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D.I.E.I. - Università degli Studi di Perugia

h-quasi planar drawings of boundedtreewidth graphs in linear area

Emilio Di Giacomo, Walter Didimo,Giuseppe Liotta, Fabrizio Montecchiani

University of Perugia

13th Italian Conference on Theoretical Computer Science19-21 September 2012, Varese, Italy

E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia

ICTCS ’12 - Varese, Italy 2

Graph Drawing and Area Requirement

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Graph G Straight-line grid drawing of G



Graph Drawing and Area Requirement

Area requirement of straight-line drawings is a widely studied topic in Graph Drawing

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Graph G Straight-line grid drawing of G

h

w



Area Requirement for planar drawings

• Area requirement problem mainly studied for planar straight-line grid drawings:– planar graphs have planar straight-line grid drawings in

O(n2) area (worst case optimal) [de Fraysseix et al.; Schnyder; 1990]

– sub-quadratic upper bounds:• trees – O(n log n) [Crescenzi et al., 1992]• outerplanar graphs – O(n1.48) [Di Battista, Frati, 2009]

– super-linear lower bound:• series-parallel graphs – Ω(n2√(log n)) [Frati, 2010]

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• Planarity imposes severe limitations on the optimization of the area

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• Planarity imposes severe limitations on the optimization of the area– Non-planar straight-line drawings in O(n) area exist

for k-colorable graphs [Wood, 2005] – no guarantee on the type and on the number of crossings

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A drawing by Wood’s technique



Beyond planarity: crossing complexity

• Non-planar drawings should be considered: – How can we “control” the crossing complexity of a

drawing?

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Crossing complexity measures

• Large Angle Crossing drawings (LAC) or Right Angle Crossing drawings (RAC), [Didimo et al., 2011]

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RAC drawing




• h-Planar drawings: at most h crossings per edge

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1-planar drawing




• h-Quasi Planar drawings: at most h-1 mutually crossing edges

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3-quasi planar drawing



The problem

• We investigate trade-offs between area requirement and crossing complexity

• We focus on h-quasi planarity as a measure of crossing complexity

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Our contribution 1/2(h-quasi planar drawings)

• General technique: Every n-vertex graph with treewidth ≤ k, has an h-quasi planar drawing in O(n) area with h depending only on k

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Our contribution 1/2(h-quasi planar drawings)

• General technique: Every n-vertex graph with treewidth ≤ k, has an h-quasi planar drawing in O(n) area with h depending only on k

• Ad-hoc techniques: Smaller values of h for specific subfamilies of planar partial k-trees (outerplanar, flat series-parallel, proper simply nested)

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Our contribution 2/2(h-quasi planarity vs h-planarity)

• Comparison: There exist n-vertex series-parallel graphs (partial 2-trees) such that every h-planar drawing requires super-linear area for any constant h– 11-quasi planar drawings in linear area always exist

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Our contribution 2/2(h-quasi planarity vs h-planarity)

• Comparison: There exist n-vertex series-parallel graphs (partial 2-trees) such that every h-planar drawing requires super-linear area for any constant h– 11-quasi planar drawings in linear area always exist

• Additional result: There exist n-vertex planar graphs such that every h-planar drawing requires quadratic area for any constant h

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What’s coming next

• Basic definitions

• Results on h-quasi planarity

• Comparison with h-planarity

• Conclusions and open problems

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BASIC DEFINITIONS

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Bounded treewidth graphs

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• What’s a k-tree?




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• What’s a k-tree? • a clique of size k is a k-tree

3-tree construction




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• What’s a k-tree? • a clique of size k is a k-tree• the graph obtained from a k-tree by adding a new vertex

adjacent to each vertex of a clique of size k is a k-tree

3-tree construction




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3-tree construction




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3-tree construction




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3-tree construction




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adjacent to each vertex of a clique of size k is a k-tree• A subgraph of a k-tree is a partial k-tree• A graph has treewidth ≤ k it is a partial k-tree

3-tree construction



Track assignment• t-track assignment of a graph G [Dujmović et al., 2004] =

t vertex coloring + total ordering <i in each color class Vi

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3-track assignment





– (Vi ,<i ) = track τi , 1 ≤ i ≤ t

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3-track assignment

track





– (Vi ,<i ) = track τi , 1 ≤ i ≤ t

– X-crossing = (u, v), (w, z): u,w V∈ i, v, z V∈ j , u <i w and z <j v, for i ≠ j

