Binary tree that stores partial sums: Name and existing implementations

问题

Consider a sequence of n positive real numbers, (a_i), and its partial sum sequence, (s_i). Given a number x ∊ (0, s_n], we have to find i such that s_i−1 < x ≤ s_i. Also we want to be able to change one of the a_i’s without having to update all partial sums. Both can be done in O(log n) time by using a binary tree with the a_i’s as leaf node values, and the values of the non-leaf nodes being the sum of the values of the respective children. If n is known and fixed, the tree doesn’t have to be self-balancing and can be stored efficiently in a linear array. Furthermore, if n is a power of two, only 2 n − 1 array elements are required. See Blue et al., Phys. Rev. E 51 (1995), pp. R867–R868 for an application. Given the genericity of the problem and the simplicity of the solution, I wonder whether this data structure has a specific name and whether there are existing implementations (preferably in C++). I’ve already implemented it myself, but writing data structures from scratch always seems like reinventing the wheel to me—I’d be surprised if nobody had done it before.

回答1:

Fenwick tree (aka Binary indexed tree) is a data structure that maintains a sequence of elements, and is able to compute cumulative sum of any range of consecutive elements in O(logn) time. Changing value of any single element needs O(logn) time as well.

回答2:

This is known as a finger tree in functional programming but apparently there are implementations in imperative languages. In the articles there is a link to a blog post explaining an implementation of this data structure in C# which could be useful to you.

来源：https://stackoverflow.com/questions/3822319/binary-tree-that-stores-partial-sums-name-and-existing-implementations