optimum interval routing in k-caterpillars and maximal outer planar networks

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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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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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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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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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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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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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

13

14

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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Example: 2-Caterpillar

1

2

3

4

56

9

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

12

3 4

5

678 9

10

11

12

13

14

15

16

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18

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21

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24 25

26

33


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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


Thank you

optimum interval routing in k-caterpillars and maximal outer planar networks

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