The Gap between Crossing Numbers and Outerplanar Crossing Numbers - PowerPoint PPT Presentation

The Gap between Crossing Numbers and Outerplanar Crossing Numbers EPSRC – GR/R37395/01: Parallel and Sequential Algorithms for Low Crossing Graph Drawing (2001-2004) EPSRC - GR/S76694/01: Outerplanar Crossing Numbers (2003-2006) Farhad Sharokhi (Denton) Ondrej Sýkora (Loughborough) László Székely (Columbia) Imrich Vr ť o (Bratislava)

Planar Crossing Number G cr(G)=4 30/03/2005 BCTCS Nottingham 2

Measure of nonplanarity −     n n 1 • [Kleitman,1971] = cr ( K ) 6 ,     6 , n 2 2     • Aesthetic and readable drawing of graph like structures • VLSI circuits [Leighton,1981] n+cr(G) ≤ A(G) ≤ O((n+cr(G))log 2 (n+cr(G))) • NP-hard [Garey, Johnson,1983] • log 3 n - approximation [Guha, 2000] 30/03/2005 BCTCS Nottingham 3

Outerplanar crossing number • Given n-vertex graph G=(V,E) • Outerplanar drawing - vertices on the corners of a convex n-gon in the plane - each edge is drawn using one straight line segment. 30/03/2005 BCTCS Nottingham 4

Outerplanar Crossing Number G ν 1 (G)=4 30/03/2005 BCTCS Nottingham 5

Book crossing numbers spine ν 1 (G)=minimum number of crossings in 1-page drawing of G 2 pages ν 2 (G)=minimum number of crossings in 2-page drawing of G ν k (G)=minimum number of crossings in k-page drawing of G 30/03/2005 BCTCS Nottingham 6

Our results General lower bound based on isoperimetric properties of G It implies that outerplanar drawings for many graphs, including the planar 2-dim. grid on n vertices have at least Ω (n log n) crossings. 30/03/2005 BCTCS Nottingham 7

Our results (continued) If there is a drawing of G with c crossings in plane ⇒ We construct an outerplanar drawing with at most 2 )log n) O((c+ ∑ v ∈ V d v crossings, d v is the degree of v 30/03/2005 BCTCS Nottingham 8

Our results (continued) • For planar graphs the drawing can be constructed in O(n log n) time. 30/03/2005 BCTCS Nottingham 9

Lower bound Let G=(V,E) satisfies f(x)-isoperimetric inequality, if for any k ≤ n/2 and any k- vertex subset U ⊂ V, there are at least f(k) edges between U and V-U f(x) is defined on non-negative integers (or sometimes on all non-negative real numbers) 30/03/2005 BCTCS Nottingham 10

Lower bound (idea) v u l(u,v) is length of (u,v) cros(u,v) ≥ f(l(u,v)+1)-d u -d v ) cros(D) ≥ ½[ ∑ (u,v) ∈ E f(length(u,v))- ∑ v ∈ V d v 2 ] 30/03/2005 BCTCS Nottingham 11

Lower bound For G=(V,E) with f(x)-isoperimetric inequality, ∆ f non-negative and decreasing ν 1 (G) ≥ -n/8 ∑ f(j) ∆ 2 f(j)-1/2 ∑ v ∈ V d v 2 ∑ is from 0 to  n/2  -2 30/03/2005 BCTCS Nottingham 12

Lower bound Define the difference function of f ∆ f(i)=f(i+1)-f(i) for any i=0,1,…,  n/2  -1 and second difference function of f ∆ 2 f(i)=( ∆ ( ∆ f))(i) for any i=0,1,…,  n/2  -2 30/03/2005 BCTCS Nottingham 13

Application of the lower bound For N × N=n grid G we have f(x)= √ 2x isoperimetric inequality Using ν 1 (G) ≥ -n/8 ∑ f(j) ∆ 2 f(j)-1/2 ∑ v ∈ V d v 2 we get ν 1 (G) ∈Ω (n log n) Similar result can be showed for triangular, hexagonal and square lattice. 30/03/2005 BCTCS Nottingham 14

Upper bound Take planar drawing of G with c crossings Change it to a planar graph H with c vertices of degree 4. Assign weight to the vertex v: d v 2 / ( ∑ v ∈ V d v 2 ) Use Gazit-Miller[1990]: Any planar (vertex) weighted graph has a (1/3,2/3) edge separator of size at most 1.6 √ ( ∑ v ∈ V d v 2 ) 30/03/2005 BCTCS Nottingham 15

Upper bound Recursively lay out the graph on a line (one-page drawing). Edge separator separates H into two graphs H 1 and H 2 . Put them on a line next each to other. Continue recursively. Draw edges and return the c “crossing vertices” to crossings. 30/03/2005 BCTCS Nottingham 16

Upper bound S(H) – maximum number of edges that go above any vertex in the obtained 1-page drawing. Similarly S(H i ). S(H) ≤ sep(H)+max{S(H 1 ),S(H 2 )} This implies 2 ) and S(H) ∈ O( √∑ v ∈ V d v c(H) ≤ c(H 1 )+c(H 2 )+2sep(H)S(H)= 2 ) ≤ c(H 1 )+c(H 2 )+ const 1 ( ∑ v ∈ V d v 2 )log n) const 2 ((c+ ∑ v ∈ V d v 30/03/2005 BCTCS Nottingham 17

Open problems and questions • Improving bounds • Proving exact results • New lower bound arguments (2-page) • Algorithms, Heuristics 30/03/2005 BCTCS Nottingham 18

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