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
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 Vrťo (Bratislava)
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Planar Crossing Number
G cr(G)=4
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Measure of nonplanarity
like structures
[Leighton,1981] n+cr(G) ≤ A(G) ≤ O((n+cr(G))log2(n+cr(G)))
, 2 1 2 6 ) (
, 6
= K cr
n
− n n
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Outerplanar crossing number
convex n-gon in the plane
straight line segment.
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Outerplanar Crossing Number
G ν1(G)=4
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Book crossing numbers
spine ν1(G)=minimum number of crossings in 1-page drawing of G 2 pages ν2(G)=minimum number of crossings in 2-page drawing of G νk(G)=minimum number of crossings in k-page drawing of G
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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.
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Our results (continued)
If there is a drawing of G with c crossings in plane ⇒ We construct an outerplanar drawing with at most
O((c+∑v∈V dv
2 )log n)
crossings, dv is the degree of v
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Our results (continued)
be constructed in O(n log n) time.
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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)
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Lower bound (idea)
v u
l(u,v) is length of (u,v)
cros(u,v) ≥ f(l(u,v)+1)-du-dv) cros(D)≥½[∑(u,v)∈Ef(length(u,v))-∑v∈Vdv
2]
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Lower bound
For G=(V,E) with f(x)-isoperimetric inequality,∆f non-negative and decreasing ν1(G)≥-n/8 ∑ f(j)∆2f(j)-1/2 ∑v∈V dv
2
∑ is from 0 to n/2-2
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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 ∆2f(i)=(∆(∆f))(i) for any i=0,1,…,n/2-2
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Application of the lower bound
For N×N=n grid G we have f(x)=√2x isoperimetric inequality Using ν1(G)≥-n/8 ∑ f(j)∆2f(j)-1/2 ∑v∈V dv
2
we get ν1(G)∈Ω (n log n) Similar result can be showed for triangular, hexagonal and square lattice.
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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: dv
2/ (∑v∈V dv 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 dv
2)
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Upper bound
Recursively lay out the graph on a line (one-page drawing). Edge separator separates H into two graphs H1 and H2. Put them on a line next each to other. Continue recursively. Draw edges and return the c “crossing vertices” to crossings.
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Upper bound
S(H) – maximum number of edges that go above any vertex in the obtained 1-page
S(H) ≤ sep(H)+max{S(H1),S(H2)} This implies S(H)∈O(√∑v∈V dv
2 ) and
c(H)≤c(H1)+c(H2)+2sep(H)S(H)= c(H1)+c(H2)+ const1(∑v∈V dv
2 )≤
const2((c+∑v∈V dv
2 )log n)
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Open problems and questions