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2007/5/29 Presenter : Wei-Ni Chu, Chuan-Ju Wang, Wei-Yang Chen

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

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2007/5/29 Presenter : Wei-Ni Chu, Chuan-Ju Wang, Wei-Yang Chen

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  1. Algorithms for the optimum communication spanning tree problemPrabha SharmaAnn 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.)

  9. Algorithm OCST I (cont.)

  10. Algorithm OCST I (cont.) • 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.)

  23. Algorithm OCST II (cont.)

  24. Algorithm OCST II (cont.) • 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

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