1 u1 t t (4) Drawing of G1,G2 18 Bipartition We call a pair of distinct vertices {s,t} in a graph G=(V,E) a separation pair of G if G has two subgraphs G1=(V1,E1) and G2=(V2,E2) such that ・ V=V1∪V2,V1∩V2={s,t}, ・ E=E1∪E2,E1∩E2=∅. Such a pair of subgraphs {G1,G2} is called a bipartition of G. s s G2 s n1=9 Bipartition t t G1 n=12 (1) Graph G (2) Subgraph G1,G2 t n2=5 19 Bipartition We call a pair of distinct vertices {s,t} in a graph G=(V,E) a separation pair of G if G has two subgraphs G1=(V1,E1) and G2=(V2,E2) such that ・ V=V1∪V2,V1∩V2={s,t}, ・ E=E1∪E2,E1∩E2=∅. Such a pair of subgraphs {G1,G2} is called a bipartition of G. t t t G2 G1 Bipartition s n=12 s s n1=3 (1) Graph G n2=11 (2) Subgraph G1,G2 20 Outline of algorithm s G s G2 G1 G2 s Bipartition t t G1 (1) Planar graph G t s (2) Subgraph G1,G2 G1 u1 s G2 u2 t t (3) Maximal planar graph G1,G2 u2 s G2 u1 G1 s G2 Combining t (5) Drawing of G u2 s G1 u1 t t (4) Drawing of G1,G2 21 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}-1, H≤max{n1,n2}-2 and such a drawing can be found in linear time. W≤max{n1, n2}-1 s G u2 s G2 G1 t Planar graph G Drawing in linear time G2 u1 G1 t Drawing of G H≤max{n1, n2}-2 22 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}-1, H≤max{n1,n2}-2 and such a drawing can be found in linear time. Assume w.l.o.g. that n1≥n2. s s s G1 t G n=12 t t G1 n1=9 n2=5 G2 23 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}-1, H≤max{n1,n2}-2 and such a drawing can be found in linear time. Assume w.l.o.g. that n1≥n2. Add dummy edges to G1 so that the resulting graph is maximal planar and has an edge (s,t). s s G1 t G n=12 t G1 n1=9 24 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}-1, H≤max{n1,n2}-2 and such a drawing can be found in linear time. Embed G1 so that the edge (s,t) lies on the outer face of G1. s s G1 t G n=12 u1 G1 t n1=9 25 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}-1, H≤max{n1,n2}-2 and such a drawing can be found in linear time. Obtain a grid drawing of G1[CK97]. s s G1 t G n=12 u1 G1 t n1=9 26 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary s bipartition of G. Then G has a grid drawing such that W≤max{n 1,n2}-1, H≤max{n1,n2}-2 and such a drawing can be found in linear time. Obtain a grid drawing of G1[CK97]. u1 G1 s H1=n1-2 Edge (u1,t) is horizontal. G1 n1=9 u1 t W1=n1-2 t n1=9 27 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}-1, H≤max{n1,n2}-2 and such a drawing can be found in linear time. u2 s s G2 s G2 t t u1 G n=12 t G1 n1=9 n2=5 G2 28 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}-1, H≤max{n1,n2}-2 and such a drawing can be found in linear time. Add n1-n2 dummy vertices to G2 so that the resulting graph has exactly n1 vertices. u2 s s G2 s G2 t t u1 G n=12 t G1 n1=9 n2=5 =n1=9 G2 29 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}-1, H≤max{n1,n2}-2 and such a drawing can be found in linear time. Add dummy edges to G2 so that the resulting graph is maximal planar and has an edge (s,t). s G2 t G n=12 s t n2=n1=9 G2 30 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}-1, H≤max{n1,n2}-2 and such a drawing can be found in linear time. Embed G2 so that the edge (s,t) lies on the outer face of G2. s G2 t G n=12 s t n2=n1=9 G2 31 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}-1, H≤max{n1,n2}-2 and such a drawing can be found in linear time. Embed G2 so that the edge (s,t) lies on the outer face of G2. s G2 t G n=12 u2 s s t t n2=n1=9 G2 n2=n1=9 G2 32 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}-1, H≤max{n1,n2}-2 and such a drawing can be found in linear time. Obtain a grid drawing of G2[CK97]. Edge (u2,s) is horizontal. s G2 t t u2 s t n2=n1=9 G2 G n=12 u2 s Theorem 1 s u1 t s u1 u2 s G1 G2 n1=9 n2=9 t u2 s t 33 t Theorem 1 G s Combine the two drawings and Erase all the dummy vertices and edges. u2 u1 34 t n=12 s u2 s G2 n2=9 G1 n1=9 u1 t t Theorem 1 G s Combine the two drawings and Erase all the dummy vertices and edges. u2 u1 35 