Prefix sum

Programming language and library implementations of scan may be either inclusive or exclusive. An inclusive scan includes input x_i when computing output y_i (i.e., ${\textstyle y_{i}=\bigoplus _{j=0}^{i}x_{j}}$) while an exclusive scan does not (i.e., ${\textstyle y_{i}=\bigoplus _{j=0}^{i-1}x_{j}}$). In the latter case, implementations either leave y₀ undefined or accept a separate "x₋₁" value with which to seed the scan. Either type of scan can be transformed into the other: an inclusive scan can be transformed into an exclusive scan by shifting the array produced by the scan right by one element and inserting the identity value at the left of the array. Conversely, an exclusive scan be transformed into an inclusive scan by shifting the array produced by the scan left and inserting the sum of the last element of the scan and the last element of the input array at the right of the array.[8]

The following table lists examples of the inclusive and exclusive scan functions provided by a few programming languages and libraries:

The directive-based OpenMP parallel programming model supports both inclusive and exclusive scan support beginning with Version 5.0.

Circuit representation of a highly parallel 16-input parallel prefix sum

Hillis and Steele present the following parallel prefix sum algorithm:[9]

for i <- 0 to floor(log2(n)) do
    for j <- 0 to n - 1 do in parallel
        if j < 2i then
            xi+1
j <- xi
j
        else
            xi+1
j <- xi
j + xi
j - 2i

In the above, the notation ${\displaystyle x_{j}^{i}}$ means the value of the jth element of array x in timestep i.

With a single processor this algorithm would run in O(n log n) time. However if the machine has at least n processors to perform the inner loop in parallel, the algorithm as a whole runs in O(log n) time, the number of iterations of the outer loop.

Circuit representation of a work-efficient 16-input parallel prefix sum

A work-efficient parallel prefix sum can be computed by the following steps.[3][10][11]

1. Compute the sums of consecutive pairs of items in which the first item of the pair has an even index: z₀ = x₀ + x₁, z₁ = x₂ + x₃, etc.
2. Recursively compute the prefix sum w₀, w₁, w₂, ... of the sequence z₀, z₁, z₂, ...
3. Express each term of the final sequence y₀, y₁, y₂, ... as the sum of up to two terms of these intermediate sequences: y₀ = x₀, y₁ = z₀, y₂ = z₀ + x₂, y₃ = w₁, etc. After the first value, each successive number y_i is either copied from a position half as far through the w sequence, or is the previous value added to one value in the x sequence.

If the input sequence has n steps, then the recursion continues to a depth of O(log n), which is also the bound on the parallel running time of this algorithm. The number of steps of the algorithm is O(n), and it can be implemented on a parallel random access machine with O(n/log n) processors without any asymptotic slowdown by assigning multiple indices to each processor in rounds of the algorithm for which there are more elements than processors.[3]

Each of the preceding algorithms runs in O(log n) time. However, the former takes exactly log₂ n steps, while the latter requires 2 log₂ n − 2 steps. For the 16-input examples illustrated, Algorithm 1 is 12-way parallel (49 units of work divided by a span of 4) while Algorithm 2 is only 4-way parallel (26 units of work divided by a span of 6). However, Algorithm 2 is work-efficient—it performs only a constant factor (2) of the amount of work required by the sequential algorithm—while Algorithm 1 is work-inefficient—it performs asymptotically more work (a logarithmic factor) than is required sequentially. Consequently, Algorithm 1 is likely to perform better when abundant parallelism is available, but Algorithm 2 is likely to perform better when parallelism is more limited.

Parallel algorithms for prefix sums can often be generalized to other scan operations on associative binary operations,[3][4] and they can also be computed efficiently on modern parallel hardware such as a GPU.[12] The idea of building in hardware a functional unit dedicated to computing multi-parameter prefix-sum was patented by Uzi Vishkin.[13]

Many parallel implementations follow a two pass procedure where partial prefix sums are calculated in the first pass on each processing unit; the prefix sum of these partial sums is then calculated and broadcast back to the processing units for a second pass using the now known prefix as the initial value. Asymptotically this method takes approximately two read operations and one write operation per item.

