### Table 3: Running time (in seconds) of Steiner tree in graph solvers (POWER3 instances). Groups: Number of groups. GeoSteiner4: Running time of GeoSteiner4. Jack3-Hanan: Running time of Jack3 using complete Hanan grid. Jack3-FST: Running time of Jack3 using reduced Hanan grid. Red: Reduction of running time in percent.

"... In PAGE 21: ... The average Hanan grid reduction is about 60%, so just 40% of the nodes and edges in the Hanan grid remain in the instance we feed to the Steiner tree in graph solver. Table3 shows the running time of the graph solvers using the three di erent types of input. In the last column the speed-up of the runtime comparing the Hanan grid and the reduced grid graph (for all the problems that were solved) can be found.... ..."

### Table 1: Beasley results for the rst set of the testbeds. There are three groups of graphs with sizes 50, 75, and 100 nodes, respectively. cost of the Steiner approximation tree produced by the heuristic to either the cost of the optimal tree, if that is available, or the cost of the best-known approximate solution to the original graph.

"... In PAGE 8: ...597 2.231 Table1 0: Simulation results using random graphs with... In PAGE 9: ...811 1.677 Table1 1: Simulation results using random graphs with 200 nodes and 50 multicast nodes. = :8 TM OSPH ADH NADH SADH WP 2 18 30 56 37 .... In PAGE 9: ...954 2.165 Table1 2: Simulation results using random graphs with 200 nodes and 50 multicast nodes. = :2 TM OSPH ADH NADH SADH WP 1 14 33 56 42 .... In PAGE 9: ...030 0.974 Table1 3: Simulation results using random graphs with 200 nodes and 100 multicast nodes. = :4 TM OSPH ADH NADH SADH WP 0 8 44 49 51 .... In PAGE 9: ...951 2.291 Table1 4: Simulation result using random graphs with 200 nodes and 100 multicast nodes. = :6 TM OSPH ADH NADH SADH WP 2 10 43 47 56 .... In PAGE 9: ...873 1.128 Table1 5: Simulation results using random graphs with 200 nodes and 100 multicast nodes. = :8 TM OSPH ADH NADH SADH WP 0 11 32 52 43 .... In PAGE 9: ...119 1.301 Table1 6: Simulation results using random graphs with 200 nodes and 100 multicast nodes. jSj TM OSPH ADH NADH SADH WP 24 53 56 42 60 10 APE 2.... In PAGE 9: ...779 2.291 Table1 7: Simulation results using random graphs with... In PAGE 10: ...207 7.451 Table1 8: Simulation results using random graphs for all random graph used (4800 graphs total). 6 CONCLUSION A exible simulation framework to study the perfor- mance of multicasting algorithms in di erent topolo- gies, was proposed.... ..."

### Table 4: Running time (in seconds) of Steiner tree in graph solvers (POWER4 instances). (The instance with 21 groups could not be solved using Jack3 for unknown technical reasons.) Groups: Number of groups. GeoSteiner4: Running time of GeoSteiner4. Jack3-Hanan: Running time of Jack3 us- ing complete Hanan grid. Jack3-FST: Running time of Jack3 using reduced Hanan grid. Red: Reduction of running time in percent.

"... In PAGE 21: ... On most instances Jack3 using reduced Hanan grid input was the fastest, but some instances could only be solved using GeoSteiner4. The results we got for the nets from the POWER4 chip were comparable (see Table4 ). The instances had between 11 and 76 groups; all but 2 could be solved.... ..."

### Table 2: Full Steiner tree generation and pruning.

1997

"... In PAGE 22: ...y Beasley and Go net [6] (Section 5.3). 5.1 Full Steiner Tree Generation and Pruning The performance of the full Steiner tree generation method is summarized in Table2 . The algorithm uses the Gabriel graph to nd subsets containing up to K = 5 terminals and bottleneck Steiner distances to prune FSTs.... ..."

Cited by 4

### Table 4: Steiner vs Shortest Paths Trees

2003

"... In PAGE 113: ... The parameters were the same as used in the romrOn case described above. The results, given in Table4 , showed that the successful delivery of packets to the receivers was very similar in both of the two cases, but other statistics gathered indicate the superiority of the use of the Steiner trees. The average amount of time a tree set remained active before a new tree set was distributed was 8% shorter in the shortest paths trees than in the Steiner tree version.... ..."

Cited by 3

### Table 1: Steiner tree approximation algorithms

1999

"... In PAGE 1: ... But only few of them have prov- ably good performance ratios. Table1 gives a survey on such results. (The algorithm of Promel and Steger [11] is a randomized algorithm, all other algorithms listed in the table are deterministic algorithms.... In PAGE 2: ... This idea naturally extends to adding shortest connections between k-tuples of terminals, for xed k. All approximation algorithms for the Steiner tree prob- lem listed in Table1 are based on this simple idea. The present approach to get better performance ratios for the Steiner tree problem in graphs is to iteratively apply a series of algorithms to the output of its predecessor.... ..."

Cited by 21

### Table 1: Steiner tree approximation algorithms

1999

"... In PAGE 1: ... But only few of them have prov- ably good performance ratios. Table1 gives a survey on such results. (The algorithm of Promel and Steger [11] is a randomized algorithm, all other algorithms listed in the table are deterministic algorithms.... In PAGE 2: ... This idea naturally extends to adding shortest connections between k-tuples of terminals, for xed k. All approximation algorithms for the Steiner tree prob- lem listed in Table1 are based on this simple idea. The present approach to get better performance ratios for the Steiner tree problem in graphs is to iteratively apply a series of algorithms to the output of its predecessor.... ..."

Cited by 21

### Table 3. Results obtained on the instances derived from Steiner series E.

"... In PAGE 18: ...ur new approach solved them in 0.2 (69.1) seconds to optimality. All the instances of K,P,C and D groups are solved in the root node of the branch-and-cut tree. Table3 presents the results of our new approach on the set E of benchmark in- stances. As before, the instance name, the number of edges of the original graph and the size of the instance after preprocessing as well as the preprocessing CPU time are listed.... ..."

### TABLE I STEINER TREE HEURISTIC STATISTICS

1992

Cited by 96

### Table 3: N is the total number of metabolic proteins in the pathway. C is the total number co-expressed genes. |V | is the size of the Steiner tree (number of nodes). If the Steiner tree is disconnected, we examined only the tree in this forest with the maximum number of leaves L. R is the average size of a Steiner tree with L leaves.

2008

"... In PAGE 16: ... The pathways chosen can be divided into two groups based on their proximity to the galactose pathway in the yeast metabolic network. Table3 lists these pathways. The Close 16 by on February 6, 2008 www.... ..."