Compact Visibility Drawing of Planar Graph

1. Compact Visibility Drawing of Planar Graph Hsueh-I Lu National Taiwan University Joint work with Ching-Chi Lin & I-Fan Sun Compact Visibility Drawing of Planar Graph

2. Outline • 開場白 • The visibility drawing problem • Previous work and our improved results • A little bit of details: • how our simple algorithm works in a lazy manner • analyzing our results using Schnyder’s realizer • Conclusion and open questions Compact Visibility Drawing of Planar Graph

3. 開場白 • Algorithmic research is (almost) always about attacking a combinatorial problem in a new (and better) way. • Whether or not you can come up with a neat solution usually depends on how you approach the problem. Compact Visibility Drawing of Planar Graph

4. A TV Game Show Compact Visibility Drawing of Planar Graph

5. Monty Hall Problem It makes a lot of difference  Compact Visibility Drawing of Planar Graph

6. Another problem Coffee and Milk Compact Visibility Drawing of Planar Graph

7. The puzzle Compact Visibility Drawing of Planar Graph

8. What’s a visibility drawing? • Representation • node  horizontal line segment • edge  vertical line segment • Requirement • adjacency is appropriately displayed • no crossing among all line segments Compact Visibility Drawing of Planar Graph

9. An example Compact Visibility Drawing of Planar Graph

10. The problem • Compact visibility drawing • Input: a planar graph G • n nodes and 3n – 6or fewer edges • Output: a visibility drawing D of G • Objective: minimizing the area of D • A classic graph drawing problem, which finds applications in VLSI layout. Compact Visibility Drawing of Planar Graph

11. Convention for measuring the drawing’s area segment endpoints  grid points Compact Visibility Drawing of Planar Graph

12. Height < #node = n • In the worst case, each row of grid points is occupied by only one horizontal line segment. Compact Visibility Drawing of Planar Graph

13. Width < #edge < 3n • In the worst case, each column of grid points is occupied only one vertical line segment. Compact Visibility Drawing of Planar Graph

14. All known results, including ours, focus on minimizing the width … Compact Visibility Drawing of Planar Graph

15. Problem formulation Otten & van Wijk [IEEE ISCS’78] No width bound (i.e., width < 3n ) Width < 2n, O(n) time Rosenstiehl & Tarjan [DCG’86] Tamassia & Tollis [DCG’86] Nummenmaa [TCS’92] Width < 1.5n, O(n) time Kant [WG’94, IJCGA’97] Open: is 1.5n worst-case optimal? If G is 4-connected Width < n, O(n) time Kant & He [TCS’97] Previous work Compact Visibility Drawing of Planar Graph

16. Our main result • Width < 22n/15, O(n) time • A negative answer to Kant’s open question. • Width improvement = 22n/15 – 3n/2 = n/30. • Our algorithm works in a easy-to-implement lazy manner, based on 3 canonical orderings for G provided by Schnyder’s realizer. Compact Visibility Drawing of Planar Graph

17. Our analysis also implies that … if G has no • degree-3 nodes or • degree-5 nodes, then the drawing produced by our lazy algorithm via realizer is no wider than 4n/3. Compact Visibility Drawing of Planar Graph

18. A by-product • A simple alternative proof to a corollary of Wagner’s Theorem on Schnyder’s realizer [Bonichon, Saëc & Mosbah, ICALP’02] Compact Visibility Drawing of Planar Graph

19. Where are we? • The visibility drawing problem • Previous work and our improved results • A little bit of details: • how our simple algorithm works in a lazy manner • analyzing our results using Schnyder’s realizer • Conclusion and open questions Compact Visibility Drawing of Planar Graph

20. Our algorithm • Additional input: • A canonical ordering of G. • Procedure: • For each y = 1 to n • Place node yfrom above and extend the drawing in a lazy manner, i.e., increasing the width of the drawing only when it is necessary. Compact Visibility Drawing of Planar Graph

21. Canonical ordering • For each node y, its neighbors in the subgraph Gy–1 induced by the smaller-indexed nodes are on the external boundary of the subgraph Gy–1. Compact Visibility Drawing of Planar Graph