t n=12 u2 s G2 n2=9 G1 n1=9 u1 t 36 Theorem 1 Theorem 1. Let G be a planar graph, and let {G1,G2} be an arbitrary bipartition of G. Then G has a grid drawing such that W≤max{n1,n2}-1, H≤max{n1,n2}-2 and such a drawing can be found in linear time. Such a drawing can be found in linear time, because drawings of G1,G2 can be found in linear time by the algorithm in CK97. W2=n1-2 n1≥n2 u2 s G2 s H1=n1-2 u1 t G1 t W1=n1-2 u2 s G2 H2=n1-2 u1 G1 t W=W1+1 =n1-1 =max{n1, n2}-1 H=H1 =n1-2 =max{n1, n2}-2 Q.E.D. Our results 37 Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. Lemma 1 Every Series-Parallel graph has a balanced bipartition (n1,n2≤2n/3). Planar graph Series-Parallel graph Theorem 2 Series-Parallel W 2 n , 2 n 1 H graph 3 3 Our results 38 Theorem 1 If a planar graph G has a balanced bipartition, then G has a grid drawing with small area. This graph has NO balanced bipartition. Lemma 1 Every Series-Parallel graph has a balanced bipartition (n1,n2≤2n/3). Planar graph Series-Parallel graph Theorem 2 Series-Parallel W 2 n , 2 n 1 H graph 3 3 39 Series-Parallel graph A Series-Parallel graph is recursively defined as follows: (1) (2) A single edge is a SP graph. G2 G1 terminal : SP graph Series connection G1 SP graph G2 G1 G2 Parallel connection SP graph 40 Series-Parallel graphs These graphs are Series-Parallel. s t Bipartition of Series-Parallel graph 41 Lemma 1. Every Series-Parallel graph G of n vertices has a bipartition {G1,G2} such that n1,n2 2n / 3 1 . Furthermore such a bipartition can be found in linear time. s G G2 G1 t SP graph G Bipartition in liner time s s G2 G1 t t Subgraph G1,G2 n1,n2 2n / 3 1 Suppose for a contradiction that a SP graph has no desired bipartition. Bipartition of Series-Parallel graph Lemma 1. Every Series-Parallel graph G of n vertices has a bipartition {G1,G2} such that n1,n2 2n / 3 1 . Furthermore such a bipartition can be found in linear time. Let {s,t} be the most balanced separation pair of G : max{n1,n2} is minimum among all bipartitions of G. n1>2n/3 G1 G11 s G1=G11・G12 G11 G12 u G12 t G2 n2<n/3 2-connected SP graph G Assume w.l.o.g. that n1≥n2. 42 Bipartition of Series-Parallel graph 43 Lemma 1. Every Series-Parallel graph G of n vertices has a bipartition {G1,G2} such that n1,n2 2n / 3 1 . Furthermore such a bipartition can be found in linear time. Let {s,t} be the most balanced separation pair of G : max{n1,n2} is minimum among all bipartitions of G. n1>2n/3 G1=G11・G12 Assume w.l.o.g. that n1≥n2 and n11≥n12. G1 n11>n/3 G11 n1> n11 u G12 n1> n11 G s t 11 n1> max{n11,n11} G2 Contradiction. n11<2n/3 n1=max{n1,n2} n2<n/3 max{n1,n2}> max{n11,n11} 2-connected SP graph G Grid drawing of Series-Parallel graph s G Lemma 1 in linear time G2 G1 t SP graph G s s G2 G1 t t Subgraph G1,G2 n1,n2 2n / 3 1 44 Grid drawing of Series-Parallel graph s G G1 SP graph G u2 s G2 t t t Subgraph G1,G2 n1,n2 2n / 3 1 s G1 G2 G1 t s s s Lemma 1 in linear time G2 45 t Subgraph G1,G2 Theorem 1 in linear time G2 u1 H≤max{n1, n2}-2 G1 t W≤max{n1, n2}-1 Grid drawing of Series-Parallel graph s G G1 SP graph G G1 t t Subgraph G1,G2 n1,n2 2n / 3 1 u2 s s G2 t G2 G1 t s s s Lemma 1 in linear time G2 46 t SP Subgraph G1,G2 n1,n2 2n / 3 1 Theorem 1 in linear time G2 u1 G1 H≤max{n1, n2}-2 2n / 3 1 t W≤max{n1, n2}-1 2n / 3 Grid drawing of Series-Parallel graph 47 Theorem 2. Every Series-Parallel graph of n vertices has a grid drawing such that W 2n / 3 , H 2n / 3 1 . Furthermore such a drawing can be found in linear time. s G1 G2 t u2 s s t SP Subgraph G1,G2 n1,n2 2n / 3 1 Theorem 1 in linear time G2 u1 G1 H≤max{n1, n2}-2 2n / 3 1 t W≤max{n1, n2}-1 2n / 3 48 Grid drawing of Series-Parallel graph u2 s s H=7 Theorem 2 in linear time t n=12 SP graph G u1 t W=8 s s G1 n1=9 t G2 t n2=5 49 Conclusions Gird drawing Width and Height Running time W≤max{n1, n2}-1 Planar graph with balanced bipartition u2 s G2 u1 O(n) H≤max{n1, n2}-2 G1 t 2 u2 s SP graph Partial 2-tree G2 u1 G1 t W n 3 2 H n 1 3 O(n) 50