GD 2014 September 25, 2014 Department of Computer Science University of Manitoba, Canada Stephane Durocher Debajyoti Mondal

a b c d f GD 2014September 25, 2014 A planar graph G k-bend drawings of G e k = 0 k = 1 k = 2

a b c d f GD 2014September 25, 2014 A planar graph G e k = 0 θ = 30 o k-bend drawings of G, with angular resolution θ k = 1 θ = 90 o k = 2 θ = 120 o

a b c d f GD 2014September 25, 2014 A planar graph G e k = 0 θ = 30 o k-bend drawings of G, with angular resolution θ and area A k = 1 θ = 90 o k = 2 θ = 120 o

a b c d f GD 2014 A planar graph G e k = 0 θ = 30 o k = 1 θ = 90 o k = 2 θ = 120 o Fixed k (θ↑, A↑) Fixed A (θ↑, k↑) θ↑ (k ?, A ?) September 25, 2014

a b c d f GD 2014 A planar graph G e k = 0 θ = 30 o k = 1 θ = 90 o k = 2 θ = 120 o Fixed k (θ↑, A↑) September 25, 2014 Fixed A (θ↑, k↑) θ↑ (k ?, A ?)

7 GD 2014 k-Bend Drawings AreaAngular R.Total BendsReference k = 00.89n 2 Ω(1/n 2 )0Brandenburg, ENDM 2008 k = 04.50n 2 Ω(1/n)0Kurowski, SOFSEM 2005 k = 0Ω(c ρn )Ω(1/ρ)0 Garg and Tamassia, IEEE SMC 1988 k = 10.45n 2 Ω(1/n 2 )2n/3Zhang, Algorithmica 2010 k = 112.5n 2 Ω(0.5/d v )3n3nDuncan and Kobourov, JGAA 2003 k = 1450n 2 Ω(1/d v )3n3nCheng, Duncan, Goodrich, Kobourov, DCG 2001 k = 2200n 2 Ω(1/d v )6n6nGoodrich and Wagner, Algorithms 2000 k = 2(6α+8/3) 2 n 2 ____ α _____ d v (α 2 +1/4) 5.5nThis Presentation k = 2(6β + 2/3) 2 n 2 ____ β _____ d v (β 2 +1) 5.5nThis Presentation September 25, 2014 (A↑, θ↑)

k-Bend Drawings AreaAngular R.Total BendsReference k = 112.5n 2 Ω(0.5/d v )3n3nDuncan and Kobourov, JGAA 2003 k = 1450n 2 Ω(1/d v )3n3nCheng, Duncan, Goodrich, Kobourov, DCG 2001 k = 2200n 2 Ω(1/d v )6n6nGoodrich and Wagner, Algorithms 2000 k = 2(6α+8/3) 2 n 2 ____ α _____ d v (α 2 +1/4) 5.5nThis Presentation k = 2(6β + 2/3) 2 n 2 ____ β _____ d v (β 2 +1) 5.5nThis Presentation

v1v1 v2v2 v3v3 v1v1 v2v2 v3v3 v4v4 v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 G3G3 G6G6 G7G7 G8G8 G4G4 G5G5 A Canonical Ordering of G [De Fraysseix, Pach, and Pollack 1988] Idea: Decomposition of a Triangulation into Trees + L-Contact Representation

v1v1 v2v2 v3v3 v1v1 v2v2 v3v3 v4v4 v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 G3G3 G6G6 G7G7 G8G8 G4G4 G5G5 A Canonical Ordering of G [De Fraysseix, Pach, and Pollack 1988] Idea: Decomposition of a Triangulation into Trees + L-Contact Representation v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8

v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v n = v 8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 A Canonical Ordering of G [De Fraysseix et al. 1988] v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 A Schnyder realizer of G [Schnyder 1990] Idea: Decomposition of a Triangulation into Trees + L-Contact Representation v7v7 v1v1 vivi T right T left T mid v2v2 v n = v 8

v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v6v6 v5v5 v1v1 v2v2 v3v3 v4v4 v7v7 v8v8 Idea: Decomposition of a Triangulation into Trees + L-Contact Representation v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 12 GD 2014September 25, 2014 Vertices are represented with L-shapes (flipped horizontally)

v6v6 v5v5 v1v1 v2v2 v3v3 v4v4 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 Game Plan!

v6v6 v5v5 v1v1 v2v2 v3v3 v4v4 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 Business Meeting & Conference Dinner

v6v6 v5v5 v1v1 v2v2 v1v1 v2v2 v3v3 v1v1 v2v2 v3v3 v4v4 v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v6v6 v5v5 v1v1 v2v2 v3v3 v4v4 v7v7 v8v8 v6v6 v5v5 v1v1 v2v2 v3v3 v4v4 v7v7 15 GD 2014September 25, 2014 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8

v6v6 v5v5 v1v1 v2v2 v1v1 v2v2 v3v3 v1v1 v2v2 v3v3 v4v4 v1v1 v2v2 v3v3 v4v4 v5v5 v1v1 v2v2 v3v3 v4v4 v6v6 v5v5 v1v1 v2v2 v3v3 v4v4 v7v7 v8v8 v6v6 v5v5 v1v1 v2v2 v3v3 v4v4 v7v7 A drawing on (W+2)×(H+2) grid, where W+H ≤ leaf(T left ) + leaf(T right ) ≤ 4n/3 16 GD 2014September 25, 2014 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8

17 GD 2014September 25, 2014 d p = 6 d q = 9 d r = 10 d s = 4 pq r s v6v6 v5v5 v1v1 v2v2 v3v3 v4v4 v7v7 v8v8 For each row l, - let r be the vertex on l with maximum degree. - insert floor (d r /2) rows to each side of l. floor (d r /2) l l

v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 18 GD 2014September 25, 2014

v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 Drawing T left 19 GD 2014September 25, 2014

v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 Drawing T right 20 GD 2014September 25, 2014

v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 Drawing T mid 21 GD 2014September 25, 2014

v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 Final Output 22 GD 2014September 25, 2014

θ v floor (d v /2) v α × d v … Because we have other options v β × d v β α Area (6α+8/3) 2 n 2 Resolution ____ α _____ d v (α 2 +1/4) Area (6β + 2/3) 2 n 2 Resolution ____ β _____ d v (β 2 +1)

Better Trade-offs: Can we achieve better trade-offs between area and angular resolution for 2-bend polyline drawings? Find such trade-offs for 1-bend and 0-bend drawings. Generalization: Examine trade-offs between area and angular resolution for more general graphs such as 1-planar graphs and thickness 2-graphs. Different Aesthetics: Examine smooth trade-offs among different aesthetic criteria for other styles of drawing graphs? 24 GD 2014September 25, 2014

Better Trade-offs? Generalization? Different Aesthetics and Styles? v6v6 v5v5 v1v1 v2v2 v3v3 v4v4 v7v7 v8v8 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 v6v6 v5v5 v1v1 v3v3 v4v4 v7v7 v8v8 v2v2 Thank You..