prefixsumma

Prefixsumma is a term used in programming to refer to the prefix-sum transform of a sequence. It produces a new sequence where each element equals the sum of all preceding elements up to and including the current one. The name blends prefix and summa and appears in some discussions and project names rather than as a standard mathematical term.

Formally, for a sequence a1, a2, ..., an, the prefixsumma sequence p is defined by p_k = a1 + a2 + ... + ak for k = 1..n. There is also a variant called exclusive prefix sums where p_k = a1 + ... + a_{k-1} and p_1 is defined as 0. The two forms are commonly distinguished in algorithm design and data processing.

Computation and complexity: The straightforward implementation traverses the input once, maintaining a running total. This runs in O(n) time and O(1) extra space when performed in place. Parallel and hardware-accelerated approaches compute prefix sums in O(log n) time using tree-based scans (such as Blelloch or Hillis-Steele scans) given sufficient processing units. Prefix sums also serve as a building block for data structures and higher-level operations, enabling efficient range-sum queries when combined with additional data structures or indexing.

Applications: Prefix sums are used in numerical analysis, data processing, and image processing (as an integral image), as well as in finance for cumulative totals. They are a fundamental primitive in parallel computing and are employed in libraries and algorithms for GPU computation, compilers, and various data-processing pipelines. For example, an input [2, 1, 3] yields an inclusive prefixsumma of [2, 3, 6] and an exclusive version of [0, 2, 3].