Reduce and Scan

Synonyms

Parallel Prefix Sums; Prefix; Prefix Reduction

Definition

Givenn inputs x ₁, …, x _n , and an associative operation ⊗, a reduction algorithm computes the output x ₁ ⊗ ⋯ ⊗ x _n .Aprefix or scan algorithm computes then outputs ${x}_{1},{x}_{1} \otimes {x}_{2},\ldots,{x}_{1} \otimes \cdots \otimes {x}_{n}$.

Discussion

The reduction problem is a subset of the prefix problem, but as we shall show, the two are closely related. The reduction and scan operators are available in APL [8]: Thus+/3 7 1 computes the value 11 and +∖3 7 1 computes the vector3 10 11.The two primitives appears in other programming languages and libraries, such as MPI [10].

There are two interesting cases to consider:

• Random access: The inputs ${x}_{1},\ldots,{x}_{n}$ are stored in consecutive location in an array.

• Linked list: The inputs ${x}_{1},\ldots, {x}_{n}$ are stored in consecutive locations in a linked list; the links connect each element to its predecessor.

We analyze these two cases using a...