collective operation

In all asymptotic runtime functions, we denote the latency $\backslash alpha$ (or startup time per message, independent of message size), the communication cost per word $\backslash beta$, the number of processing units $p$ and the input size per node $n$. In cases where we have initial messages on more than one node we assume that all local messages are of the same size. To address individual processing units we use $p\_i\; \backslash in\; \backslash \{\; p\_0,\; p\_1,\; \backslash dots,\; p\_\{p\; -\; 1\}\; \backslash \}$.

If we do not have an equal distribution, i.e. node $p\_i$ has a message of size $n\_i$, we get an upper bound for the runtime by setting $n\; =\; \backslash max(n\_0,\; n\_1,\; \backslash dots,\; n\_\{p-1\})$.

A distributed memory model is assumed. The concepts are similar for the shared memory model. However, shared memory systems can provide hardware support for some operations like broadcast ({{section link||Broadcast}}) for example, which allows convenient concurrent read.Sanders, Mehlhorn, Dietzfelbinger, Dementiev 2019, p. 395 Thus, new algorithmic possibilities can become available.

{{Main|Broadcast (parallel pattern)}}

File:Broadcast (collective operation).png

The broadcast patternSanders, Mehlhorn, Dietzfelbinger, Dementiev 2019, pp. 396-401 is used to distribute data from one processing unit to all processing units, which is often needed in SPMD parallel programs to dispense input or global values. Broadcast can be interpreted as an inverse version of the reduce pattern ({{section link||Reduce}}). Initially only root $r$ with $id$ $0$ stores message $m$. During broadcast $m$ is sent to the remaining processing units, so that eventually $m$ is available to all processing units.

Since an implementation by means of a sequential for-loop with $p-1$ iterations becomes a bottleneck, divide-and-conquer approaches are common. One possibility is to utilize a binomial tree structure with the requirement that $p$ has to be a power of two. When a processing unit is responsible for sending $m$ to processing units $i..j$, it sends $m$ to processing unit $\backslash left\; \backslash lceil\; (i+j)/2\; \backslash right\; \backslash rceil$ and delegates responsibility for the processing units $\backslash left\; \backslash lceil\; (i+j)/2\; \backslash right\; \backslash rceil\; ..\; j$ to it, while its own responsibility is cut down to $i..\backslash left\; \backslash lceil\; (i+j)/2\; \backslash right\; \backslash rceil-1$.

Binomial trees have a problem with long messages $m$. The receiving unit of $m$ can only propagate the message to other units, after it received the whole message. In the meantime, the communication network is not utilized. Therefore pipelining on binary trees is used, where $m$ is split into an array of $k$ packets of size $\backslash left\; \backslash lceil\; n/k\; \backslash right\; \backslash rceil$. The packets are then broadcast one after another, so that data is distributed fast in the communication network.

Pipelined broadcast on balanced binary tree is possible in $\backslash mathcal\{O\}(\backslash alpha\; \backslash log\; p\; +\; \backslash beta\; n)$, whereas for the non-pipelined case it takes $\backslash mathcal\{O\}((\backslash alpha\; +\; \backslash beta\; n)\; \backslash log\; p)$ cost.

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{{Main|Reduce (parallel pattern)}}

File:Reduce.png

The reduce patternSanders, Mehlhorn, Dietzfelbinger, Dementiev 2019, pp. 402-403 is used to collect data or partial results from different processing units and to combine them into a global result by a chosen operator. Given $p$ processing units, message $m\_i$ is on processing unit $p\_i$ initially. All $m\_i$ are aggregated by $\backslash otimes$ and the result is eventually stored on $p\_0$. The reduction operator $\backslash otimes$ must be associative at least. Some algorithms require a commutative operator with a neutral element. Operators like $sum$, $min$, $max$ are common.

Implementation considerations are similar to broadcast ({{section link||Broadcast}}). For pipelining on binary trees the message must be representable as a vector of smaller object for component-wise reduction.

Pipelined reduce on a balanced binary tree is possible in $\backslash mathcal\{O\}(\backslash alpha\; \backslash log\; p\; +\; \backslash beta\; n)$.

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File:All-Reduce.png

The all-reduce patternSanders, Mehlhorn, Dietzfelbinger, Dementiev 2019, pp. 403-404 (also called allreduce) is used if the result of a reduce operation ({{section link||Reduce}}) must be distributed to all processing units. Given $p$ processing units, message $m\_i$ is on processing unit $p\_i$ initially. All $m\_i$ are aggregated by an operator $\backslash otimes$ and the result is eventually stored on all $p\_i$. Analog to the reduce operation, the operator $\backslash otimes$ must be at least associative.