An implementation of a parallel prefix sum algorithm, like other parallel algorithms, has to take the parallelization architecture of the platform into account. More specifically, multiple algorithms exist which are adapted for platforms working on shared memory as well as algorithms which are well suited for platforms using distributed memory, relying on message passing as the only form of interprocess communication.

The following algorithm assumes a shared memory machine model; all processing elements (PEs) have access to the same memory. A version of this algorithm is implemented in the Multi-Core Standard Template Library (MCSTL),[14][15] a parallel implementation of the C++ standard template library which provides adapted versions for parallel computing of various algorithms.

In order to concurrently calculate the prefix sum over ${\displaystyle n}$ data elements with ${\displaystyle p}$ processing elements, the data is divided into ${\displaystyle p+1}$ blocks, each containing ${\displaystyle {\frac {n}{p+1}}}$ elements (for simplicity we assume that ${\displaystyle p+1}$ divides ${\displaystyle n}$). Note, that although the algorithm divides the data into ${\displaystyle p+1}$ blocks, only ${\displaystyle p}$ processing elements run in parallel at a time.

In a first sweep, each PE calculates a local prefix sum for its block. The last block does not need to be calculated, since these prefix sums are only calculated as offsets to the prefix sums of succeeding blocks and the last block is by definition not succeeded.

The ${\displaystyle p}$ offsets which are stored in the last position of each block are accumulated in a prefix sum of their own and stored in their succeeding positions. For ${\displaystyle p}$ being a small number, it is faster to do this sequentially, for a large ${\displaystyle p}$, this step could be done in parallel as well.

A second sweep is performed. This time the first block does not have to be processed, since it does not need to account for the offset of a preceding block. However, in this sweep the last block is included instead and the prefix sums for each block are calculated taking the prefix sum block offsets calculated in the previous sweep into account.

function prefix_sum(elements) {
    n := size(elements)
    p := number of processing elements
    prefix_sum := [0...0] of size n
    
    do parallel i = 0 to p-1 {
        // i := index of current PE
        from j = i * n / (p+1) to (i+1) * n / (p+1) - 1 do {
            // This only stores the prefix sum of the local blocks
            store_prefix_sum_with_offset_in(elements, 0, prefix_sum)
        }
    }
    
    x = 0
    
    for i = 1 to p {                        // Serial accumulation of total sum of blocks 
        x +=  prefix_sum[i * n / (p+1) - 1] // Build the prefix sum over the first p blocks
        prefix_sum[i * n / (p+1)] = x       // Save the results to be used as offsets in second sweep
    }
    
    do parallel i = 1 to p {
        // i := index of current PE
        from j = i * n / (p+1) to (i+1) * n / (p+1) - 1 do {
            offset := prefix_sum[i * n / (p+1)]
            // Calculate the prefix sum taking the sum of preceding blocks as offset
            store_prefix_sum_with_offset_in(elements, offset, prefix_sum)
        }
    }
    
    return prefix_sum
}

Improvement: In case that the number of blocks are too much that makes the serial step time-consuming by deploying a single processor, the Hillis and Steele algorithm can be used to accelerate the second phase.

The Hypercube Prefix Sum Algorithm[16] is well adapted for distributed memory platforms and works with the exchange of messages between the processing elements. It assumes to have ${\displaystyle p=2^{d}}$ processor elements (PEs) participating in the algorithm equal to the number of corners in a ${\displaystyle d}$-dimensional hypercube.

Different hypercubes for varying number of nodes

Throughout the algorithm, each PE is seen as a corner in a hypothetical hyper cube with knowledge of the total prefix sum ${\displaystyle \sigma }$ as well as the prefix sum ${\displaystyle x}$ of all elements up to itself (according to the ordered indices among the PEs), both in its own hypercube.

