By Sion, Atallah
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Extra resources for Attacking Digital Watermarks
4. For each of the phases s o m e c h a n g e s or e x t e n s i o n s t o the original algorithm w e r e n e c e s s a r y . The correspondence between the coordinate system used by the constraints and the layout algorithm is as follows. In the x-direction, coordinate units correspond to subsequent positions. In the y-direction the levels are assigned subsequent numbers. So the constraint "A is the left neighbor of B" can be defined by the two equations "Ax + 1 = Bx" and "Ay = By". The following text assumes that the orientation of the graph is top-to-bottom.
2. 10: Constraint example: PERT chart 52 CHAPTER 4. LAYOUT ALGORITHMS AND LAYOUT CONSTRAINTS I~ n ~ d $~'tm~ ~ l o ~ m t ~lrmmnt I J. 3. LAYOUT STABILITY 53 Name of Graph Layout Time Without Constraints With Constraints Family tree P E R T chart UNIX tools 1 sec. 1 sec. 4 sec. 1 sec. 1 sec. 10 sec. 6 Results The three examples shown above demonstrate that the single-dimensional layout constraints provided by the proposed solution can significantly improve the layout of the graph. 1 shows the amount of time taken by the layout of each of these graphs, both with and without layout constraints measured on a Sun 3 workstation.
9 G r a p h i c a l a b s t r a c t i o n : Application-specific abstraction mechanisms are used to reduce the complexity of the large (possibly infinite) reachability graphs. The user can choose to view the graph incrementally. In that case, the user controls which portions of the graph should be explored further. 1. 3: PROSPEC editor 9 Persistence: For communication between the tools, a standard internal representation of the graph is used. 9 Extendibility: None. ParaGraph [BCLO0] is a graph editor designed to support massively parallel programming environments.