Optimum Interval Routing in k-Caterpillars and Maximal Outer Planar Networks

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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

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Outline of the talk

• Research Context
• Message Passing Networks
• Explicit vs. Implicit Routing
• Interval Routing Scheme
• Main Contributions
• Optimal Interval Routing in
• K-Caterpillars
• Maximal Outer Planar Nets.
• Open Question, References

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Message Passing Networks

• Co-operating parallel processes share computation
  by way of message passing
• Example MPI processes interface provides
• MPI_Send()
• MPI_Recv()
• Different from the shared memory multiprocessing

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Routing Schemes

• Explicit Routing
• Routing Tables
• Implicit Routing
• Labeling nodes of
• chain,
• mesh,
• hypercube,
• CCC, etc

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Compare the following two Labeling Schemes for a
chain
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Observation1

• 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

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Observation2

• 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

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Observation3

• 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

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Interval Routing

• A type of implicit routing
• Introduced by Santoro
• SK1985, The Computer Journal
• Work by Van Leeuwan, Fraigniaud
• LT1987, The Computer Journal
• FG1998, Algorithmica
• Not optimal in general
• PR1991, The Computer Journal
• Present Research
• GSA2003, PCDN 2003

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Interval Routing Scheme-Main Idea
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Interval Routing Scheme-Main Idea
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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

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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

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Example 4-tree
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Definition Caterpillar

• A Caterpillar is a tree which results into a path
  when all the leaves are removed

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Example Caterpillar
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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 (k1)-complete sub-graphs)
  when all the k-leaves (nodes with degree k) are
  removed

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Example 2-Caterpillar
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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

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Example Maximal Outer Planar Network
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MOP as Intersection Graph of sub-trees of a tree
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Definition Median

• A node is a median if the average distance from
  every other node is minimized.

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Dual of the Example Maximal Outer Planar Network
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MST of Example MOP rooted at the Median
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Conclusion

• New optimal algorithm for k-caterpillars and
  maximal outer planar networks.

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References

• SK1985 Labeling and Implicit Routing in
  Networks, Nocola Santoro and Ramez Khatib, The
  Computer Journal, Vol 28, No.1, 1985.
• LT1987 Interval Routing, J. Van Leeuwen and
  R.B.Tan, The Computer Journal, Vol 30, No.4,
  1987.
• FG1998 Interval Routing Schemes, P. Fraigniaud
  and C. Gavoille, Algorithmica, (1998) 21
  155-182.
• PR1991 Short Note on efficiency of Interval
  Routing, P. Ruzicka, The Computer Journal, Vol
  34, No.5, 1991.
• GSA2003 Gur Saran Adhar, PCDN2003

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Thank you