## CS 662 Theory of Parallel Algorithms Networks

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San Diego State University -- This page last updated March 28, 1996, 1996

## Interconnection Networks

Static Interconnection Networks
Processors are connected directly to other processors via point to point communication links
Dynamic Interconnection Networks
Processors are connected dynamically via switches

### Static Interconnection Networks

#### Evaluating Static Interconnection Networks

Distance
Shortest path between two processors
Diameter
Maximum distance between any two processors

Connectivity
Measure of the multiplicity between two processors
Arc Connectivity
Minimum number of communication links that can be removed to break it into two disconnected networks
Bisection Width
Minimum number of communication links that can be removed to break it into two equal sized disconnected networks

Channel Width
Number of bits that can be sent simultaneously over a communication link
Number of wires in the communication link

Channel Rate
Peak rate at which a single wire can deliver bits

Channel bandwidth
Peak rate at which a communication link can deliver bits

Bisection bandwidth
Bisection Width * Channel bandwidth

Cost

Degree of a Processor
Number of communication links connected to a processor

Expandability of network
How much work is needed to add processors to an existing network

#### Types of Static Interconnection Networks

Completely-Connected network

Star Network

Linear Array and Ring
CDC Cyperplus
Mesh Network

Mesh with wraparound

Three-dimensional Mesh

Two-Dimension Mesh Machines
DAP, Paragon
Three-Dimension Mesh Machines
Cray T3D, J-Machine
Tree Network

Dyanamic Tree

Dyanamic Fat Tree

CM-5 uses a fat tree network
Hypercube

A d-dimensional hypercube consists of P = processors
Recursive Definition

A zero-dimensional hypercube is a single processor

A 1-dimensional hypercube is constructed by connecting two zero-dimensional hypercubes

A (d+1)-dimensional hypercube is constructed by connecting the corresponding processors of two d-dimensional hypercubes
Hypercube Examples

Binary Bit Definition

Label the processors 0 through in binary.

Two processors are directly connected if and only if the binary representation of their labels differ at exactly one bit position

Hamming Distance

Let s and t be processors in a d-dimensional hypercube.

The total number of bit positions at which these labels of s and t differ is the Hamming distance between them

The length of the shortest path between s and t is equal to the Hamming distance between them

Characteristics of Networks with p Processors
 Network Diameter Bisection Arc Number of Width Connectivity Links Completey-connected 1 p-1 p(p-1)/2 Star 2 1 1 p-1 Complete binary tree 2lg((p+1)/2) 1 1 p-1 Linear array p-1 1 1 p-1 ring 2 2 p 2-D mesh no wrap 2 2-D mesh with wrap p/2 log p 2p Hypercube log(p) p/2 log p (plog(p))/2