SESSION GRAPH BASED AND TREE METHODS + RELATED ...

32 Int'l Conf. Foundations of Computer Science | FCS'12 |
At the end of this phase (lines 23 23- 30), there are
omissions of repeated edges of �, and repeated nodes of �.
Moreover, Steiner nodes with the deg less than 2 are also
omitted from �, , and their edges from �. Finally, all the
edges in the set � are the edges of the Steiner tree, tree and the
summation of their weights is the cost of the Steiner tree.
3.4 Time Complexity Analysis
In MSTG algorithm, the most time complexity belongs
to the computations using Dijkstra’s algorithm algorithm. By using
Fibonacci-Heap Heap for implementing Dijkstra’s algorithm, it
has O�� � � log �� time complexity [3].
In the preprocessing phase of the MSTG algorithm,
Dijkstra’s algorithm hasn’t been used. . In the first phase,
Dijkstra’s algorithm has been used for � times and in the
second phase, at the worst condition it has been used for �
times; therefore because � ≤ �, the time complexity of our
algorithm is O���� � � log ��.
4 Experimental Results
The implementation of the proposed algorithm that
called MSTG has been done by Visual C# C#, and it has been
examined on some well-known data sets like the Beasley’s Beasley
data set [12]. The configuration of the system that has been
used for this examination was a 2.50 GHz CPU and a 3 GB
RAM. The results of running the MSTG algorithm on the
sets B, C and D of Beasley’s data set are respectively in
tables 1, 2 and 3. The rate of this algorithm is computed
from the ratio of the cost of MSTG to the optimum cost and
also the time of execution has been show shown in “hour: minute:
second: mille second”.
Table 1: The results of MSTG algorithm on the set B
Graph
Number
1 B
2 B
3 B
4 B
5 B
6 B
7 B
8 B
9 B
10 B
11 B
12 B
13 B
14 B
15 B
16 B
17 B
18 B
Nodes
Count
50
50
50
50
50
50
75
75
75
75
75
75
100
100
100
100
100
100
Edges
Count
63
63
63
100
100
100
94
94
94
150
150
150
125
125
125
200
200
200
Terminals
Count
9
13
25
9
13
25
13
19
38
13
19
38
17
25
50
17
25
50
Optimum
Cost
82
83
138
59
61
122
111
104
220
86
88
174
165
235
318
127
131
218
MSTG
Result
82
83
138
59
61
122
111
104
220
86
92
174
170
235
321
132
131
218
Rate
(Opt/MSTG Opt/MSTG)
1
1
1
1
1
1
1
1
1
1
1.045
1
1.03
1
1.009
1.039
1
1
Time
(h:m:s:ms)
0: 0: 0: 10
0: 0: 0: 16
0: 0: 0: 33
0: 0: 0: 15
0: 0: 0: 20
0: 0: 0: 50
0: 0: 0: 28
0: 0: 0: 37
0: 0: 0: 101
0: 0: 0: 54
0: 0: 0: 66
0: 0: 0: 131
0: 0: 0: 69
0: 0: 0: 123
0: 0: 0: 220
0: 0: 0: 125
0: 0: 0: 168
0: 0: 0:350
Table 2: The results of MSTG algorithm on the set C
Graph
Number
1 C
2 C
3 C
4 C
5 C
6 C
7 C
8 C
9 C
10 C
11 C
12 C
13 C
14 C
15 C
16 C
17 C
18 C
19 C
20 C
Nodes
Count
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
Edges
Count
625
625
625
625
625
1000
1000
1000
1000
1000
2500
2500
2500
2500
2500
12500
12500
12500
12500
12500
Terminals
Count
5
10
83
125
250
5
10
83
125
250
5
10
83
125
250
5
10
83
125
250
Optimum
Cost
85
144
754
1079
1579
55
102
509
707
1093
32
46
258
323
556
11
18
113
146
267
Table 3: The results of MSTG algorithm on the set D
Graph
Number
1 D
2 D
3 D
4 D
5 D
6 D
7 D
8 D
9 D
10 D
11 D
12 D
13 D
14 D
15 D
16 D
17 D
18 D
19 D
20 D
Nodes
Count
Edges
Count
1000 1250
1000 1250
1000 1250
1000 1250
1000 1250
1000 2000
1000 2000
1000 2000
1000 2000
1000 2000
1000 5000
1000 5000
1000 5000
1000 5000
1000 5000
1000 25000
1000 25000
1000 25000
1000 25000
1000 25000
Terminals
Count
5
10
167
250
500
5
10
167
250
500
5
10
167
250
500
5
10
167
250
500
Optimum
Cost
106
220
1565
1935
3250
67
103
1072
1448
2110
29
42
500
667
1116
13
23
223
310
537
MSTG
Result
85
144
760
1090
1581
55
102
519
720
1100
33
49
262
330
561
12
20
119
152
268
MSTG
Result
107
220
1578
1951
3265
73
105
1097
1470
2120
29
42
517
676
1129
15
24
237
328
542
Rate
(Opt/MSTG)
1
1
1.008
1.01
1.001
1
1
1.019
1.018
1.006
1.031
1.065
1.015
1.021
1.009
1.09
1.111
1.0531
1.041
1.003
Rate
(Opt/MSTG)
1.009
1
1.008
1.008
1.004
1.09
1.019
1.023
1.015
1.005
1
1
1.034
1.013
1.012
1.154
1.043
1.063
1.058
1.009
Time
(h:m:s:ms)
0: 0: 0: 547
0: 0: 1: 306
0: 0: 4: 203
0: 0: 7: 578
0: 0: 17: 409
0: 0: 0: 969
0: 0: 1: 281
0: 0: 7: 640
0: 0: 10: 625
0: 0: 22: 110
0: 0: 1: 265
0: 0: 1: 579
0: 0: 7: 547
0: 0: 11: 500
0: 0: 25: 656
0: 0: 1: 901
0: 0: 2: 891
0: 0: 6: 937
0: 0: 9: 312
0: 0: 37: 16
Time
(h:m:s:ms)
0: 0: 2: 890
0: 0: 4: 47
0: 0: 42: 515
0: 0: 52: 016
0: 4: 4: 484
0: 0: 5: 782
0: 0: 7: 719
0: 0: 50: 907
0: 1: 23: 578
0: 5: 9: 631
0: 0: 6: 734
0: 0: 7: 375
0: 0: 55: 94
0: 1: 24: 375
0: 5: 27: 762
0: 0: 28: 690
0: 0: 30: 336
0: 1: 47: 301
0: 2: 38: 931
0: 4: 56: 388
In Fig. 4, there are two samples of obtained Steiner
trees from MSTG algorithm on graphs B1 B and C15 that has
been drawn with random vertices.
(a)
(b)
Fig.4: Two samples of MSTG algorithm on graph
B1 (a) and graph C15 (b)