1. Fractal Approaches for Visualizing Huge Hierarchies Hideki Koike, Hirotaka Yoshihara Department of Communications and Systems University of Electro-Communications Chofu, Tokyo 182, Japan 元智資工所 系統實驗室 楊錫謦 1999/8/4

2. Outline Introduction Problems with Huge Hierarchies Fractal approach Discussion & Conclusion

3. Introduction Visualization systems for huge data structures(hierarchical) have a potential usefulness. It is meaningless to visualize such huge data with a current tree visualization framework because of the exponential explosion in size of a tree and the increase of visual element.

4. Problems with Huge Hierarchies(Cont.) : 1. Exponential explosion in size 。 For general tree layout The width of the tree increases exponentially corresponding to the depth of the tree. d n = N d n-1 (for normal tree layout) d n : the width which is necessary to draw all subtree under the tree at depth n. N : the branch degree of a tree.

5. Problems with Huge Hierarchies(cont.) : to generalize: d n = d 1 N n-1 The width increases exponentially. 。 For Cone Tree

6. Problems with Huge Hierarchies(cont.) : r n-1 / ( r n - r n-1 ) = sin½θ & θ= 2π/ N to generalize: r n = r 1 [ 1 + 1 / sin(π/ N) ] n-1 The size also increases exponentially.

7. Problems with Huge Hierarchies(cont.) : 2. Scrolling is useless Why scrolling? -- The exponential explosion in size causes overflow off the screen. Why is scrolling not practical use? -- When the users focus on a node, they can’t see the minimum information in a scrollable window. -- It’s hard to trace links with scrolling to see a node’s neighbors.

8. Problems with Huge Hierarchies(cont.) : 3. Increase of Visual Elements It should be also considered how the increase of visual elements affects users’ cognitive load as well as the system’s response time.

9. Fractal approach : A word represents a wide variety of self- similar objects. 1. Fractal Views There exists a relation: D = - log r x N x between the branching factor, N, and the scale factor, r, at node x.

10. Fractal approach(cont.) : Fractal views calculate the degree of importance which is called fractal value ( Fv ) of each node, and decide which nodes should be displayed or erased based on these fractal values. The Fv x is calculated with the following equation. Fv focus = 1 Fv child-of-x = r x * Fv x (r x = CN x -1/D ) C is a constant value which satisfies 0 < C ≦ 1. For simplify calculation, C is set to 1 in our system.

11. Fractal approach(cont.) : The most important property of fractal view is the ability to control the amount of data displayed. The Fv x is calculated with the following equation. Fv focus = 1 Fv child-of-x = r x * Fv x (r x = CN x -1/D ) C is a constant value which satisfies 0 < C ≦ 1. For simplify calculation, C is set to 1 in our system.

12. Fractal approach(cont.) : The propagated value, Fv, of node at depth n is: Fv = r n = N -n/D ===> M = ( Fv -D - 1/N ) / ( 1 - 1/N ) ===> The number of nodes which have a value greater or equal to the threshold is nearly constant without a relation to the branching factor. In fisheye views, the degree of importance (DOI) of each node is calculated, with a distance from the root (-API) and a distance from the focus (D). DOI = API - D

13. Fractal approach(cont.) : 2. Fractal Tree Layout Using the self-similarity characteristic of a fractal, it is possible to standardize the view at every level of the tree. ==> In a fractal tree, when a part of the subtree is magnified, a similar view appears as well.

14. Fractal approach(cont.) : In the fractal tree layout, each node size is proportional to its fractal value.Thus, the number of recognizable nodes is almost constant. 3. Fractal Pruning All the displayed nodes, whether they are recognizable or not, reduce the system’s response time, these unrecognizable nodes as well as the nodes outside the viewing area should be erased.

15. Discussion & Conclusion: 。 Unsolved problem: Strictly speaking, this framework is not a 3D layout algorithm, because it does not check the overlap of cones. Solution: 1. A smaller fractal dimension minimizes the overlap of cones. 2. Self-avoidance -- the angle of each branch is decided independently.

16. Discussion & Conclusion(Cont.):

17. Discussion & Conclusion: 。 Conclusion 1. The geometrical characteristic of a fractal, self-similarity, madeit possible to visually interact with a huge tree in the same manner at every level of the tree. 2. The fractal dimension made it possible to keep the number of displayed nodes nearly constant. 3. Many problems remain unsolved.