Prefix Sum

In computer science, the prefix sum, scan, or cumulative sum of a sequence of numbers x₀, x₁, x₂, ... is a second sequence of numbers y₀, y₁, y₂, ..., the sums of prefixes (running totals) of the input sequence:

y₀ = x₀

y₁ = x₀ + x₁

y₂ = x₀ + x₁+ x₂

...

For instance, the prefix sums of the natural numbers are the triangular numbers:

input numbers	1	2	3	4	5	6	...
prefix sums	1	3	6	10	15	21	...

Prefix sums are trivial to compute in sequential models of computation, by using the formula y_i = y_{i − 1} + x_i to compute each output value in sequence order. However, despite their ease of computation, prefix sums are a useful primitive in certain algorithms such as counting sort, and they form the basis of the scan higher-order function in functional programming languages. Prefix sums have also been much studied in parallel algorithms, both as a test problem to be solved and as a useful primitive to be used as a subroutine in other parallel algorithms.

Abstractly, a prefix sum requires only a binary associative operator ⊕, making it useful for many applications from calculating well separated pair decompositions of points to string processing.

Mathematically, the operation of taking prefix sums can be generalized from finite to infinite sequences; in that context, a prefix sum is known as a partial sum of a series. Prefix summation or partial summation form linear operators on the vector spaces of finite or infinite sequences; their inverses are finite difference operators.