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GRAPH UNION (Topic in Concept)

Cesim Erten Kadir Has University cesim@khas.edu.tr

Melih Sözdinler Işık Universiy melihs@isikun.edu.tr

Figure 1: 101 nodes 195 edges with weights and final layout.

Introduction Graph Union is the topic aiming to represent large graphs by using visualization techniques. With this technique we are aiming to have high performance even

use Coffman Graham Layering. However the problem is that it

branches and each edges and nodes represent the relations among

accepts the input graph as unit weighted edges. The adopted

employees in these graphs.

version has not been proposed yet.

4. Other Applications 3.Assigning node positions in layer The most inportant phase should be that part. Readability

representing large graphs. We are proposing a technique that graph is represented as subgraphs such as seen in the

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of the output graph depends on this phase.

Complier Flow Graph VLSI Design Social Network Visualization.

a.Adding dummy nodes

Figure 3. order

graph in such a way that there is no performance problem to

important phase. With that phase, each consecutive layers could

draw these graphs.

be bipartite graph that is the input of cossing minimizatio phase.

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to do crossing minimization. Adding dummy node is

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crossing minimization. In our previous papers we proposed

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weighted versions of each algorithm in literature. When we apply

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crossing minimization, we used layer by layer sweep method.

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4.Assigning edge bends

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During the third phase, we have decided the positions of

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each original node and dummy nodes. But dummy nodes are not

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The technique we are purely based on the Sugiyama Layout if the subgraphs are previously identified as in Figure 3. The Sugiyama layout has the following procedure and the sample outputs are at Figure 2 and Figure 1.

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needed after third phase. So we remove them and assign their

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5.Finalizing the layout

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between layers and nodes etc. It depends on the application area.

Application Areas

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increase output readability. In that phase, we increase the length

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The finalizing the layout is just the phase in order to

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Technique

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Figure 2: 51 Nodes 60 edges 25185 edge crossings.

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positions as bends of the graph.

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Indeed we get the graph with increased readability.

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bipartite graph.

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reduction between layers[1,2,5,6]. We are aiming to do weighted

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There are several algorithms in literature in orer to crossing

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b.Crossing minimization

accepting one layer as free, we run our algorithms over that

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With this method, we take two layers of layered graph, and

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graph. We are aiming to determine each subgraphs in union

continues with Sugiyama et al’s popular method [4].

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Since there is no algorithm that runs over multiple layers in

Each graph in Figure 1 represents the subgraph of union

Layered drawing idea first started with Warfield [3] and

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Figure 3: Each graph is represented as a layered drawing by using Sugiyama Layout with weighted values.

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Cycle Removal

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Layering Assignment

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Assigning node positions in layer a. Adding dummy nodes b. Crossing minimization

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Assigning edge bends

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Finalizing the layout

1. Process Scheduling Processes come with fashion of layered drawing. By modifying the original algorithm of layering phase, we can receive the suggested output for scheduling. However, since the environment is living and real time, updating the layered graph is important.

2. Protein Sequencing 1.Cycle Removal Cycle removal is essential for the Sugiyama Layout. In order to run layering algorithm, the whole graph should be acycic. Feedback Arcset Problem approach by Demetrescu et.al. could solve that problem. The problem in its original has unit weight acceptance. We have to adopt it to weighted feedback arcset problem.

2.Layering Assignment There are two main algorithms presented before. These are Coffman Graham Layering and Longest Path Layering. We

By identifying the relations among different proteins, we can produce the layering drawing according to their relations.

3. Company Social Graph By considering company as a union graph such as each departments have own subgraphs as relations with the employees. We can design a tool in order to produce social graph that shows interactions between company branches as well as inter employee relations. For example, we accept each subgraphs in Figure 3 as

References [1] A. Yamaguchi and A. Sugimoto. An approximation algorithm for the two-layered graph drawing problem. In Proceedings of 5th Annual Intl. Conf. on Computing and Combinatorics (COCOON ’99), pages 81–91. LNCS, Springer, 1999. [2] C. Demestrescu and I. Finocchi. Breaking cycles for minimizing crossings. J. Exp. Algorithmics, 6:2, 2001. [3] J. Warfield, Crossing Theory and hierarchy mapping, IEEE Transactions on Systems, Man, and Cybonetrics, SMc 7(7):502-523,1977. [4] K. Sugiyama, S. Tagawa, and M. Toda. Methods for visual understanding of hierarchical system structures. IEEE Transactions on Syst. Man. and Cybernetics, 11(2):109–125, 1981. [5]O. A. Çakiroğlu, C. Erten, Ö. Karatas, and M. Sözdinler. Crossing minimization in weighted bipartite graphs. In LNCS 4525, Proc. of Workshop on Experimental Algorithms WEA, pages 122–135, 2007. [6] X. Y. Li and M. F. Stallmann. New bounds on the barycenter heuristic for bipartite graph drawing. Inf. Process. Lett., 82(6):293–298, 2002.

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