Algorithms for graph visualization · Straight-line Drawing of SP-Graphs ss Q-Nodes (Induction...

Algorithms for graph visualizationDivide and Conquer - Series-Parallel Graphs

WINTER SEMESTER 2013/2014

Tamara Mchedlidze – MARTIN NOLLENBURG

www.kit.eduKIT – Universitat des Landes Baden-Wurttemberg und

nationales Forschungszentrum in der Helmholtz-Gemeinschaft

Algorithmen zur Visualisierung von Graphen

Tamara Mchedlidze

Institut fur Theoretische Informatik

Lehrstuhl Algorithmik I

Series-parallel GraphsGraph G is series-parallel, if

Series composition:Identify t1 and s2,s1 is the source of G, t2 is the sink of G

Parallel composition:Identify s1, s2 and set it to be source of GIdentify t1, t2 and set it to be sink of G

t1 = s2

s1 = s2

t1 = t2

ss It consists of two series-parallel graphs G1, G2 with sources s1, s2 andsinks t1, t2 which are combined using one of the following rules:

ss It contains a single edge (s, t) (s-source, t-sink)

Tamara Mchedlidze

Series-parallel Graphs. Decomposition Tree.

In order to proof this statement we can use a decomposition tree of G,which is a binary tree T with nodes of three types: S,P and Q-type.

LemmaSeries-parallel graphs are acyclic and planar.

Tamara Mchedlidze

ss If G is a single edge, then the corresponding node is Q-node

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ss If G is a single edge, then the corresponding node is Q-nodess If G is a parallel composition of G1 (with tree T1) and G2 (with tree T2), then theroot of T is P-node and T1 is its left subtree, T2 is its right subtree

Q G1 G2

Tamara Mchedlidze

ss If G is a single edge, then the corresponding node is Q-nodess If G is a parallel composition of G1 (with tree T1) and G2 (with tree T2), then theroot of T is P-node and T1 is its left subtree, T2 is its right subtreess If G is a series composition of G1 (with tree T1) and G2 (with tree T2), then the rootof T is S-node and T1 is its left subtree, T2 is its right subtree

Q G1 G2

T1 T2G1

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Series-parallel Graphs. Decomposition Example.

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Q Q Q Q

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Series-parallel Graphs. Applications.

Flowcharts PERT-Diagrams(Program Evaluation and Review Technique)

Tamara Mchedlidze

Series-parallel Graphs. Applications.

Flowcharts PERT-Diagrams(Program Evaluation and Review Technique)

Computational Complexity: Linear time algorithms for NP-hard problems(e.g. Maximum Matching, Maximum Independent Set, Hamiltonian Comple-tion)

Tamara Mchedlidze

Straight-line Drawing of SP-Graphs

ss Draw graph G inside a right-angled isosceles bounding triangle ∆(G) G

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ss Q-Nodes (Induction base):

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ss S-Nodes (series composition)

∆(G2)

∆(G1)

∆(G)

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∆(G2)

∆(G1)

ss P-Nodes (parallel composition)

∆(G1)

∆(G2)

∆(G)

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∆(G2)

∆(G1)

∆(G2)

∆(G)

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∆(G2)

∆(G1)

∆(G2)

∆(G)

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∆(G2)

∆(G1)

∆(G2)

∆(G)

change embedding!

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∆(G2)

∆(G1)

∆(G2)

∆(G)

change embedding!

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∆(G2)

∆(G1)

∆(G2)

∆(G)

change embedding!

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∆(G2)

∆(G1)

∆(G2)

∆(G)

change embedding!

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Straight-line Drawing of SP-Graphsss What makes parallel composition possible without creating crossings?

Tamara Mchedlidze

does not contain any vertex

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does not contain any vertexss This condition can be preserved during the induction step.

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ss The area of the drawing is?

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ss The area of the drawing is? O(m2),m is the number of edges

Tamara Mchedlidze

ss The area of the drawing is? O(m2),m is the number of edges

TheoremA series-parallel graph G (with variable embedding) admits an upward straigh-linedrawing with O(n2) area. The isomorphic componencts of G have congruent drawingsup to a translation.

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Lower Bound for the Area

Theorem [Bertolazzi et al. 94]There exists a 2n-vertex series-parallel graph Gn such that any upward pla-nar drawing of Gn respecting embedding requires area Ω(4n).

