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CS 662 Theory of Parallel Algorithms

Networks

[To Lecture Notes Index]

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

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

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
- Total number of communication links

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

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 |