### Table 2. Probabilistic Cell-Graphs

### TABLE II PROBABILISTIC CELL-GRAPHS

### Table 1: Experimental results for some DIMACS vertex coloring graphs, some probabilistic networks and frequency assignment graph celar03-pp-001

"... In PAGE 12: ... The program did not use parallelism. In Table1 the results of our experiments on a number of graphs are reported. Besides instance name, number of vertices, number of edges, and the computed treewidth, we report on the CPU time in seconds and the maximum number of sets (S, r), considered at once, max |TW| =maxi=0,.... ..."

### Table 1: Experimental results for some DIMACS vertex coloring graphs, some probabilistic networks and frequency assignment graph celar03-pp-001

2006

"... In PAGE 12: ... The program did not use parallelism. In Table1 the results of our experiments on a number of graphs are reported. Besides instance name, number of vertices, number of edges, and the computed treewidth, we report on the CPU time in seconds and the maximum number of sets (S, r), considered at once, max |T W | = maxi=0,.... ..."

Cited by 7

### Table 1: Experimental results for some DIMACS vertex coloring graphs, some probabilistic networks and frequency assignment graph celar03-pp-001

"... In PAGE 12: ... The program did not use parallelism. In Table1 the results of our experiments on a number of graphs are reported. Besides instance name, number of vertices, number of edges, and the computed treewidth, we report on the CPU time in seconds and the maximum number of sets (S, r), considered at once, max |T W | = maxi=0,.... ..."

### Table 2 (a): The used processing time in seconds for graphs from probabilistic networks and frequency assignment instances.

"... In PAGE 15: ... Table 1(b) gives graphs from the DIMACS coloring benchmark. Table2 (a) and Table 2(b) give the processing times in seconds. These are given for each of the heuristics, and each of the graphs that were used in Table 1(a) and Table 1 (b).... In PAGE 17: ... Table2 (b): The used processing time in seconds for DIMACS vertex coloring instances. The upper bound produced by the heuristic is equal to the best one The upper bound produced by the heuristic is worse than the best one S Heuristic name No.... ..."

### Table 4.1: Available statistics on the explanation graphs, on learning, and on the probabilistic inference other than learning

### Table 1 (a): The upper bounds produced by the heuristics described in this paper for some graphs from probabilistic networks and frequency assignment instances.

