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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.
A perfect binary tree is a binary tree in which all interior nodes have two children and all leaves have the same depth or same level (the level of a node defined as the number of edges or links from the root node to a node). [18] A perfect binary tree is a full binary tree.
Such a tree has exactly + leaf nodes, which can be identified (in the order given by the binary tree) with the + input weights. The goal of the problem is to find a tree, among all of the possible trees with internal nodes, that minimizes the weighted sum of the external path lengths. These path lengths are the numbers of steps from the root to ...
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.
The nodes are thus in a one-to-one correspondence with finite (possibly empty) sequences of positive numbers, which are countable and can be placed in order first by sum of entries, and then by lexicographic order within a given sum (only finitely many sequences sum to a given value, so all entries are reached—formally there are a finite ...
This unsorted tree has non-unique values (e.g., the value 2 existing in different nodes, not in a single node only) and is non-binary (only up to two children nodes per parent node in a binary tree). The root node at the top (with the value 2 here), has no parent as it is the highest in the tree hierarchy.
A binary tree that is not in extended form may be converted into an extended binary tree by treating all its nodes as internal, and adding an external node for each missing child of an internal node. In the other direction, an extended binary tree with at least one internal node may be converted back into a non-extended binary tree by removing ...
Using an inorder binary tree, a prefix sum can be computed by first performing an up-phase in which each interior node computes a partial sum = for left- and rightmost leaves l and r, followed by a down-phase in which prefixes of the form = are sent down the tree and allow each processor to finish computing its prefix sum.