optimum interval routing in k-caterpillars and maximal outer planar networks
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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
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PCDN Innsbruck, Austria Feb., 2003
Outline of the talk
Research Contexto Message Passing Networkso Explicit vs. Implicit Routingo Interval Routing Scheme
Main Contributionso Optimal Interval Routing in
K-Caterpillars Maximal Outer Planar Nets. Open Question, References
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PCDN Innsbruck, Austria Feb., 2003
Message Passing Networks
Co-operating parallel processes share computation by way of message passingo Example: MPI processes interface
provides– MPI_Send();– MPI_Recv();
Different from the shared memory multiprocessing
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PCDN Innsbruck, Austria Feb., 2003
Routing Schemes
Explicit RoutingRouting Tables
Implicit RoutingLabeling nodes of
• chain, • mesh, • hypercube,• CCC, etc…
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Compare the following two Labeling Schemes for a chain
5 2 3 1 N 4N-1
3 N-11 2 4 5 N
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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
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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
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PCDN Innsbruck, Austria Feb., 2003
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
3 N-11 2 4 5 N
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PCDN Innsbruck, Austria Feb., 2003
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
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Interval Routing Scheme-Main Idea
{S(i)
(i)
L(s) < j <= L(s+1)
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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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PCDN Innsbruck, Austria Feb., 2003
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
0 0
0 0
1
2
3
4 5
6
7
8 9
10
11
*
1112
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15
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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 (k+1)-
complete sub-graphs) when all the k-leaves (nodes with degree k) are removed
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PCDN Innsbruck, Austria Feb., 2003
Example: 2-Caterpillar
1
2
3
4
56
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A[1,2]
B[1,2]
C[1,2] D[2,3]
E[2,3] F[3,4]
G[5,8] H[7,9]
I[7,9]
J[7,8]
K[6,8]L[6,8]
1
23
4
5
6 9
78
7 8
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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
R
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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
R
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MST of Example MOP rooted at the Median
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3 4
5
678 9
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Conclusion
New optimal algorithm for k-caterpillars and maximal outer planar networks.
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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
PCDN Innsbruck, Austria Feb., 2003
Thank you
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