All-reduce can be interpreted as a reduce operation with a subsequent broadcast ({{section link||Broadcast}}). For long messages a corresponding implementation is suitable, whereas for short messages, the latency can be reduced by using a hypercube ({{section link|Hypercube (communication pattern)|All-Gather/ All-Reduce}}) topology, if $p$ is a power of two. All-reduce can also be implemented with a butterfly algorithm and achieve optimal latency and bandwidth.{{Cite journal |last=Yuan |first=Xin |date=February 2009 |title=Bandwidth optimal all-reduce algorithms for clusters of workstations |url=https://www.cs.fsu.edu/~xyuan/paper/09jpdc.pdf |journal=Journal of Parallel and Distributed Computing |volume=69 |issue=2}}

All-reduce is possible in $\backslash mathcal\{O\}(\backslash alpha\; \backslash log\; p\; +\; \backslash beta\; n)$, since reduce and broadcast are possible in $\backslash mathcal\{O\}(\backslash alpha\; \backslash log\; p\; +\; \backslash beta\; n)$ with pipelining on balanced binary trees. All-reduce implemented with a butterfly algorithm achieves the same asymptotic runtime.

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{{Main|Prefix sum}}

File:Prefix-Sum (Scan).png

The prefix-sum or scan operationSanders, Mehlhorn, Dietzfelbinger, Dementiev 2019, pp. 404-406 is used to collect data or partial results from different processing units and to compute intermediate results by an operator, which are stored on those processing units. It can be seen as a generalization of the reduce operation ({{section link||Reduce}}). Given $p$ processing units, message $m\_i$ is on processing unit $p\_i$. The operator $\backslash otimes$ must be at least associative, whereas some algorithms require also a commutative operator and a neutral element. Common operators are $sum$, $min$ and $max$. Eventually processing unit $p\_i$ stores the prefix sum $m\_\{i\text{'}\}$. In the case of the so-called exclusive prefix sum, processing unit $p\_i$ stores the prefix sum $m\_\{i\text{'}\}$. Some algorithms require to store the overall sum at each processing unit in addition to the prefix sums.

For short messages, this can be achieved with a hypercube topology if $p$ is a power of two. For long messages, the hypercube ({{section link|Hypercube (communication pattern)|Prefix sum}}, {{section link|Prefix sum|Distributed memory: Hypercube algorithm}}) topology is not suitable, since all processing units are active in every step and therefore pipelining can't be used. A binary tree topology is better suited for arbitrary $p$ and long messages ({{section link|Prefix sum|Large Message Sizes: Pipelined Binary Tree}}).

Prefix-sum on a binary tree can be implemented with an upward and downward phase. In the upward phase reduction is performed, while the downward phase is similar to broadcast, where the prefix sums are computed by sending different data to the left and right children. With this approach pipelining is possible, because the operations are equal to reduction ({{section link||Reduce}}) and broadcast ({{section link||Broadcast}}).

Pipelined prefix sum on a binary tree is possible in $\backslash mathcal\{O\}(\backslash alpha\; \backslash log\; p\; +\; \backslash beta\; n)$.

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{{Main|Barrier (computer science)}}

The barrierSanders, Mehlhorn, Dietzfelbinger, Dementiev 2019, p. 408 as a collective operation is a generalization of the concept of a barrier, that can be used in distributed computing. When a processing unit calls barrier, it waits until all other processing units have called barrier as well. Barrier is thus used to achieve global synchronization in distributed computing.

One way to implement barrier is to call all-reduce ({{section link||All-Reduce}}) with an empty/ dummy operand. We know the runtime of All-reduce is $\backslash mathcal\{O\}(\backslash alpha\; \backslash log\; p\; +\; \backslash beta\; n)$. Using a dummy operand reduces size $n$ to a constant factor and leads to a runtime of $\backslash mathcal\{O\}(\backslash alpha\; \backslash log\; p)$.

File:Gather.png

The gather communication patternSanders, Mehlhorn, Dietzfelbinger, Dementiev 2019, pp. 412-413 is used to store data from all processing units on a single processing unit. Given $p$ processing units, message $m\_i$ on processing unit $p\_i$. For a fixed processing unit $p\_j$, we want to store the message $m\_1\; \backslash cdot\; m\_2\; \backslash cdot\; \backslash ldots\; \backslash cdot\; m\_p$ on $p\_j$. Gather can be thought of as a reduce operation ({{section link||Reduce}}) that uses the concatenation operator. This works due to the fact that concatenation is associative. By using the same binomial tree reduction algorithm we get a runtime of $\backslash mathcal\{O\}(\backslash alpha\; \backslash log\; p\; +\; \backslash beta\; p\; n)$. We see that the asymptotic runtime is similar to the asymptotic runtime of reduce $\backslash mathcal\{O\}(\backslash alpha\; \backslash log\; p\; +\; \backslash beta\; n)$, but with the addition of a factor p to the term $\backslash beta\; n$. This additional factor is due to the message size increasing in each step as messages get concatenated. Compare this to reduce where message size is a constant for operators like $min$.

