Prefix Sum - Scan Higher Order Function

Scan Higher Order Function

In functional programming terms, the prefix sum may be generalized to any binary operation (not just the addition operation); the higher order function resulting from this generalization is called a scan, and it is closely related to the fold operation. Both the scan and the fold operations apply the given binary operation to the same sequence of values, but differ in that the scan returns the whole sequence of results from the binary operation, whereas the fold returns only the final result. For instance, the sequence of factorial numbers may be generated by a scan of the natural numbers using multiplication instead of addition:

input numbers	1	2	3	4	5	6	...
prefix products	1	2	6	24	120	720	...

In Haskell, there are two variants of scan, called scanl and scanl1, differing slightly in their argument signature, and the prefix sum operation may be written scanl1 (+). The corresponding suffix operations are also available as scanr and scanr1. The procedural Message Passing Interface libraries provide an operation MPI_Scan for computing a scan operation between networked processing units. The C++ language has a standard library function partial_sum; despite its name, it takes a binary operation as one of its arguments, and performs a scan with that operation.