1. Algorithms for the optimum communication spanning tree problem Prabha Sharma Ann Oper Res (2006) 143:203-209 2007/5/29 Presenter : Wei-Ni Chu, Chuan-Ju Wang, Wei-Yang Chen

2. Outline Introduction Notation Algorithm OCST I Example Algorithm OCST II

3. Introduction Optimum Communication Spanning Tree Problem (OCSTP) –Given a graph. –A set of requirements, which represent the volume of communication between the nodes and. –A set of distances for all. –The cost of communication for the tree T

4. Introduction (cont.) Johnson, Lenstra and Rinooy Kan(1978) have shown the OCSTP even with all is NP-hard. Hu(1974) considered the OCSTP on a complete graph. –Optimal Requirement Spanning Tree –Optimal Distance Spanning tree He proved that if the satisfy a generalized triangle inequality, the optimum tree will be a star tree.

5. Introduction (cont.) Concept of a near optimum tree –A tree is said to be near optimum if its communication cost is better than the cost of all trees which differ from it in one arc only. Two algorithms for constructing near optimum trees are given. –Algorithm OCSTP I is a pseudo-polynomial algorithm. –Algorithm OCSTP II construct a near optimum tree in.

6. Notation

7. Algorithm OCST I Begin by keeping all equal to the minimum. By Hu’s (1974) result, the optimal communication spanning tree in this case is a cut tree and can be constructed in time. Increase the value of each tree arc one at a time and maintain a near optimal tree. When all arcs have attained their true values we stop.

8. Algorithm OCST I (cont.)

10. Algorithm OCST I is an algorithm –Step1 –Step3 Computation for each may be repeated at most times. For each computation for, has to be computed for each and for each and tree with minimum cost has to be found. Total computations for steps3 are, where is equal to the largest of the values.

11. Algorithm OCST I (cont.) We define We know that each iteration It means

12. Algorithm OCST I (cont.) Because and, is near optimum with.

13. Algorithm OCST I (cont.) Problem –Is always ? –Why the computation for each may be repeated at most times? –In each iteration, can be removed?

14. Example Consider our problem on a complete graph –Three nodes A,B,C –,, –We use sum-requirement to compute the communication cost of the tree.

15. Example (cont.) Under this case, optimal solution can be easily found.

16. Example (cont.) Apply OCST I Step 1: –Construct a tree with for all

17. Example (cont.) –Apply Hu’s Optimal Requirement Spanning Tree, and we will get an optimal tree T. Step 2: –Assign

18. Example (cont.) Step 3: –In the first iteration, –And compute cost for all possible adjacent trees

19. Example (cont.) –Choose – –,go to step 3 –We found that no, such that. –We are done. Step4: –Output T as a near optimal tree.

20. Algorithm OCST II Generalized triangle inequality –Let be the distances associated with three sides of any triangle formed by three nodes in the n- node network (n>4). –Let. –If there exist a positive not larger than such that for all triangles in the network, we say that the distances of the network satisfy the generalized triangle inequality.

21. Algorithm OCST II (cont.) Hu (1974) proved that if the distances satisfy the generalized triangle inequality and the are all equal, the OCST is a star tree. Begin by keeping all equal to the smallest value. One at a time is increased and critical values such that for all in. The process has to be repeated till all non-tree arcs attain their true values.

22. Algorithm OCST II (cont.)

24. Validity of OCST II

25. Algorithm OCST II (cont.) Validity of OCST II –The critical value is chosen such that –Also are adjacent trees. Thus, it follows that is a near optimum tree for –By lemma, it follows that for the, if, then can be increased to its true value without disturbing the near optimality of

26. Algorithm OCST II (cont.) Algorithm OCST II is an algorithm –For each arc not in the star tree, there are at most (n-1) critical values, since for each critical value, the path is one arc less than the number of arcs in –To compute all the critical values,where h < n –There are m-(n-1) non-tree arcs in the beginning, and step 3 may have to be performed for all these arcs. –Total complexity of the algorithm is