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X-crossing





– (Vi ,<i ) = track τi , 1 ≤ i ≤ t

– X-crossing = (u, v), (w, z): u,w V∈ i, v, z V∈ j , u <i w and z <j v, for i ≠ j

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NOT an X-crossing



Track layout

• (c, t)-track layout of G = t-track assignment + edge c-coloring: no two edges of the same color form an X-crossing

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(2,3)-track layout



Track layout


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(2,3)-track layout



Track layout


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(2,3)-track layout



THE GENERAL TECHNIQUE: COMPUTING COMPACT H-QUASI PLANAR DRAWINGS OF K-TREES

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Ingredients of the result

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• assume to have a (c,t)-track layout: we show how to compute a [c(t-1)+1]-quasi planar drawing in O(t3 n) area




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• we prove that every partial k-tree has a (2,t)-track layout where t depends on k but it does not depend on n




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• we prove that every partial k-tree has a (2,t)-track layout where t depends on k but it does not depend on n

every partial k-tree has a O(1)-quasi planar drawing in area O(n)



An example

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INPUT: A partial k-tree G



An example

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G = 2-treeINPUT: A partial k-tree G



An example

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1. Compute a (2,tk)-track layout of G



An example

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1) = (2,t)-track layout of G t = 4



An example

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2. Construct an hk-quasi planar drawing from

OUTPUT: The drawing



An example

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2) = h-quasi planar drawing of G h ≤ c(t-1)+1 = 2(4-1)+1 = 7



An example

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2. Construct an hk-quasi planar drawing from

OUTPUT: The drawing



(c,t)-track layout h-quasi planar drawing

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• Lemma 1: every n-vertex graph G admitting a (c,t)-track layout, also admits an h-quasi planar drawing in O(t3n) area, where h = c(t − 1) + 1



An example

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place the verticesalong segments




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any edge connecting a vertex on a segment i to a vertex on a segment j (i < j) do not overlap with any vertex on a segment k s.t. i < k <j




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any edge connecting a vertex on a segment i to a vertex on a segment j (i < j) do not overlap with any vertex on a segment k s.t. i < k <j




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O(t2n)

t

A = O(t3n)



(c,t)-track layout h-quasi planar drawing: upper bound on h

• We prove that at most c(t − 1) edges mutually cross

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• We prove that at most c(t − 1) edges mutually cross– every edge (u,v) with u ϵ si and v ϵ sj is completely

contained in a parallelogram Πi,j

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si

parallelogram Πi,j

sj




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at most c mutually crossing edges in each parallelogram




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+at most t − 1 parallelograms mutually overlap (to prove)




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+at most t − 1 parallelograms mutually overlap (to prove)

at most c(t − 1) mutually crossing edges in our drawing

=




• Simplified (but consistent) model– segments = points

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• Simplified (but consistent) model– segments = points– parallelograms = curves

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• An overlap occurs iff1 - two curves form a crossing

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• An overlap occurs iff2 - two curves share an endpoint and the other two

endpoints are either before or after the one in common

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• Simplified (but consistent) model– an overlap occurs iff

1 - two curves form a crossing 2 - two curves share an endpoint and the other two

endpoints are either before or after the one in common

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4 mutually overlapping parallelograms




• To prove: at most t − 1 parallelograms mutually overlap • Proof by induction on t

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• To prove: at most t − 1 parallelograms mutually overlap • Proof by induction on t– t = 2: one parallelogram, OK

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• To prove: at most t − 1 parallelograms mutually overlap • Proof by induction on t– t = 2: one parallelogram, OK– t > 2:• Ot = biggest set of mutually overlapping

parallelograms in Γt

– suppose by contradiction that |Ot| > t – 1• By induction |Ot-1| ≤ t - 2

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1 2 i1 i2 ip ip + 1 t-1 t

• Ot = P U R




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• P = subset of parallelograms of Ot having st as a side– t − 2+ |P| ≥ t |P| ≥ 2

1 2 i1 i2 ip ip + 1 t-1 t




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• P = subset of parallelograms of Ot having st as a side– t − 2+ |P| ≥ t |P| ≥ 2