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Proof:

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Proof:

sn−1

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Proof:

sn−1

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Proof:

sn−1

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Proof:

sn−1

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Proof:

sn−1

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Proof:

sn−1

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Proof:

sn−1

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Proof:

sn−1

ss We have that: Area(Π) > 2 ·Area(Gn)

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Proof:

sn−1

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Proof:

sn−1

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Proof:

sn−1

ss We have that: Area(Π) > 2 ·Area(Gn)ss Area(Gn+1) ≥ 2 ·Area(Π)

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Proof:

sn−1

ss We have that: Area(Π) > 2 ·Area(Gn)ss Area(Gn+1) ≥ 2 ·Area(Π)ss Area(Gn+1) ≥ 4 ·Area(Gn)

Tamara Mchedlidze

Property of the Algorithm

Algorithm

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Algorithm

nicer???

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Algorithm

nicer???

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Algorithm

nicer???

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Algorithm

nicer???

a b c d

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Algorithm

nicer???

a b c d

hss Graph G = (a, b, c, d, e, f, g, h,(a, h), (a, e), (b, g), (b, f), (c, g), (c, f), (d, e),(d, h), (e, f), (h, g))

Tamara Mchedlidze

Algorithm

nicer???

a b c d

ss Let G′ be G where b→ c→ b, a→ d→ a.

Tamara Mchedlidze

Algorithm

nicer???

a b c d

ss Let G′ be G where b→ c→ b, a→ d→ a.

ss G and G′ are isomorphic.

Tamara Mchedlidze

Graph AutomorphismDefinition: Automorphism of a digraph

An automorphism of a directed graph G = (V,E) is a permutation of the vertex setwhich preserves adjacency of the vertices and either preserves or reverses all thedirections of the edges:ss (u, v) ∈ E ⇔ (π(u), π(v)) ∈ E, orss (u, v) ∈ E ⇔ (π(v), π(u)) ∈ E

Tamara Mchedlidze

Graph Automorphism

ss The set of all automorphisms (direction preserving and reversing)forms the automorphism group of G.

Definition: Automorphism of a digraphAn automorphism of a directed graph G = (V,E) is a permutation of the vertex setwhich preserves adjacency of the vertices and either preserves or reverses all thedirections of the edges:ss (u, v) ∈ E ⇔ (π(u), π(v)) ∈ E, orss (u, v) ∈ E ⇔ (π(v), π(u)) ∈ E

Tamara Mchedlidze

Graph Automorphism

ss Finding an automorphism group of a graph is isomorphismcomplete, that is equivalent to testing whether two graphs areisomorphic.

Tamara Mchedlidze

Graph Automorphism

ss Finding an automorphism group of a graph is isomorphismcomplete, that is equivalent to testing whether two graphs areisomorphic.ss For planar graphs, graphs with bounded degree isomorphismproblem has polynomial-time algorithms.

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Geometric Automorphism

ss Different types of automorphism:

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Automorphism 1 → 2 → 3 → 4 →1 is geometrically representable, while1 → 2 → 3 → 1 is not.

Automorphism 1 → 2 → 3 →1 is geometrically representable, while1 → 2 → 3 → 4 → 1 is not.

Automorphism 1 → 2 → 3 → 1,4 → 5 → 4 is not geometrically rep-resentable.

Tamara Mchedlidze

ss An automorphism group P of a graph is geometric, if there exists adrawing of G that displays each element of P as a symmetry.ss For general graphs it is NP-hard to find a geometric automorphismof a graph.ss For planar graphs, planar geometric automorphisms can be found inpolynomial time. For outerplanar graphs and trees in linear time.

Automorphism 1 → 2 → 3 → 4 →1 is geometrically representable, while1 → 2 → 3 → 1 is not.

Automorphism 1 → 2 → 3 →1 is geometrically representable, while1 → 2 → 3 → 4 → 1 is not.

Automorphism 1 → 2 → 3 → 1,4 → 5 → 4 is not geometrically rep-resentable.

Tamara Mchedlidze

Symmetries in SP-Graphsπvert

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Symmetries in SP-Graphsπvert πhor

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Symmetries in SP-Graphsπvert πhor πrot

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Symmetries in SP-Graphsπvert πhor πrot πvert, πhor, πrot

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ss A geometric automorphism group P of a graph G is upward planar, if there existsan upward planar drawing of G that displays each element of P as a symmetry.

