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A Criterion for Cost Optimal Construction of Irregular Networks

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### A Criterion forCost Optimal Construction ofIrregular Networks

Geir Horn, Olav Lysne and Tor Skeie

Classical Question

Given a set of switches, and a set of nodes what is the best performance you can get when connecting these components?

G. Horn et al: Cost Optimal Construction of Irregular Networks (CAC'03)

Our Reverse Question

Given a set of nodeshow should the networkand its switches be constructed to best support these nodes?

G. Horn et al: Cost Optimal Construction of Irregular Networks (CAC'03)

Our Question Today

Given the need to connect a set of nodes, and a traffic pattern:

- What is the least number of switches necessary?
- What is the minimal size of each switch necessary?
- What topology should be used?

G. Horn et al: Cost Optimal Construction of Irregular Networks (CAC'03)

Assumptions

- Uniform traffic distribution
- Uniform distribution of nodes over the switches
- No parallel links

G. Horn et al: Cost Optimal Construction of Irregular Networks (CAC'03)

The Switch Size

Number of nodes hosted

Number of links to the network

Intuitively:

G. Horn et al: Cost Optimal Construction of Irregular Networks (CAC'03)

The Number of Switches

Maximum number of links to the network on any switch

G. Horn et al: Cost Optimal Construction of Irregular Networks (CAC'03)

Combined Optimum

Minimise the maximal switch degree

- Smallest possible switches
- Least number of switches

G. Horn et al: Cost Optimal Construction of Irregular Networks (CAC'03)

The Solution

- Size of biggest switch,

- Number of links on each switch to the network

Algorithmic mapping

- Input:
- Even number of nodes,

- Output:
- Number of switches,

G. Horn et al: Cost Optimal Construction of Irregular Networks (CAC'03)

It is always possible to construct a networkwith these switches

Example Topology

Complete

Part

3

5

1

4

2

6

G. Horn et al: Cost Optimal Construction of Irregular Networks (CAC'03)

Topological Properties

- Complete graphs (networks) whenever

- Exponentially growing number of isomorphic classes for network sizes immediately following a complete network
- Example: is complete, and has 22 isomorphic classes

G. Horn et al: Cost Optimal Construction of Irregular Networks (CAC'03)

- Always a complete part of the network
- Maximum shortest path is of length two

Simulated Topologies

1

2

3

G. Horn et al: Cost Optimal Construction of Irregular Networks (CAC'03)

5

7

4

6

Complete part

Saturation point = 63.5%

Saturation point = 63.5%

Saturation point = 57.2%

63.5%

63.5%

57.2%

Domain of Applicability

G. Horn et al: Cost Optimal Construction of Irregular Networks (CAC'03)

Conclusions & Further work

- Work-in-progress report
- Optimality = Simultaneously minimise
- The number of switches
- The size of the switches
- Algorithmic solution

G. Horn et al: Cost Optimal Construction of Irregular Networks (CAC'03)

- Further work:
- Solutions for fixed size switches?
- How to select the best performing network?
- Scalability for clustered traffic?

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