D.I.E.I. - Università degli Studi di Perugia h-quasi planar drawings of bounded treewidth graphs in linear area Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani University of Perugia 13 th Italian Conference on Theoretical Computer Science 19-21 September 2012, Varese, Italy
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D.I.E.I. - Università degli Studi di Perugia h-quasi planar drawings of bounded treewidth graphs in linear area Emilio Di Giacomo, Walter Didimo, Giuseppe.
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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
19/09/2012
Graph G Straight-line grid drawing of G
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 3
Graph Drawing and Area Requirement
Area requirement of straight-line drawings is a widely studied topic in Graph Drawing
19/09/2012
Graph G Straight-line grid drawing of G
h
w
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 4
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]
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 5
Area Requirement for planar drawings
• Planarity imposes severe limitations on the optimization of the area
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 6
Area Requirement for planar drawings
• 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
19/09/2012
A drawing by Wood’s technique
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 7
Beyond planarity: crossing complexity
• Non-planar drawings should be considered: – How can we “control” the crossing complexity of a
drawing?
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 8
Crossing complexity measures
• Large Angle Crossing drawings (LAC) or Right Angle Crossing drawings (RAC), [Didimo et al., 2011]
19/09/2012
RAC drawing
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 9
Crossing complexity measures
• h-Planar drawings: at most h crossings per edge
19/09/2012
1-planar drawing
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 10
Crossing complexity measures
• h-Quasi Planar drawings: at most h-1 mutually crossing edges
19/09/2012
3-quasi planar drawing
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 11
The problem
• We investigate trade-offs between area requirement and crossing complexity
• We focus on h-quasi planarity as a measure of crossing complexity
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 12
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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E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 13
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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E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 14
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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E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 15
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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E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 16
What’s coming next
• Basic definitions
• Results on h-quasi planarity
• Comparison with h-planarity
• Conclusions and open problems
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 17
BASIC DEFINITIONS
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 18
Bounded treewidth graphs
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• What’s a k-tree?
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 19
Bounded treewidth graphs
19/09/2012
• What’s a k-tree? • a clique of size k is a k-tree
3-tree construction
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 20
Bounded treewidth graphs
19/09/2012
• 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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 21
Bounded treewidth graphs
19/09/2012
• 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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 22
Bounded treewidth graphs
19/09/2012
• 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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 23
Bounded treewidth graphs
19/09/2012
• 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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 24
Bounded treewidth graphs
19/09/2012
• 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• 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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 25
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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 26
Track assignment• t-track assignment of a graph G [Dujmović et al., 2004] =
t vertex coloring + total ordering <i in each color class Vi
– (Vi ,<i ) = track τi , 1 ≤ i ≤ t
19/09/2012
3-track assignment
track
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
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Track assignment• t-track assignment of a graph G [Dujmović et al., 2004] =
t vertex coloring + total ordering <i in each color class Vi
– (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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 28
Track assignment• t-track assignment of a graph G [Dujmović et al., 2004] =
t vertex coloring + total ordering <i in each color class Vi
– (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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
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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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 30
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
19/09/2012
(2,3)-track layout
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 31
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
19/09/2012
(2,3)-track layout
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 32
THE GENERAL TECHNIQUE: COMPUTING COMPACT H-QUASI PLANAR DRAWINGS OF K-TREES
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 33
Ingredients of the result
19/09/2012
• 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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 34
Ingredients of the result
19/09/2012
• 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
• 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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 35
Ingredients of the result
19/09/2012
• 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
• 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)
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 36
An example
19/09/2012
INPUT: A partial k-tree G
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 37
An example
19/09/2012
G = 2-treeINPUT: A partial k-tree G
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 38
An example
19/09/2012
INPUT: A partial k-tree G
1. Compute a (2,tk)-track layout of G
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 39
An example
19/09/2012
1) = (2,t)-track layout of G t = 4
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 40
An example
19/09/2012
INPUT: A partial k-tree G
1. Compute a (2,tk)-track layout of G
2. Construct an hk-quasi planar drawing from
OUTPUT: The drawing
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 41
An example
19/09/2012
2) = h-quasi planar drawing of G h ≤ c(t-1)+1 = 2(4-1)+1 = 7
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 42
An example
19/09/2012
INPUT: A partial k-tree G
1. Compute a (2,tk)-track layout of G
2. Construct an hk-quasi planar drawing from
OUTPUT: The drawing
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 43
(c,t)-track layout h-quasi planar drawing
19/09/2012
• 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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 44
An example
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 45
(c,t)-track layout h-quasi planar drawing