Tamara Mchedlidze

ss A geometric automorphism group P of a graph G is upward planar, if there existsan upward planar drawing of G that displays each element of P as a symmetry.ss How does a geometric automorphism group for a series-parallel graph look like?

Tamara Mchedlidze

Theorem (Hong, Eades, Lee ’00)

An upward planar automorphism group of a series-parallel digraph is eitherss idss id, π with π ∈ πvert, πhor, πrotss id, πvert, πhor, πrot.

Tamara Mchedlidze

Vertical Automorphism

Q Q Q Q

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Q Q Q Q

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Q Q Q Q

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code = 1 code=1code = 2

ss code(G) - two graphs at the samelevel have the same code iff theyare isomorphicss tuple(G) - codes of the children

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tuple(G) =< 1, 1, 2 >

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tuple(G) =< 1, 1, 2 >

Why sorted?

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tuple(G) =< 1, 1, 2 >

Why sorted?

compare :(

1 1 222 3 3 3< > < 3 2 2 1 1 3 3 3 >

Tamara Mchedlidze

tuple(G) =< 1, 1, 2 >

Why sorted?

compare :)

Tamara Mchedlidze

Vertical AutomorphismAlgorithm constructing a Canonical Labeling

ss Set tuple(Gi) = 〈0〉 for all Q-nodes Gi of G.ss For each t = max depth(G), . . . , 0ss For each S- or P-node G′ at depth t with children G1, . . . , Gkset tuple(G′) = 〈code(G1), . . . , code(Gk)〉. If G′ is a P-node,sort tuple(G′) in non-decreasing order.ss Sort all the nodes at depth t lexicografically according to tuples.ss For each component G′ at depth t, compute code(G′) as fol-lows. Assign the integer 1 to those components representedby the first distinct tuple, assign 2 to the components with thesecond type of tuple, and etc.

Tamara Mchedlidze

LemmaTwo nodes u and v at the same depth of the decomposition tree of G repre-sent isomorphic subgraphs of G iff code(u) = code(v).

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〈0〉〈0〉

〈0〉

〈0〉〈0〉〈0〉〈0〉

〈0〉

〈0〉〈0〉

Tamara Mchedlidze

〈0〉〈0〉

〈0〉

〈0〉〈0〉〈0〉〈0〉

〈0〉

〈0〉〈0〉

Tamara Mchedlidze

〈0〉〈0〉

〈0〉

〈0〉〈0〉〈0〉〈0〉

〈0〉

〈0〉〈0〉

〈1, 1〉〈1, 1〉

Tamara Mchedlidze

〈0〉〈0〉

〈0〉

〈0〉〈0〉〈0〉〈0〉

〈0〉

〈0〉〈0〉

〈1, 1〉〈1, 1〉

Tamara Mchedlidze

〈0〉〈0〉

〈0〉

〈0〉〈0〉〈0〉〈0〉

〈0〉

〈0〉〈0〉

〈1, 1〉〈1, 1〉

〈1, 1〉

Tamara Mchedlidze

〈0〉〈0〉

〈0〉

〈0〉〈0〉〈0〉〈0〉

〈0〉

〈0〉〈0〉

〈1, 1〉〈1, 1〉

〈1, 1〉

Tamara Mchedlidze

〈0〉〈0〉

〈0〉

〈0〉〈0〉〈0〉〈0〉

〈0〉

〈0〉〈0〉

〈1, 1〉〈1, 1〉

〈1, 1〉

〈1, 1〉〈1, 1〉〈1, 1, 2〉

Tamara Mchedlidze

〈0〉〈0〉

〈0〉

〈0〉〈0〉〈0〉〈0〉

〈0〉

〈0〉〈0〉

〈1, 1〉〈1, 1〉

〈1, 1〉

〈1, 1〉〈1, 1〉〈1, 1, 2〉

Tamara Mchedlidze

〈0〉〈0〉

〈0〉

〈0〉〈0〉〈0〉〈0〉

〈0〉

〈0〉〈0〉

〈1, 1〉〈1, 1〉

〈1, 1〉

〈1, 1〉〈1, 1〉〈1, 1, 2〉

〈1, 1, 2〉 1

Tamara Mchedlidze

G is an S-node

ss Let G be composed out of G1 . . . Gn through series or parallelcomposition, tuple(G) contains the codes of G1, . . . , Gn.ss How can we use tuple(G) do decide whether G has a verticalautomorphism?

