Hypercube (communication pattern)

Most of the communication primitives presented in this article share a common template.Foster, I.(1995). Designing and Building Parallel Programs: Concepts and Tools for Parallel Software Engineering. Addison Wesley; {{ISBN|0201575949}}. Initially, each processing element possesses one message that must reach every other processing element during the course of the algorithm. The following pseudo code sketches the communication steps necessary. Hereby, Initialization, Operation, and Output are placeholders that depend on the given communication primitive (see next section).

Input: message $m$.

Output: depends on Initialization, Operation and Output.

Initialization

$s\; :=\; m$

for do

$y\; :=\; i\; \backslash text\{\; XOR\; \}\; 2^k$

Send $s$ to $y$

Receive $m$ from $y$

Operation$(s,\; m)$

endfor

Output

Each processing element iterates over its neighbors (the expression $i\; \backslash text\{\; XOR\; \}\; 2^k$ negates the $k$-th bit in $i$'s binary representation, therefore obtaining the numbers of its neighbors). In each iteration, each processing element exchanges a message with the neighbor and processes the received message afterwards. The processing operation depends on the communication primitive.

File:Hypergraph Communication Pattern.png

In the beginning of a prefix sum operation, each processing element $i$ owns a message $m\_i$. The goal is to compute $\backslash bigoplus\_\{0\; \backslash le\; j\; \backslash le\; i\}\; m\_j$, where $\backslash oplus$ is an associative operation. The following pseudo code describes the algorithm.

Input: message $m\_i$ of processor $i$.

Output: prefix sum $\backslash bigoplus\_\{0\; \backslash le\; j\; \backslash le\; i\}\; m\_j$ of processor $i$.

$x\; :=\; m\_i$

$\backslash sigma\; :=\; m\_i$

for $0\; \backslash le\; k\; \backslash le\; d\; -\; 1$ do

$y\; :=\; i\; \backslash text\{\; XOR\; \}\; 2^k$

Send $\backslash sigma$ to $y$

Receive $m$ from $y$

$\backslash sigma\; :=\; \backslash sigma\; \backslash oplus\; m$

if bit $k$ in $i$ is set then $x\; :=\; x\; \backslash oplus\; m$

endfor

The algorithm works as follows. Observe that hypercubes of dimension $d$ can be split into two hypercubes of dimension $d\; -\; 1$. Refer to the sub cube containing nodes with a leading 0 as the 0-sub cube and the sub cube consisting of nodes with a leading 1 as 1-sub cube. Once both sub cubes have calculated the prefix sum, the sum over all elements in the 0-sub cube has to be added to the every element in the 1-sub cube, since every processing element in the 0-sub cube has a lower rank than the processing elements in the 1-sub cube. The pseudo code stores the prefix sum in variable $x$ and the sum over all nodes in a sub cube in variable $\backslash sigma$.

This makes it possible for all nodes in 1-sub cube to receive the sum over the 0-sub cube in every step.

This results in a factor of $\backslash log\; p$ for $T\_\backslash text\{start\}$ and a factor of $n\backslash log\; p$ for $T\_\backslash text\{byte\}$: $T(n,p)\; =\; (T\_\backslash text\{start\}\; +\; nT\_\backslash text\{byte\})\backslash log\; p$.

File:Hypergraph Communication Steps for Prefix Sum.png

All-gather operations start with each processing element having a message $m\_i$. The goal of the operation is for each processing element to know the messages of all other processing elements, i.e. $x\; :=\; m\_0\; \backslash cdot\; m\_1\; \backslash dots\; m\_p$ where $\backslash cdot$ is concatenation. The operation can be implemented following the algorithm template.

Input: message $x\; :=\; m\_i$ at processing unit $i$.

Output: all messages $m\_1\; \backslash cdot\; m\_2\; \backslash dots\; m\_p$.

$x\; :=\; m\_i$

for do

$y\; :=\; i\; \backslash text\{\; XOR\; \}\; 2^k$

Send $x$ to $y$

Receive $x\text{'}$ from $y$

$x\; :=\; x\; \backslash cdot\; x\text{'}$

endfor

With each iteration, the transferred message doubles in length. This leads to a runtime of $T(n,p)\; \backslash approx\; \backslash sum\_\{j=0\}^\{d-1\}(T\_\backslash text\{start\}\; +\; n\; \backslash cdot\; 2^jT\_\backslash text\{byte\})=\; \backslash log(p)\; T\_\backslash text\{start\}\; +\; (p-1)nT\_\backslash text\{byte\}$.