• The algorithm starts by assuming every PE is the single corner of a zero dimensional hyper cube and therefore ${\displaystyle \sigma }$ and ${\displaystyle x}$ are equal to the local prefix sum of its own elements.
• The algorithm goes on by unifying hypercubes which are adjacent along one dimension. During each unification, ${\displaystyle \sigma }$ is exchanged and aggregated between the two hyper cubes which keeps the invariant that all PEs at corners of this new hyper cube store the total prefix sum of this newly unified hyper cube in their variable ${\displaystyle \sigma }$. However, only the hyper cube containing the PEs with higher index also adds this ${\displaystyle \sigma }$ to their local variable ${\displaystyle x}$, keeping the invariant that ${\displaystyle x}$ only stores the value of the prefix sum of all elements at PEs with indices smaller or equal to their own index.

In a ${\displaystyle d}$-dimensional hyper cube with ${\displaystyle 2^{d}}$ PEs at the corners, the algorithm has to be repeated ${\displaystyle d}$ times to have the ${\displaystyle 2^{d}}$zero-dimensional hyper cubes be unified into one ${\displaystyle d}$-dimensional hyper cube. Assuming a duplex communication model where the ${\displaystyle \sigma }$ of two adjacent PEs in different hyper cubes can be exchanged in both directions in one communication step, this means ${\displaystyle d=\log _{2}p}$ communication startups.

i := Index of own processor element (PE)
m := prefix sum of local elements of this PE
d := number of dimensions of the hyper cube

x = m;     // Invariant: The prefix sum up to this PE in the current sub cube
σ = m;     // Invariant: The prefix sum of all elements in the current sub cube

for (k=0; k <= d-1; k++) {
    y = σ @ PE(i xor 2^k)  // Get the total prefix sum of the opposing sub cube along dimension k
    σ = σ + y              // Aggregate the prefix sum of both sub cubes

    if (i & 2^k) {
        x = x + y  // Only aggregate the prefix sum from the other sub cube, if this PE is the higher index one.
    }
}

The Pipelined Binary Tree Algorithm[17] is another algorithm for distributed memory platforms which is specifically well suited for large message sizes.

Like the hypercube algorithm, it assumes a special communication structure. The processing elements (PEs) are hypothetically arranged in a binary tree (e.g. a Fibonacci Tree) with infix numeration according to their index within the PEs. Communication on such a tree always occurs between parent and child nodes.

The infix numeration ensures that for any given PE_j, the indices of all nodes reachable by its left subtree ${\displaystyle \mathbb {[l...j-1]} }$ are less than ${\displaystyle j}$ and the indices ${\displaystyle \mathbb {[j+1...r]} }$ of all nodes in the right subtree are greater than ${\displaystyle j}$. The parent's index is greater than any of the indices in PE_j's subtree if PE_j is a left child and smaller if PE_j is a right child. This allows for the following reasoning:

Information exchange between processing elements during upward (blue) and downward (red) phase in the Pipelined Binary Tree Prefix Sum algorithm.

• The local prefix sum ${\displaystyle \mathbb {\oplus [l..j-1]} }$ of the left subtree has to be aggregated to calculate PE_j's local prefix sum ${\displaystyle \mathbb {\oplus [l..j]} }$.
• The local prefix sum ${\displaystyle \mathbb {\oplus [j+1..r]} }$ of the right subtree has to be aggregated to calculate the local prefix sum of higher level PE_h which are reached on a path containing a left children connection (which means ${\displaystyle h>j}$).
• The total prefix sum ${\displaystyle \mathbb {\oplus [0..j]} }$ of PE_j is necessary to calculate the total prefix sums in the right subtree (e.g. ${\displaystyle \mathbb {\oplus [0..j..r]} }$ for the highest index node in the subtree).
• PE_j needs to include the total prefix sum ${\displaystyle \mathbb {\oplus [0..l-1]} }$ of the first higher order node which is reached via an upward path including a right children connection to calculate its total prefix sum.

Note the distinction between subtree-local and total prefix sums. The points two, three and four can lead to believe they would form a circular dependency, but this is not the case. Lower level PEs might require the total prefix sum of higher level PEs to calculate their total prefix sum, but higher level PEs only require subtree local prefix sums to calculate their total prefix sum. The root node as highest level node only requires the local prefix sum of its left subtree to calculate its own prefix sum. Each PE on the path from PE₀ to the root PE only requires the local prefix sum of its left subtree to calculate its own prefix sum, whereas every node on the path from PE_p-1 (last PE) to the PE_root requires the total prefix sum of its parent to calculate its own total prefix sum.