Tamara Mchedlidze

G is an S-node

Lemma (Hong, Eades, Lee ’00)If G is an S-node, then G has a vertical automorphismiff each of G1, . . . , Gk has a vertical automorphism.

Tamara Mchedlidze

G is an S-node

ss Assume G has a vertical automorphism α

Proof:

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G is an S-node

ss Assume G has a vertical automorphism αss Then α “fixes” all the components

Proof:

Tamara Mchedlidze

G is an S-node

ss Assume G has a vertical automorphism αss Then α “fixes” all the componentsss Therefore each of the series components has avertical automorhism

Proof:

Tamara Mchedlidze

G is an S-node

ss Assume G has a vertical automorphism αss Then α “fixes” all the componentsss Therefore each of the series components has avertical automorhismss If each of G1, . . . , Gn has a vertical isomorphism,arrange them as in Figure.

Proof:

Tamara Mchedlidze

G is P-node, tuple(G) =< 1 . . . 1︸︷︷︸even

, 2 . . . 2︸︷︷︸even

, · · · >

Lemma (Hong, Eades, Lee ’00)If G is a P-node, consider a partition of Cj = Gi : 1 ≤ i ≤ k, code(Gi) = j,j = 1, . . . , k into classes of isomorphic graphs.ss If ∀j , |Cj | are even⇒ has a vertical automorphism.ss If there exists a unique j, such that |Cj | is odd⇒ G has a vertical automorphism

iff graphs of Cj have a vertical automorphism.ss If there exists |Ci|, |Cj | with i 6= j, both odd ⇒ G does not have a vertical auto-morphism.

ss Arrange components as in Figure.

Proof:

Tamara Mchedlidze

tuple(G) =< 1 . . . 1︸︷︷︸odd

, 2 . . . 2︸︷︷︸even

, 3 . . . 3︸︷︷︸even

, · · · >

Proof:

Tamara Mchedlidze

tuple(G) =< 1 . . . 1︸︷︷︸odd

, 2 . . . 2︸︷︷︸even

, 3 . . . 3︸︷︷︸even

, · · · >

ss Any vertical automorphism “fixes” amember of Cj , therefore it has avertical automorphism.

Proof:

Tamara Mchedlidze

tuple(G) =< 1 . . . 1︸︷︷︸odd

, 2 . . . 2︸︷︷︸even

, 3 . . . 3︸︷︷︸even

, · · · >

ss Any vertical automorphism “fixes” amember of Cj , therefore it has avertical automorphism.ss Conversly, arrange as in figure.

Proof:

Tamara Mchedlidze

tuple(G) =< 1 . . . 1︸︷︷︸odd

, 2 . . . 2︸︷︷︸odd

, 3 . . . 3︸︷︷︸even

, · · · >

Proof:

Tamara Mchedlidze

tuple(G) =< 1 . . . 1︸︷︷︸odd

, 2 . . . 2︸︷︷︸odd

, 3 . . . 3︸︷︷︸even

, · · · >

ss Any vertical automorphism has to“fix” two distinct components.

Proof:

Tamara Mchedlidze

tuple(G) =< 1 . . . 1︸︷︷︸odd

, 2 . . . 2︸︷︷︸odd

, 3 . . . 3︸︷︷︸even

, · · · >

ss Any vertical automorphism has to“fix” two distinct components.ss In both components we can find apath on which some vertices arealigned on the axis. Contradictsplanarity.

Proof:

Tamara Mchedlidze

Vertical AutomorphismTheorem (Hong, Eades, Lee ’00)

Given a decomposition tree of a series-parallel graph and its canon-ical labeling. Let G be a component which consists from G1, . . . , Gkthrough series or parallel composition.ss If G is an S-node, then G has a vertical automorphism iff each of

G1, . . . , Gk has a vertical automorphism.ss If G is a P-node, consider a partition of Cj = Gi : 1 ≤ i ≤k, code(Gi) = j, j = 1, . . . , k into classes of isomorphic graphs.ss If ∀j , |Cj | are even⇒ has a vertical automorphism.ss If there exists a unique j, such that |Cj | is odd⇒ G has a verti-

cal automorphism iff graphs of Cj have a vertical automorphism.ss If there exists |Ci|, |Cj | with i 6= j, both odd⇒ G does not havea vertical automorphism.

Tamara Mchedlidze

Algorithms for graph visualization · Straight-line Drawing of SP-Graphs ss Q-Nodes (Induction...

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