On Balloon Drawings of Rooted Trees

1. On Balloon Drawings of Rooted Trees Chun-Cheng Lin and Hsu-Chun Yen Dept. of Electrical Engineering, National Taiwan University

2. Definition. The balloon drawing of a rooted tree is a drawing having the following constraints: all the children of the same parent are placed on the circumference of the cycle centered at their parent, there exist no edge crossings in the drawing, and with respect to the root, the deeper an edge is, the shorter its drawing length becomes. Balloon Drawing of Rooted Trees

3. Two models of balloon drawing (I) • The fractal model – top-down method ( Koike & Yoshihara, 1993 ) • Displaying self-similarly • Evenly angle degree • Edge length formula • rm =  rm-1 r3 r2 r1 120o 120o 120o

4. The subtree with nonuniform sizes (SNS) model – Bottom-up method ( Carriere & Kazman, 1995 ) Allow the subtrees with the same parent to reside in circles of different sizes r2 r r1 1 r3 r4 Two models of balloon drawing (II) outer circle inner circle

5. Comparison • The balloon drawings of the same ordered tree under different models are given: • The drawing based on the SNS model results in a clearer display on large subtrees than that under the fractal model. the SNS model the fractal model Q. For an unordered rooted tree (i.e., changing the order of subtrees is allowed), what is a good balloon drawing under the SNS model depend on? A. Angular resolution and aspect ratio of angle. Goal: optimize them.

6. αmin αmax AngResl =αmin AspRatio = αmax / αmin Preliminaries • Unordered tree • The order of subtrees is not significant. • Angular resolution( denoted by AngResl ) • the minimum degree of two neighboring edges incident to the common vertex. • Range: ( 0o, 360o )  Larger AngResl is better. • Aspect ratio( denoted by AspRatio ) • the ratio of the maximum degree to the minimum degree of the angles incident to a vertex • Range: ( 1,  ) • Smaller AspRatio approaching to one gives a more balanced display

7. m1 m2 M1 M2 Reduction to star graphs • Definition. Theballoon drawing of a star graph with children of nonuniform sizeis a drawing in which • circles associated with different children of the root do not overlap, and • all the children of the root are placed on the circumference of a circle centered at the root. The balloon drawing under the SNS model (for the nozero level) (for the level zero)

8. m1 m1 min min m2 M1 max max M1 M2 m2 M2 Balloon Drawing of Unordered Trees • Changing the order of subtrees affects the angular resolution as well as the aspect ratio of the drawing. • Example. Larger AngResl and smaller AspRatio. swap M2 and m2

9. Independence Optimizing AngResl and AspRatio on each level is Independent. Swapping any two subtrees inside the outer circle doesn’t affect the optimization on other levels.

10. slice 2 slice 3  2  3 slice 1 c2 c3 1  1 c1 co 1’ c4  4 slice 4

11. Subindex difference = 1 • Procedure 1 OptBalloonDrawing • Order {1,…,n} in ascending order as either m1, m2, …, mk-1, mk, Mk, Mk-1, …, M2, M1 if n is even, or m1, m2, …, mk-1, mk, mid, Mk, Mk-1, …, M2, M1 if n is odd, where mi (resp. Mi) is the i-th minimum (resp. maximum) among all, and mid is the median if n is odd. • Output a drawing witnessed by the following circular permutation:  = (M1, m2, M3, m4, … (, mid) …, M4, m3, M2, m1) e.g. (n = 10) (the drawing has 10 slices) {1,…,10} m1 < m2 < m3 < m4 < m5 < M5 < M4 < M3 < M2 < M1 m5 M5 M4 m4 m3 M3 m2 M2 m1 M1

12. Theorem.Procedure 1 achieves optimality in angular resolution as well as in aspect ratio for star graphs. • Basic idea to prove the correctness of the algorithm: • Claim.optAngResl must be (Mi + mi-1)/2 for some i  {2,…,k}, which can be generated by Procedure 1. • Similarly, the minimum of the largest angle must be (Mj-1 + mj)/2 for j  {2,…,k}, which can be generated by Procedure 1. • Since the permutation  , generated by Procedure 1, simultaneously has both the maximum degree of the smallest angle and the minimum degree of the largest angle of any drawing,  also witnesses the optAspRatio. Q.E.D

13. An experimental result

14. (of uneven angle type) Balloon drawing with uneven angles • Area of balloon drawing • The size of the cycle enclosing the drawing • The drawing under the SNS model may not be minimal (of even angle type)

15. The Aspect Ratio (resp. Angular resolution) problem • Given the initial drawing of a star graph (with uneven angles) and a real number r, determine how to flip the drawing of subtrees so that AspRatior (resp. AngResl r). slice 3 slice 2 slice 4 slice 5 slice 1

16. Matching • Matching • A set of edges such that any two edges shares no common node. • Maximum matching • A matching of the maximum cardinality • Perfect matching • For a graph with n nodes, the largest possible matching consists of n/2 edges • The maximum matching problem for bipartite graphs with n vertices and m edges can be found in time. Perfect!!

17. A1 A4 A2 A3 • Theorem. Both the Aspect Ratio Problem and the Angular Resolution Problem can be solved in O(n2.5) time. pf. Consider the Aspect Ratio Problem ; the other problem can be proved by a slight modification. A1 A4 A2 A3

18. Assume is the smallest angle. The nodes with odd index are placed on the upper level. • Algorithm. • Iteratively selects an pair (x,y) where x {bi, b’i} and y  {bi 1, b’i 1} such that x+y is assumed to be the ‘smallest’ angle. • A bipartite graph G(x,y) is constructed in such a way that a drawing respecting the aspect ratio r exists iff G(x,y) has a perfect matching. Take the following example for illustration: b’4 b1 b4 b’1 A bipartite graph G(x,y) Perfect matching  a balloon drawing Delete the edges (s,t) where s + t > r 

19. Local magnetic spring model • Magnetic spring model (Sugiyama and Misue, 1995) • The graph is placed on a global magnetic field • Edge  magnetized spring • Our local magnetic spring model

20. Experimental results and applications • Experimental results • Applications

21. Thank you for your attention.