h-quasi planar drawings of bounded treewidth graphs in linear area

1. 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 Science 19-21 September 2012, Varese, Italy

2. Graph Drawing and Area Requirement Graph G Straight-linegriddrawing of G

3. Graph Drawing and Area Requirement w Area requirement of straight-line drawings is a widely studied topic in Graph Drawing h Graph G Straight-linegriddrawing of G

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]

5. Area Requirement for planar drawings • Planarity imposes severe limitations on the optimization of the area

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 A drawing by Wood’s technique

7. Beyond planarity: crossing complexity • Non-planar drawings should be considered: • How can we “control” the crossing complexity of a drawing?

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

9. Crossing complexity measures • h-Planar drawings: at most h crossings per edge 1-planar drawing

10. Crossing complexity measures • h-Quasi Planar drawings: at most h-1 mutually crossing edges 3-quasi planar drawing

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

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

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)

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-lineararea for any constant h • 11-quasi planar drawings in linear area always exist

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-lineararea 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

16. What’s coming next • Basic definitions • Results on h-quasi planarity • Comparison with h-planarity • Conclusions and open problems

17. BASIC DEFINITIONS

18. Bounded treewidth graphs • What’s a k-tree?

19. Bounded treewidth graphs • What’s a k-tree? • a clique of size k is a k-tree 3-tree construction

20. Bounded treewidth graphs • 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

21. Bounded treewidth graphs • 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

22. Bounded treewidth graphs • 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

23. Bounded treewidth graphs • 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

24. Bounded treewidth graphs • 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

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 3-track assignment

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 ) = trackτi, 1 ≤ i ≤ t track 3-track assignment

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 ) = trackτi, 1 ≤ i ≤ t • X-crossing = (u, v), (w, z): u,w ∈ Vi, v, z ∈ Vj , u <i w and z <j v, for i ≠ j X-crossing

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 ) = trackτi, 1 ≤ i ≤ t • X-crossing = (u, v), (w, z): u,w ∈ Vi, v, z ∈ Vj , u <i w and z <j v, for i ≠ j NOT anX-crossing

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 (2,3)-track layout

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 (2,3)-track layout

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 (2,3)-track layout

32. THE GENERAL Technique: Computing compact h-quasi planar drawings of k-trees

33. Ingredients of the result • 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

34. Ingredients of the result • 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

35. Ingredients of the result • 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)

36. An example INPUT: A partial k-tree G

37. An example INPUT: A partial k-tree G G = 2-tree

38. An example INPUT: A partial k-tree G • Compute a (2,tk)-track layout  of G

39. An example 1) = (2,t)-track layout of G t = 4

40. An example INPUT: A partial k-tree G • Compute a (2,tk)-track layout  of G • Construct an hk-quasi planar drawing  from  OUTPUT: The drawing 

41. An example 2)  = h-quasi planar drawing of G h ≤ c(t-1)+1 = 2(4-1)+1 = 7

42. An example INPUT: A partial k-tree G • Compute a (2,tk)-track layout  of G • Construct an hk-quasi planar drawing  from  OUTPUT: The drawing 

43. (c,t)-track layout  h-quasi planar drawing • 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

44. An example

45. (c,t)-track layout  h-quasi planar drawing place the vertices alongsegments

46. (c,t)-track layout  h-quasi planar drawing anyedgeconnecting a vertex on a segment i to a vertex on a segment j (i < j) do notoverlapwithanyvertex on a segment k s.t. i < k <j

47. (c,t)-track layout  h-quasi planar drawing anyedgeconnecting a vertex on a segment i to a vertex on a segment j (i < j) do notoverlapwithanyvertex on a segment k s.t. i < k <j

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

49. (c,t)-track layout  h-quasi planar drawing A = O(t3n) t O(t2n)

50. (c,t)-track layout  h-quasi planar drawing: upper bound on h • We prove that at most c(t − 1) edges mutually cross