## CS 662 Theory of Parallel Algorithms Networks part 3

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San Diego State University -- This page last updated April 18, 1996

### Shuffle-Exchange Network

The r-dimensional shuffle-exchange graph has:
nodes
edges
Label the nodes from 0 to -1 in binary

Two nodes U and V are connected directly by an edge if either:
U and V differ in only in the last bit (exchange edge)
U is a left or right cyclic shift of V (shuffle edge)
2-dimensional shuffle-exchange
3-dimensional shuffle-exchange

Shuffle-Exchange Simulation of Hypercube Algorithms

A normal hypercube algorithm is one in which:
only one dimension of edges is used per step
only consecutive dimensions are used in consecutive steps
each step can consist of computation and a communication down edges of a single dimension

Theorem
An normal algorithm that runs in T(n) steps on an r-dimension hypercube can be run in 2T(n) - 1 steps on an r-dimension shuffle-exchange

proof:
Assume that the first dimension of edges used in the algorithm is K
Map the process at hypercube node to shuffle-exchange node
Now processes linked by K-dimension edges in hypercube are connected by exchange edges in shuffle-exchange
So run first step of algorithm
Now next step of algorithm either uses K+1 or K-1 dimension edges of hypercube
If K+1, map process stored in node to node by using shuffle edges.
This take one step
Now run the next step of algorithm
If K-1, map process stored in node to node by using shuffle edges in reverse direction
Hence each "step" of the hypercube algorithm takes 2 "steps" on the shuffle-exchange except the first step

Example - Hypercube Broadcast on Shuffle-Exchange