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Fig. 6. Graph of 2500 nodes shown on levels 3 and 5 of the decomposition tree

Figure 6 Graph of 2500 nodes shown on levels 3 and 5 of the decomposition tree

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\ clustering of a graph, G = (V, F) consists of a partition V = Vz U V2 U....L /, of the node set of G. Graph theoretic clustering is the process of formin, ‘lusters based on the structure of the graph [22,29,23,6,24,30]. The usual aim i o form clusters that exhibit a high cohesiveness and a low coupling; these may b neasured, for example, by the number of intra-cluster edges and the number o nter-cluster edges respectively. For example, the graph with the adjacency tabl n Figure 1 has been divided into two clusters with only one edge coupling them Methods for graph theoretic clustering, motivated by applications in paralle computing, data mining, and VLSI design are well developed [6,24,30]. Fig. 1. Graph Theoretic, single linkage clustering example
\ clustering of a graph, G = (V, F) consists of a partition V = Vz U V2 U....L /, of the node set of G. Graph theoretic clustering is the process of formin, ‘lusters based on the structure of the graph [22,29,23,6,24,30]. The usual aim i o form clusters that exhibit a high cohesiveness and a low coupling; these may b neasured, for example, by the number of intra-cluster edges and the number o nter-cluster edges respectively. For example, the graph with the adjacency tabl n Figure 1 has been divided into two clusters with only one edge coupling them Methods for graph theoretic clustering, motivated by applications in paralle computing, data mining, and VLSI design are well developed [6,24,30]. Fig. 1. Graph Theoretic, single linkage clustering example
Fig. 2. Voronoi diagram based on pseudnodes, real nodes shown as points
Fig. 2. Voronoi diagram based on pseudnodes, real nodes shown as points
Fig. 3. Nono-tree space decomposition and data structure A systema tic and recursive division of space into a series of cells, called a tree code in the astrophysics literature [2,25,18,4,1], can be used to speed up n-body force calcu ations. 1 The force on an individual particle from other particles close by is, on average, evaluated by direct particle-to-particle interaction, whereas the force due to more di were first developed stant particles is included as a group contribution. Tree codes in the context of galactic simulations involving hundreds of billions of bodies. Tree codes may be based on regular (Quad-tree, Oct-tree) and irregular ( decomposi Voronot) space decompositions. A nono-tree (a nine-way recursive tion), is i in linear time. lustrated in Figure 3. Regular trees such as this can be built
Fig. 3. Nono-tree space decomposition and data structure A systema tic and recursive division of space into a series of cells, called a tree code in the astrophysics literature [2,25,18,4,1], can be used to speed up n-body force calcu ations. 1 The force on an individual particle from other particles close by is, on average, evaluated by direct particle-to-particle interaction, whereas the force due to more di were first developed stant particles is included as a group contribution. Tree codes in the context of galactic simulations involving hundreds of billions of bodies. Tree codes may be based on regular (Quad-tree, Oct-tree) and irregular ( decomposi Voronot) space decompositions. A nono-tree (a nine-way recursive tion), is i in linear time. lustrated in Figure 3. Regular trees such as this can be built
Fig. 4. Comparing node 5 and 6 with the North-West pseudonode, where s/d=0.80 and s/d=1.46 respectively Using a quad-tree space decomposition and a value of 1.0 for @ the fixed- tolerance parameter, some examples are given for the node set shown on the left of Figure 4.
Fig. 4. Comparing node 5 and 6 with the North-West pseudonode, where s/d=0.80 and s/d=1.46 respectively Using a quad-tree space decomposition and a value of 1.0 for @ the fixed- tolerance parameter, some examples are given for the node set shown on the left of Figure 4.
Table 1. Experimental Comparison of tree-code Vs direct force calculation
Table 1. Experimental Comparison of tree-code Vs direct force calculation
Fig. 5. Cluster measure for levels 4, 6, 8 in the clustering
Fig. 5. Cluster measure for levels 4, 6, 8 in the clustering
Fig. 7. Graph of 2500 nodes shown on level 6 and the lowest level of the decomposition tree
Fig. 7. Graph of 2500 nodes shown on level 6 and the lowest level of the decomposition tree
Fig. 8. Cycle of 400 nodes shown on levels 2 and 3 with the space decompositior overlaid
Fig. 8. Cycle of 400 nodes shown on levels 2 and 3 with the space decompositior overlaid
Fig. 9. Cycle of 400 nodes shown on level 5 and the lowest level, with the space de- composition overlaid
Fig. 9. Cycle of 400 nodes shown on level 5 and the lowest level, with the space de- composition overlaid
Fig. 11. Graph of 4700 nodes shown on level 8 and the lowest level of the decomposition tree Fig. 12. Graph of 1400 nodes shown on levels 4 and 6 of the decomposition tree
Fig. 11. Graph of 4700 nodes shown on level 8 and the lowest level of the decomposition tree Fig. 12. Graph of 1400 nodes shown on levels 4 and 6 of the decomposition tree
Fig. 10. Graph of 4700 nodes shown on levels 3 and 5 of the decomposition tree
Fig. 10. Graph of 4700 nodes shown on levels 3 and 5 of the decomposition tree
Fig. 13. Graph of 1400 nodes shown on level 8 and the lowest level of the decomposition tree
Fig. 13. Graph of 1400 nodes shown on level 8 and the lowest level of the decomposition tree
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Computer ScienceTheoretical Computer ScienceGraph DrawingLevels of AbstractionFast AlgorithmDecomposition method
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