22. “Lazy” extension

23. “Lazy” extension

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25. “Lazy” extension

26. “Lazy” extension

27. “Lazy” extension

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30. “Lazy” extension

31. Time complexity • It is not difficult to implement the lazy algorithm to run in O(n) time • e.g., by using data structure like doubly linked list so that each of those less than 3n edges can be drawn in O(1) time. Compact Visibility Drawing of Planar Graph

32. The challenge We use Schnyder’s Realizer. How do we choose a canonical ordering with which our lazy algorithm produces a compact visibility drawing? Compact Visibility Drawing of Planar Graph

33. A realizer, obtainable in linear time, partitions the internal edges of G into three edge-disjoint trees, each rooted an external node. The preordering traversal of each tree gives a canonical ordering. How to use realizer? Compact Visibility Drawing of Planar Graph

34. Schnyder’s Realizer [SODA’90] Compact Visibility Drawing of Planar Graph

35. A realizer, obtainable in linear time, partitions the internal edges of G into three edge-disjoint trees, each rooted an external node. The preordering traversal of each tree gives a canonical ordering. We can prove that our lazy algorithm yields a drawing with the improved width bound using one of those three canonical orderings. How to use realizer? Compact Visibility Drawing of Planar Graph

36. An upper bound on width • Width < 3n – Σymindeg(y), where • indeg(y) = the number of neighbors of node y with smaller indices, • outdeg(y) = the number of neighbors of node y with larger indices, and • mindeg(y) = min(indeg(y), outdeg(y)). Compact Visibility Drawing of Planar Graph

37. mindeg(8) = min(2, 4) = 2 mindeg(9) = min(4, 1) = 1 For example Compact Visibility Drawing of Planar Graph

38. Width < 3n – Σymindeg(y). Why? Compact Visibility Drawing of Planar Graph

39. At least mindeg(y) visible points  indeg(y) Observation 1 • At the end of the iteration for drawing node y (i.e., all edges (x, y) with x < y are drawn), y has at least indeg(y) ≥mindeg(y) visible points. y Compact Visibility Drawing of Planar Graph

40. Observation 2 • Drawing an edge (y, z) with y < z either • increases the width by 1 (i.e., y is not visible) or • decreases the number of visible points of y by 1. Compact Visibility Drawing of Planar Graph

41. Combining Obs 1 & 2 • In the first mindeg(y) iterations for drawing edges (y, z) with y < z, the drawing is not extended. • Therefore, when drawing those 3n edges, • each edge extends the drawing by at most 1 unit. • At least Σymindeg(y) edges do not extend the drawing. Compact Visibility Drawing of Planar Graph

42. So, … Width ≤ 3n – Σymindeg(y). Compact Visibility Drawing of Planar Graph

43. Two examples of applying the upper bound on width: • Width ≤ 2n for any canonical ordering • Width ≤ 4n / 3 via realizer, if G has no degree-3 nodes. Compact Visibility Drawing of Planar Graph

44. Example 1: Width ≤ 2n • Since mindeg(y) ≥ 1, • we have Σymindeg(y) ≥ n, and thus • width is at most 3n – n = 2n. Compact Visibility Drawing of Planar Graph

45. Example 2: no deg-3 node • Each node has degree at least 4. • it must have • exactly 3 outgoing edges, and • at least 1 incoming edges y Compact Visibility Drawing of Planar Graph

46. Sum of mindeg(y) in three directions ≥ 2+2+1 = 5 ≥ 2 y ≥ 2 ≥ 1 Compact Visibility Drawing of Planar Graph

47. Width ≤ 4n/3 • On average, each node contributes 5/3 to mindeg(y) among three orderings. • Therefore, one of those three orderings gives a drawing whose width is at most (3– 5/3)n = 4n/3. Compact Visibility Drawing of Planar Graph

48. With a more careful analysis, • we can prove that one of those three canonical orderings defined by Schnyder’s realizer yields a visibility drawing no wider than 22n/15. Compact Visibility Drawing of Planar Graph

49. Our analysis is almost tight • The visibility drawing of the following graph produced by our algorithm has width 4n/3 – O(1). (22n/15 – 4n/3 = 2n/15) Compact Visibility Drawing of Planar Graph

50. Conclusion • Based upon Schnyder’s realizer, we give an O(n)-time algorithm for visibility drawing that is no wider than 22n/15. Compact Visibility Drawing of Planar Graph