SESSION GRAPH BASED AND TREE METHODS + RELATED ...

26 Int'l Conf. Foundations of Computer Science | FCS'12 |
slave processor performs a fixed number of GRASP iterations.
Once all processors have finished their computations, the best
solution is collected by the master processor. The results of
computational experiments illustrate the effectiveness of the
proposed parallel GRASP procedure for the Steiner problem
in graphs. Verhoeven and Severens proposed sequential and
parallel local search methods for the Steiner tree problem
based on a novel neighborhood. They claimed their approach
is “better" than those known in the literature. Computational
results indicated that good speedups could be obtained without
loss in solution quality [18].
4 Minimum Spanning Tree Heuristic
Algorithm
In the minimum spanning tree heuristic (MSTH)
suggested by Takahashi and Matsuyama [19], the solution
푇 ���� is obtained by deleting from the minimum spanning
tree for 퐺 non-terminals of degree 1 (one at a time). The
worst-case time complexity of the minimum spanning tree
heuristic is 푂 ( 푒 + 푣 푙표푔 푣). The worst-case error ratio
|푇 ����| ⁄ |푇 �(푁)| is tightly bounded by 푣 − 푛 + 1. Hence,
the minimum spanning tree heuristic can be considered as
inferior to the shortest paths heuristic in the worst-case sense.
It also performs poorly on average. The proposed heuristic
algorithm in this paper consists of eight steps. In the first step,
after assuming that 푇 ����� is equal with null, we
obtained 푇 ���� by graph 퐺.
5 Improve Minimum Spanning Tree
Heuristic Algorithm
5.1 Description of the IMSTH Algorithm
The proposed heuristic algorithm in this paper consists
of eight steps. In the first step, after assuming that 푇 ����� is
equal with null, we obtained 푇 ���� by graph 퐺.
In the second step of algorithm, we assume two divisions of
edges and paths between two terminals in 푇 ����:
� Existence direct edge in 푇 ����
� Not existence direct edge in 푇 ���� and existence
direct edge in graph.
In the third step, for the first category, select the direct
edges of two terminals which obtained by 푇 ���� and add into
푇 �����. in the fourth step, for the second category, make
compare between paths and direct edges of two terminals:
� If direct edge be shortest, add into 푇 �����.
� If path be shortest, add into 푇 �����.
In the next step, we study 푇 �����, to determine all disjoin
terminals. In the sixth step obtained the shortest paths between
disjoin terminal and other terminals in the graph with Dijkstra
algorithm. Select shortest path between reached paths and add
into 푇 ����� .
In the next step, we study 푇 �����, for connectivity. If we
have forest, determine terminals with degree 1, then seek
shortest path from this terminal to other terminals, and add it
into 푇 ����� .
At the end, study 푇 for cycle. If we have cycle, delete cycle.
Fig. 1 shows our suggested algorithm.
Suggested algorithm: (INPUT: 퐺 = (푉, 퐸) where 푤: 퐸 → 푅 + ,
푁 ⊆ 푉set of Terminals, OUTPUT: 푇 ����� approximate optimal tree) {
Step 0: 푇����� → ∅
Step 1: Compute 푇 ���� of the graph 퐺.
Step 2: Assume two divisions of edges and paths, between two
terminals in 푇 ����:
a) Existence direct edge in 푇 ����
b) Not existence direct edge in 푇 ���� and existence direct edge in
graph
Step 3: For the first category, select the direct edges of two terminals
which obtained by 푇 ���� and add into 푇 �����.
Step 4: For the second category, make compare between paths and
direct edges of two terminals:
a) If direct edge be shortest, add into 푇 �����.
b) If path be shortest, add into 푇 �����.
Step 5: Study 푇 �����, to determine all disjoin terminals.
Step 6: Determine the shortest paths between disjoin terminal and other
terminals in the 푇 ����� with Dijkstra algorithm. Select shortest path
between reached paths and add into 푇 ����� .
Step7: Study 푇 �����, for connectivity. If we have forest, determine
terminals with degree 1, seek shortest path from this terminal to other
terminals, and add it into the 푇 ����� .
Step 8: Study 푇 ����� for cycle. If we have cycle, delete cycle.}
Fig. 1: Suggested algorithm for SPG in graph
5.2 Illustrative Example
In the example graph shown in Figure 2a, vertices 푣0
, 푣3 ,푣4, 푣6 and 푣8 are terminal vertices. We construct 푇 ����
by this graph (Fig. 2b). The cost of 푇 ���� is equal with 808.
It can be observed that between two terminals 푣3, 푣6
and 푣0, 푣8 exists a direct edge in 푇 ����, so the direct
edges< 푣3, 푣6> and < 푣0, 푣8> is selected and add into 푇 퐼푀푆푇퐻
. at the other hand, between two terminals 푣3 and 푣4 a direct
edge in graph exists that is shortest than the paths between this
two terminals, so in the fourth step the direct edge < 푣3, 푣4>
is selected and add into the 푇 ����� (Fig. 2c). In the next step,
we study 푇 ����� for disjoin terminals but don’t find any
disjoin terminal.
Now Study 푇 �����, for connectivity. It is observed that we
have forest. so determine terminals with degree 1, seek
shortest path from this terminal to other terminals, and add it
into the 푇 ����� (Fig. 2d). At the end, the cost of 푇 ����� is
equal with 738(Fig. 2e). Fig. 2 illustrates steps of our
algorithm.