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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, [1] [2] and they form the basis of the scan higher-order function in functional programming languages.
In the beginning of a prefix sum operation, each processing element owns a message .The goal is to compute , where is an associative operation. The following pseudo code describes the algorithm.
A Fenwick tree or binary indexed tree (BIT) is a data structure that stores an array of values and can efficiently compute prefix sums of the values and update the values. It also supports an efficient rank-search operation for finding the longest prefix whose sum is no more than a specified value.
List ranking can equivalently be viewed as performing a prefix sum operation on the given list, in which the values to be summed are all equal to one. The list ranking problem can be used to solve many problems on trees via an Euler tour technique, in which one forms a linked list that includes two copies of each edge of the tree, one in each direction, places the nodes of this list into an ...
Eventually processing unit stores the prefix sum ′ <= ′. In the case of the so-called exclusive prefix sum, processing unit stores the prefix sum ′ < ′. Some algorithms require to store the overall sum at each processing unit in addition to the prefix sums.
The communication time of the algorithm is βm + αlog p + √ 4αβmlog p, [7] so the startup latency is only one half of the startup latency of the two-tree broadcast. The drawback of the ESBT broadcast is that it does not work for other values of p and it cannot be adapted for (non-commutative) reduction or prefix sum.
In computer science, a segmented scan is a modification of the prefix sum with an equal-sized array of flag bits to denote segment boundaries on which the scan should be performed. [ 1 ] Example
In fact, prefix sum can be used on any set of values and binary operation which form a group: the binary operation must be associative, every value must have an inverse, and there exists an identity value. With a bit of thought, we can find some exceptional cases where prefix sum becomes incapable or inefficient.