In computer science, the prefix sum, cumulative sum, inclusive scan, or simply scan 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 yi = yi − 1 + xi 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,[1][2]
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.[3][4][5]
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.[6][7]
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.
^Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), "8.2 Counting Sort", Introduction to Algorithms (2nd ed.), MIT Press and McGraw-Hill, pp. 168–170, ISBN 0-262-03293-7.
^Cole, Richard; Vishkin, Uzi (1986), "Deterministic coin tossing with applications to optimal parallel list ranking", Information and Control, 70 (1): 32–53, doi:10.1016/S0019-9958(86)80023-7
^Cite error: The named reference lf80 was invoked but never defined (see the help page).
^Cite error: The named reference tv85 was invoked but never defined (see the help page).
^Lakshmivarahan, S.; Dhall, S.K. (1994), Parallelism in the Prefix Problem, Oxford University Press, ISBN 0-19508849-2.
^Blelloch, Guy (2011), Prefix Sums and Their Applications (Lecture Notes)(PDF), Carnegie Mellon University.
^Callahan, Paul; Kosaraju, S. Rao (1995), "A Decomposition of Multi-Dimensional Point Sets with Applications to k-Nearest-Neighbors and n-Body Potential Fields", Journal of the ACM, 42 (1): 67–90, doi:10.1145/200836.200853, S2CID 1818562.
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