1. PCDN Innsbruck, Austria Feb., 2003 Optimum Interval Routing in k-Caterpillars and Maximal Outer Planar Networks Gur Saran Adhar Department of Computer Science University of North Carolina at Wilmington, USA

2. PCDN Innsbruck, Austria Feb., 2003 2 Outline of the talk  Research Context oMessage Passing Networks oExplicit vs. Implicit Routing oInterval Routing Scheme  Main Contributions oOptimal Interval Routing in K-Caterpillars Maximal Outer Planar Nets. Open Question, References

3. PCDN Innsbruck, Austria Feb., 2003 3 Message Passing Networks  Co-operating parallel processes share computation by way of message passing oExample: MPI processes interface provides –MPI_Send(); –MPI_Recv();  Different from the shared memory multiprocessing

4. PCDN Innsbruck, Austria Feb., 2003 4 Routing Schemes  Explicit Routing Routing Tables  Implicit Routing Labeling nodes of chain, mesh, hypercube, CCC, etc…

5. PCDN Innsbruck, Austria Feb., 2003 5 Compare the following two Labeling Schemes for a chain

6. PCDN Innsbruck, Austria Feb., 2003 6 Observation:1  First labeling defines a total order on the nodes in the chain  Second labeling does not define a total order  Each node receives a unique label

7. PCDN Innsbruck, Austria Feb., 2003 7 Observation:2  A chain (one-path) is an alternating sequence of: node (a complete set of size one) followed by an edge (a complete set of size two).  Adjacent edges share exactly one node

8. PCDN Innsbruck, Austria Feb., 2003 8 Observation:3  A chain represents an intersection relationship between INTERVALS on a real line.  A chain is a special tree and the individual INTERVALS its sub-trees  A route is essentially linking the sub- trees

9. PCDN Innsbruck, Austria Feb., 2003 9 Interval Routing  A type of implicit routing  Introduced by Santoro –SK:1985, The Computer Journal  Work by Van Leeuwan, Fraigniaud –LT:1987, The Computer Journal –FG:1998, Algorithmica  Not optimal in general –PR:1991, The Computer Journal  Present Research –GSA:2003, PCDN 2003

10. PCDN Innsbruck, Austria Feb., 2003 10 Interval Routing Scheme-Main Idea

11. PCDN Innsbruck, Austria Feb., 2003 11 Interval Routing Scheme-Main Idea

12. PCDN Innsbruck, Austria Feb., 2003 12 Recursive Definition: tree  Basis: one node is a tree  Recursive Step: adding a new node by joining to one node in the graph already constructed also results in a tree

13. PCDN Innsbruck, Austria Feb., 2003 13 Recursive Definition: K-tree  Basis: A Complete graph on k nodes is a K-tree  Recursive Step: adding a new node to every node in a complete sub-graph of order k in the graph already constructed also results in a K-tree

14. PCDN Innsbruck, Austria Feb., 2003 14 Example: 4-tree

15. PCDN Innsbruck, Austria Feb., 2003 15 Definition: Caterpillar  A Caterpillar is a tree which results into a path when all the leaves are removed

16. PCDN Innsbruck, Austria Feb., 2003 16 Example: Caterpillar

17. PCDN Innsbruck, Austria Feb., 2003 17 Definition: K-Caterpillar  A K-Caterpillar is a k-tree which results into a k-path (an alternating sequence of k complete sub-graphs followed by (k+1)-complete sub-graphs) when all the k-leaves (nodes with degree k) are removed

18. PCDN Innsbruck, Austria Feb., 2003 18 Example: 2-Caterpillar

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21. PCDN Innsbruck, Austria Feb., 2003 21 Definition: Maximal Outer Planar Network (MOP)  A network is outer planar if it can be embedded on a plane so that all nodes lie on the outer face  A outer planar network is maximal outer planar which has maximum number of edges

22. PCDN Innsbruck, Austria Feb., 2003 22 Example: Maximal Outer Planar Network

23. PCDN Innsbruck, Austria Feb., 2003 23 MOP as Intersection Graph of sub- trees of a tree

24. PCDN Innsbruck, Austria Feb., 2003 24 Definition: Median  A node is a median if the average distance from every other node is minimized.

25. PCDN Innsbruck, Austria Feb., 2003 25 Dual of the Example Maximal Outer Planar Network

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32. PCDN Innsbruck, Austria Feb., 2003 32 MST of Example MOP rooted at the Median

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34. PCDN Innsbruck, Austria Feb., 2003 34 Conclusion  New optimal algorithm for k-caterpillars and maximal outer planar networks.

35. PCDN Innsbruck, Austria Feb., 2003 35 References [SK:1985] Labeling and Implicit Routing in Networks, Nocola Santoro and Ramez Khatib, The Computer Journal, Vol 28, No.1, 1985. [LT:1987] Interval Routing, J. Van Leeuwen and R.B.Tan, The Computer Journal, Vol 30, No.4, 1987. [FG:1998] Interval Routing Schemes, P. Fraigniaud and C. Gavoille, Algorithmica, (1998) 21: 155-182. [PR:1991] Short Note on efficiency of Interval Routing, P. Ruzicka, The Computer Journal, Vol 34, No.5, 1991. {GSA:2003] Gur Saran Adhar, PCDN’2003

36. PCDN Innsbruck, Austria Feb., 2003 Thank you