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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. - PowerPoint PPT Presentation
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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]
– 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]
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 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
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 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)
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 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
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 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
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 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
19/09/2012
• 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
19/09/2012
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
ICTCS ’12 - Varese, Italy 27
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
19/09/2012
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
19/09/2012
NOT an X-crossing
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 29
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 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
ICTCS ’12 - Varese, Italy 50
(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
• Simplified (but consistent) model– segments = points
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 56
(c,t)-track layout h-quasi planar drawing: upper bound on h
• Simplified (but consistent) model– segments = points– parallelograms = curves
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 57
(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
ICTCS ’12 - Varese, Italy 63
(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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 64
(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
ICTCS ’12 - Varese, Italy 65
(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
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
ICTCS ’12 - Varese, Italy 67
(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
ICTCS ’12 - Varese, Italy 68
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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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
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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
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
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E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
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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
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
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…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
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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*
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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
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3h
….
E. Di Giacomo, W. Didimo, G. Liotta, F. Montecchiani D.I.E.I. - Università degli Studi di Perugia
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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
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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 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
o(n) aspect ratio?
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