19/09/2012
place the verticesalong segments
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 46
(c,t)-track layout h-quasi planar drawing
19/09/2012
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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 47
(c,t)-track layout h-quasi planar drawing
19/09/2012
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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 48
(c,t)-track layout h-quasi planar drawing
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 49
(c,t)-track layout h-quasi planar drawing
19/09/2012
O(t2n)
t
A = O(t3n)
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
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(c,t)-track layout h-quasi planar drawing: upper bound on h
• We prove that at most c(t − 1) edges mutually cross
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 51
(c,t)-track layout h-quasi planar drawing: upper bound on h
• 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
19/09/2012
si
parallelogram Πi,j
sj
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 52
(c,t)-track layout h-quasi planar drawing: upper bound on h
19/09/2012
at most c mutually crossing edges in each parallelogram
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 53
(c,t)-track layout h-quasi planar drawing: upper bound on h
19/09/2012
at most c mutually crossing edges in each parallelogram
+at most t − 1 parallelograms mutually overlap (to prove)
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 54
(c,t)-track layout h-quasi planar drawing: upper bound on h
19/09/2012
at most c mutually crossing edges in each parallelogram
+at most t − 1 parallelograms mutually overlap (to prove)
at most c(t − 1) mutually crossing edges in our drawing
=
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 55
(c,t)-track layout h-quasi planar drawing: upper bound on h
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
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(c,t)-track layout h-quasi planar drawing: upper bound on h
• An overlap occurs iff1 - two curves form a crossing
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 58
(c,t)-track layout h-quasi planar drawing: upper bound on h
• An overlap occurs iff2 - two curves share an endpoint and the other two
endpoints are either before or after the one in common
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 59
(c,t)-track layout h-quasi planar drawing: upper bound on h
• 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
19/09/2012
4 mutually overlapping parallelograms
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 60
(c,t)-track layout h-quasi planar drawing: upper bound on h
• To prove: at most t − 1 parallelograms mutually overlap • Proof by induction on t
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 61
(c,t)-track layout h-quasi planar drawing: upper bound on h
• To prove: at most t − 1 parallelograms mutually overlap • Proof by induction on t– t = 2: one parallelogram, OK
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 62
(c,t)-track layout h-quasi planar drawing: upper bound on h
• 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
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
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(c,t)-track layout h-quasi planar drawing: upper bound on h
19/09/2012
1 2 i1 i2 ip ip + 1 t-1 t
• Ot = P U R
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(c,t)-track layout h-quasi planar drawing: upper bound on h
19/09/2012
• 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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
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(c,t)-track layout h-quasi planar drawing: upper bound on h
19/09/2012
• 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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 66
(c,t)-track layout h-quasi planar drawing: upper bound on h
19/09/2012
• 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
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(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:
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
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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:
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 69
Some values
19/09/2012
K h_k (our result) h_k [Di Giacomo et al., 2005]1 3 32 11 153 299 5415
(1,t)-track layouts
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 70
COMPARING H-QUASI PLANARITY WITH H-PLANARITY
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 71
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
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 72
Area lower bound for h-planar drawings of partial 2-trees: sketch of proof
19/09/2012
a graph G
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 73
Area lower bound for h-planar drawings of partial 2-trees: sketch of proof
19/09/2012
l
….
G* = l-extension of G
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 74
Area lower bound for h-planar drawings of partial 2-trees: sketch of proof
• 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.
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 75
Area lower bound for h-planar drawings of partial 2-trees: sketch of proof
• 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.
19/09/2012
if 2 edges of G cross…
u
vw
z
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 76
Area lower bound for h-planar drawings of partial 2-trees: sketch of proof
• 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
19/09/2012
…one vertex will be inside a triangle
u
vw
z
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 77
Area lower bound for h-planar drawings of partial 2-trees: sketch of proof
• 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
19/09/2012
…at least one edge of thetriangle will receive h+1 crossings…!!!
h
u
vw
z
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 78
Area lower bound for h-planar drawings of partial 2-trees: sketch of proof
• 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
19/09/2012
G
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 79
Area lower bound for h-planar drawings of partial 2-trees: sketch of proof
• 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*
19/09/2012
G3h….
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 80
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
19/09/2012
3h
….
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 81
CONCLUSIONS AND OPEN PROBLEMS
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 82
Conclusions and remarks
• We studied h-quasi planar drawings of partial k-trees in linear area– drawings with optimal area and controlled crossing
complexity
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 83
Conclusions and remarks
• We studied h-quasi planar drawings of partial k-trees in linear area– drawings with optimal area and controlled crossing
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)?
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 84
Conclusions and remarks
• We studied h-quasi planar drawings of partial k-trees in linear area– drawings with optimal area and controlled crossing
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)?• O(n) area and h ϵ O(1) can be simultaneously achieved
for some families of planar graphs
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 85
Conclusions and remarks
• We studied h-quasi planar drawings of partial k-trees in linear area– drawings with optimal area and controlled crossing
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)?• 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
19/09/2012
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 86
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