The same principle can be applied to the All-Reduce operations, but instead of concatenating the messages, it performs a reduction operation on the two messages. So it is a Reduce operation, where all processing units know the result. Compared to a normal reduce operation followed by a broadcast, All-Reduce in hypercubes reduces the number of communication steps.

Here every processing element has a unique message for all other processing elements.

Input: message $m\_\{ij\}$ at processing element $i$ to processing element $j$.

for $d\; >\; k\; \backslash geq\; 0$ do

Receive from processing element $i\; \backslash text\{\; XOR\; \}\; 2^k$:

all messages for my $k$-dimensional sub cube

Send to processing element $i\; \backslash text\{\; XOR\; \}\; 2^k$:

all messages for its $k$-dimensional sub cube

endfor

With each iteration a messages comes closer to its destination by one dimension, if it hasn't arrived yet. Hence, all messages have reached their target after at most $d\; =\; \backslash log\{p\}$ steps. In every step, $p\; /\; 2$ messages are sent: in the first iteration, half of the messages aren't meant for the own sub cube. In every following step, the sub cube is only half the size as before, but in the previous step exactly the same number of messages arrived from another processing element.

This results in a run-time of $T(n,p)\; \backslash approx\; \backslash log\{p\}\; (T\_\backslash text\{start\}\; +\; \backslash frac\{p\}\{2\}nT\_\backslash text\{byte\})$.

The ESBT-broadcast (Edge-disjoint Spanning Binomial Tree) algorithm{{cite journal|last1=Johnsson|first1=S.L.|last2=Ho|first2=C.-T.|title=Optimum broadcasting and personalized communication in hypercubes|journal=IEEE Transactions on Computers|volume=38|issue=9|year=1989|pages=1249–1268|issn=0018-9340|doi=10.1109/12.29465}} is a pipelined broadcast algorithm with optimal runtime for clusters with hypercube network topology. The algorithm embeds $d$ edge-disjoint binomial trees in the hypercube, such that each neighbor of processing element $0$ is the root of a spanning binomial tree on $2^d\; -\; 1$ nodes. To broadcast a message, the source node splits its message into $k$ chunks of equal size and cyclically sends them to the roots of the binomial trees. Upon receiving a chunk, the binomial trees broadcast it.

In each step, the source node sends one of its $k$ chunks to a binomial tree. Broadcasting the chunk within the binomial tree takes $d$ steps. Thus, it takes $k$ steps to distribute all chunks and additionally $d$ steps until the last binomial tree broadcast has finished, resulting in $k\; +\; d$ steps overall. Therefore, the runtime for a message of length $n$ is $T(n,\; p,\; k)\; =\; \backslash left(\backslash frac\{n\}\{k\}\; T\_\backslash text\{byte\}\; +\; T\_\backslash text\{start\}\; \backslash right)\; (k\; +\; d)$. With the optimal chunk size $k^*\; =\; \backslash sqrt\{\backslash frac\{nd\; \backslash cdot\; T\_\backslash text\{byte\}\}\{T\_\backslash text\{start\}\}\}$, the optimal runtime of the algorithm is $T^*(n,\; p)\; =\; n\; \backslash cdot\; T\_\backslash text\{byte\}\; +\; \backslash log(p)\; \backslash cdot\; T\_\backslash text\{start\}\; +\; \backslash sqrt\{n\; \backslash log(p)\; \backslash cdot\; T\_\backslash text\{start\}\; \backslash cdot\; T\_\backslash text\{byte\}\}$.

File:HypergraphESBT.png

This section describes how to construct the binomial trees systematically. First, construct a single binomial spanning tree von $2^d$ nodes as follows. Number the nodes from $0$ to $2^d\; -\; 1$ and consider their binary representation. Then the children of each nodes are obtained by negating single leading zeroes. This results in a single binomial spanning tree. To obtain $d$ edge-disjoint copies of the tree, translate and rotate the nodes: for the $k$-th copy of the tree, apply a XOR operation with $2^k$ to each node. Subsequently, right-rotate all nodes by $k$ digits. The resulting binomial trees are edge-disjoint and therefore fulfill the requirements for the ESBT-broadcasting algorithm.

• Hypercube internetwork topology

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Category:Parallel computing