• R = Ot \ P– they must have a side sj , 1 ≤ j ≤ i1 and a side sl , ip + 1

≤ l ≤ t − 1 they are present in Γt-1

– |Ot| = |R| + |P| and |Ot| ≥ t |R| ≥ t − |P|

1 2 i1 i2 ip ip + 1 t-1 t




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• Let ih + 1 ≤ l ≤ t − 1 be the greatest index among the segments in R– parallelograms Πi2,l ,…, Πip,l and all the parallelograms in R

mutually overlap• they form a bundle of mutually overlapping

parallelograms in Γt−1 whose size is at least t − |P| + |P| − 1 > t - 2, a contradiction, OK

1 2 i1 i2 ip ip + 1 t-1 t



(2, tk)-track layout of k-trees

• Theorem 1: Every partial k-tree admits a (2, tk)-track layout, where tk is given by the following set of equations:

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Putting results together

• Theorem 2: Every partial k-tree with n vertices admits a hk -quasi planar grid drawing in O(tk

3n) area, where hk = 2(tk − 1) + 1 and tk is given by the following set of equations:

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Some values

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K h_k (our result) h_k [Di Giacomo et al., 2005]1 3 32 11 153 299 5415

(1,t)-track layouts



COMPARING H-QUASI PLANARITY WITH H-PLANARITY

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Area lower bound for h-planar drawings of partial 2-trees

• Theorem 6: Let h be a positive integer, there exist n-vertex series-parallel graphs such that any h-planar straight-line drawing requires Ω(n2√(log n)) area

• Hence, h-planarity is more restrictive than h-quasi planarity in terms of area requirement for partial 2-trees

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Area lower bound for h-planar drawings of partial 2-trees: sketch of proof

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a graph G




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l

….

G* = l-extension of G




• Lemma 5: Let h be a positive integer, and let G be a planar graph. In any h-planar drawing of the 3h-extension G* of G, there are no two edges of G crossing each other.

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• Lemma 5: Let h be a positive integer, and let G be a planar graph. In any h-planar drawing of the 3h-extension G* of G, there are no two edges of G crossing each other.

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if 2 edges of G cross…

u

vw

z




• Lemma 5: Let h be a positive integer, and let G be a planar graph. In any h-planar drawing of the 3h-extension G* of G, there are no two edges of G crossing each other

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…one vertex will be inside a triangle

u

vw

z




• Lemma 5: Let h be a positive integer, and let G be a planar graph. In any h-planar drawing of the 3h-extension G* of G, there are no two edges of G crossing each other

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…at least one edge of thetriangle will receive h+1 crossings…!!!

h

u

vw

z




• Consider the n-vertex graph G of the family of series-parallel graphs described in [Frati, 2010] – Ω(n2√(log n)) area may be required in planar s.l. drawings

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G




• Construct the 3h-extension G* of G– n* = 3m + n = Θ(n) – G* is a series-parallel graph– G must be drawn planarly in any h-planar drawing

of G*

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G3h….



Extending the lower bound to planar graphs

• Theorem 7: Let ε > 0 be given and let h(n) : N → N be a function such that h(n) ≤ n0.5− ε n ∀ ϵ N. For every n > 0 there exists a graph G with Θ(n) vertices such that any h(n)-planar straight-line grid drawing of G requires Ω(n1+ 2ε) area

– Ω(n2) area necessary if h is a constant

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3h

….



CONCLUSIONS AND OPEN PROBLEMS

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Conclusions and remarks

• We studied h-quasi planar drawings of partial k-trees in linear area– drawings with optimal area and controlled crossing

complexity

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complexity• Interesting also in the case of planar graphs– Are there h-quasi planar drawings of planar graphs in o(n2)

area where h ϵ o(n)?

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area where h ϵ o(n)?• O(n) area and h ϵ O(1) can be simultaneously achieved

for some families of planar graphs

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area where h ϵ o(n)?• O(n) area and h ϵ O(1) can be simultaneously achieved

for some families of planar graphs• Theorem 8: Every planar graph with n vertices admits a

O(log16 n)-quasi planar grid drawing in O(n log48 n) area

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Some open problems

• h-quasi planar drawings of planar graphs:– is it possible to achieve both O(n) area and h ϵ O(1)?

• h-quasi planar drawings of partial k-trees:– studying area - aspect ratio trade offs: O(n) area and

o(n) aspect ratio?

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