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File:All-Gather.png

The all-gather communication pattern is used to collect data from all processing units and to store the collected data on all processing units. Given $p$ processing units $p\_i$, message $m\_i$ initially stored on $p\_i$, we want to store the message $m\_1\; \backslash cdot\; m\_2\; \backslash cdot\; \backslash ldots\; \backslash cdot\; m\_p$ on each $p\_j$.

It can be thought of in multiple ways. The first is as an all-reduce operation ({{section link||All-Reduce}}) with concatenation as the operator, in the same way that gather can be represented by reduce. The second is as a gather-operation followed by a broadcast of the new message of size $pn$. With this we see that all-gather in $\backslash mathcal\{O\}(\backslash alpha\; \backslash log\; p\; +\; \backslash beta\; p\; n)$ is possible.

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File:Scatter.png

The scatter communication patternSanders, Mehlhorn, Dietzfelbinger, Dementiev 2019, p. 413 is used to distribute data from one processing unit to all the processing units. It differs from broadcast, in that it does not send the same message to all processing units. Instead it splits the message and delivers one part of it to each processing unit.

Given $p$ processing units $p\_i$, a fixed processing unit $p\_j$ that holds the message $m\; =\; m\_1\; \backslash cdot\; m\_2\; \backslash cdot\; \backslash ldots\; \backslash cdot\; m\_p$. We want to transport the message $m\_i$ onto $p\_i$. The same implementation concerns as for gather ({{section link||Gather}}) apply. This leads to an optimal runtime in $\backslash mathcal\{O\}(\backslash alpha\; \backslash log\; p\; +\; \backslash beta\; p\; n)$.

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{{Main|All-to-all (parallel pattern)}}

All-to-allSanders, Mehlhorn, Dietzfelbinger, Dementiev 2019, pp. 413-418 is the most general communication pattern. For , message $m\_\{i,\; j\}$ is the message that is initially stored on node $i$ and has to be delivered to node $j$. We can express all communication primitives that do not use operators through all-to-all. For example, broadcast of message $m$ from node $p\_k$ is emulated by setting $m\_\{i,\; j\}\; =\; m$ for $i\; =\; k$ and setting $m\_\{l,\; j\}$ empty for $l\; \backslash neq\; k$.

Assuming we have a fully connected network, the best possible runtime for all-to-all is in $\backslash mathcal\{O\}(p\; (\backslash alpha\; +\; \backslash beta\; n))$ . This is achieved through $p$ rounds of direct message exchange. For $p$ power of 2, in communication round $k$ , node $p\_i$ exchanges messages with node $p\_j,\; j\; =\; i\; \backslash oplus\; k$ .

If the message size is small and latency dominates the communication, a hypercube algorithm can be used to distribute the messages in time $\backslash mathcal\{O\}(\backslash log\; p\; (\backslash alpha\; +\; \backslash beta\; p\; n))$ .File:All-to-All.png

This tableSanders, Mehlhorn, Dietzfelbinger, Dementiev 2019, p. 394 gives an overview over the best known asymptotic runtimes, assuming we have free choice of network topology.

Example topologies we want for optimal runtime are binary tree, binomial tree, hypercube.

In practice, we have to adjust to the available physical topologies, e.g. dragonfly, fat tree, grid network (references other topologies, too).

More information under Network topology.

For each operation, the optimal algorithm can depend on the input sizes $n$. For example, broadcast for short messages is best implemented using a binomial tree whereas for long messages a pipelined communication on a balanced binary tree is optimal.

The complexities stated in the table depend on the latency $\backslash alpha$ and the communication cost per word $\backslash beta$ in addition to the number of processing units $p$ and the input message size per node $n$. The # senders and # receivers columns represent the number of senders and receivers that are involved in the operation respectively. The # messages column lists the number of input messages and the Computations? column indicates if any computations are done on the messages or if the messages are just delivered without processing. Complexity gives the asymptotic runtime complexity of an optimal implementation under free choice of topology.

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{{Reflist}}

{{cite book|last1=Sanders|first1=Peter|title=Sequential and Parallel Algorithms and Data Structures - The Basic Toolbox|last2=Mehlhorn|first2=Kurt|last3=Dietzfelbinger|first3=Martin|last4=Dementiev|first4=Roman|date=2019|publisher=Springer Nature Switzerland AG|isbn=978-3-030-25208-3|authorlink1=Peter Sanders (computer scientist)|authorlink2=Kurt Mehlhorn}}

Category:Parallel computing

Category:Algorithms

Category:Distributed computing