This leads to a two-phase algorithm:

Upward Phase
Propagate the subtree local prefix sum ${\displaystyle \mathbb {\oplus [l..j..r]} }$ to its parent for each PE_j.

Downward phase
Propagate the exclusive (exclusive PE_j as well as the PEs in its left subtree) total prefix sum ${\displaystyle \mathbb {\oplus [0..l-1]} }$ of all lower index PEs which are not included in the addressed subtree of PE_j to lower level PEs in the left child subtree of PE_j. Propagate the inclusive prefix sum ${\displaystyle \mathbb {\oplus [0..j]} }$ to the right child subtree of PE_j.

Note that the algorithm is run in parallel at each PE and the PEs will block upon receive until their children/parents provide them with packets.

k := number of packets in a message m of a PE
m @ {left, right, parent, this} := // Messages at the different PEs

x = m @ this

// Upward phase - Calculate subtree local prefix sums
for j=0 to k-1: // Pipelining: For each packet of a message
    if hasLeftChild:
        blocking receive m[j] @ left // This replaces the local m[j] with the received m[j]
        // Aggregate inclusive local prefix sum from lower index PEs
        x[j] = m[j] ⨁ x[j]

    if hasRightChild:
        blocking receive m[j] @ right
        // We do not aggregate m[j] into the local prefix sum, since the right children are higher index PEs
        send x[j] ⨁ m[j] to parent
    else:
        send x[j] to parent

// Downward phase
for j=0 to k-1:
    m[j] @ this = 0

    if hasParent:
        blocking receive m[j] @ parent
        // For a left child m[j] is the parents exclusive prefix sum, for a right child the inclusive prefix sum
        x[j] = m[j] ⨁ x[j] 
    
    send m[j] to left  // The total prefix sum of all PE's smaller than this or any PE in the left subtree
    send x[j] to right // The total prefix sum of all PE's smaller or equal than this PE

If the message m of length n can be divided into k packets and the operator ⨁ can be used on each of the corresponding message packets separately, pipelining is possible.[17]

If the algorithm is used without pipelining, there are always only two levels (the sending PEs and the receiving PEs) of the binary tree at work while all other PEs are waiting. If there are p processing elements and a balanced binary tree is used, the tree has ${\displaystyle \log _{2}p}$ levels, the length of the path from ${\displaystyle PE_{0}}$ to ${\displaystyle PE_{\mathbb {root} }}$ is therefore ${\displaystyle \log _{2}p-1}$ which represents the maximum number of non parallel communication operations during the upward phase, likewise, the communication on the downward path is also limited to ${\displaystyle \log _{2}p-1}$ startups. Assuming a communication startup time of ${\displaystyle T_{\mathbb {start} }}$ and a bytewise transmission time of ${\displaystyle T_{\mathbb {byte} }}$, upward and downward phase are limited to ${\displaystyle (2\log _{2}p-2)(T_{\mathbb {start} }+n\cdot T_{\mathbb {byte} })}$ in a non pipelined scenario.

Upon division into k packets, each of size ${\displaystyle {\tfrac {n}{k}}}$ and sending them separately, the first packet still needs ${\displaystyle (\log _{2}p-1)\left(T_{\mathbb {start} }+{\frac {n}{k}}\cdot T_{\mathbb {byte} }\right)}$ to be propagated to ${\displaystyle PE_{\mathbb {root} }}$ as part of a local prefix sum and this will occur again for the last packet if ${\displaystyle k>\log _{2}p}$. However, in between, all the PEs along the path can work in parallel and each third communication operation (receive left, receive right, send to parent) sends a packet to the next level, so that one phase can be completed in ${\displaystyle 2\log _{2}p-1+3(k-1)}$ communication operations and both phases together need ${\displaystyle (4\cdot \log _{2}p-2+6(k-1))\left(T_{\mathbb {start} }+{\frac {n}{k}}\cdot T_{\mathbb {byte} }\right)}$ which is favourable for large message sizes n.

The algorithm can further be optimised by making use of full-duplex or telephone model communication and overlapping the upward and the downward phase.[17]