"... In PAGE 15: ... The results of implementing these algorithms on different graphs are given in the following tables. Table1 (a) and Table 1(b) show a comparison between different methods illustrated in this paper from the point view of their best upper bound values. Table 1(a) consists of instances of probabilistic networks and frequency assignment problems.... In PAGE 15: ... Table 1(a) and Table 1(b) show a comparison between different methods illustrated in this paper from the point view of their best upper bound values. Table1 (a) consists of instances of probabilistic networks and frequency assignment problems. Table 1(b) gives graphs from the DIMACS coloring benchmark.... In PAGE 15: ... Table 1(a) consists of instances of probabilistic networks and frequency assignment problems. Table1 (b) gives graphs from the DIMACS coloring benchmark. Table 2(a) and Table 2(b) give the processing times in seconds.... In PAGE 15: ... Table 2(a) and Table 2(b) give the processing times in seconds. These are given for each of the heuristics, and each of the graphs that were used in Table1 (a) and Table 1 (b). Times are rounded to the nearest integer.... In PAGE 15: ... Table 2(a) and Table 2(b) give the processing times in seconds. These are given for each of the heuristics, and each of the graphs that were used in Table 1(a) and Table1 (b). Times are rounded to the nearest integer.... In PAGE 16: ...16 The main differences between the results obtained by MFEO1 and those obtained by RATIO2 are as follows: ! The results of applying the MFEO1 heuristic on graphs of probabilistic networks and frequency assignment instances in Table1 (a) are better than or equal to those produced by any other heuristic in that table. However, this is not the case for some of the instances from the DIMACS coloring benchmark, see Table 1(b).... In PAGE 16: ...16 The main differences between the results obtained by MFEO1 and those obtained by RATIO2 are as follows: ! The results of applying the MFEO1 heuristic on graphs of probabilistic networks and frequency assignment instances in Table 1(a) are better than or equal to those produced by any other heuristic in that table. However, this is not the case for some of the instances from the DIMACS coloring benchmark, see Table1 (b). We notice when we consider Table 1(b) that the upper bounds for some graphs are better when using RATIO2 than those obtained when using MFEO1.... In PAGE 16: ... However, this is not the case for some of the instances from the DIMACS coloring benchmark, see Table 1(b). We notice when we consider Table1 (b) that the upper bounds for some graphs are better when using RATIO2 than those obtained when using MFEO1. ! The results of MFEO1 are more stable, always better than or equal to that produced by any other heuristic, except for RATIO2.... In PAGE 16: ...Table 1 (a): The upper bounds produced by the heuristics described in this paper for some graphs from probabilistic networks and frequency assignment instances. S Graph name |V| |E| LB MF EMF MFEO1 MFEO2 RATIO1 RATIO2 1 anna 138 986 11 12 12 12 12 12 12 2 david 87 812 11 13 13 13 13 13 13 3 dsjc125_1 125 736 17 65 65 64 64 64 66 4 dsjc125_5 125 3891 55 111 111 110 110 110 109 5 dsjc250_1 250 3218 3 177 177 177 177 177 177 6 games120 120 1276 10 39 39 39 42 41 38 7 LE450_5A 450 5714 3 315 315 310 315 315 304 8 mulsol_i_4 175 3946 32 32 32 32 32 32 32 9 myciel4 23 71 10 11 11 10 10 10 10 10 myciel5 47 236 8 21 21 20 20 20 20 11 myciel6 95 755 20 35 35 35 35 35 35 12 myciel7 191 2360 31 66 66 66 66 66 66 13 queen5_5 25 320 12 18 18 18 18 18 19 14 school1 385 19095 80 225 225 225 225 225 209 Table1 (b): The upper bounds produced by the heuristics described in this paper for some DIMACS vertex coloring instances. The second aspect of interest of the implementation of the heuristics is the processing ... In PAGE 17: ...17 (b) give the details. Table 3 summarizes the behavior of the heuristics given in this paper on the graphs used in Table1 (a) and Table 1(b). For each heuristic, it gives the number of times from each set of instances when the heuristic gives the best result, and when its result is smaller than the best result of all five heuristics.... ..."

### Table 3: Probabilistic network characteristics

"... In PAGE 18: ...2 Probabilistic networks We continue our computational study with 8 real-life probabilistic networks from the field of medicine. Table3 shows the origin of the instances and the size of the network. The most effi- cient way to compute the inference in a probabilistic network is by the use of the junction-tree propagation algorithm of Lauritzen and Spiegelhalter [24].... In PAGE 18: ... The moralization of a network (or directed graph) D = (V, A) is the undirected graph G = (V, E) obtained from D by adding edges between every pair of non-adjacent vertices that have a common successor, and then dropping arc directions. The size of the edge set E is also reported in Table3 . After the application of pre-processing... In PAGE 18: ...Table 3: Probabilistic network characteristics techniques for computing the treewidth [8], an additional four instances to conduct our heuris- tics on are available. The size of these four instances is reported in Table3 as well. After [8], henceforth we refer to these instances as instancename {3,4}.... ..."

### Table 2: Comparison of performance on practical examples;; the probabilistic

1998

"... In PAGE 8: ... We also tried a modi ed version of the EA which rst runs APGAN and then inserts the computed topological sort into the initial population. Table2 shows the results of applying GDPPO to the schedules generated by the various heuristics on several practical SDF graphs;; the satellite re- ceiver example is taken from [16], whereas the other examples are the same as considered in [3]. The probabilistic algorithms ran once on each graph and were aborted after 3000 tness evaluations.... In PAGE 8: ... Additionally, an exhaustive searchwithamaximum run-time of 1 hour was carried out;; as it only com- pleted in two cases 3 , the search spaces of these problems seem to be rather complex. In all of the practical benchmark examples in Table2 the results achieved by the EA equal or surpass those generated by RPMC. Compared to APGAN on these practical examples, the EA is neither inferior nor superior;; it shows both better and worse performance in two cases each.... ..."

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