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I have a three-dimensional fenwick tree data structure.
I need to calculate sum on some segment from (x0, y0, z0) to (x, y, z)
What is the formula for inclusion-exclusion? For instance, for 2D variant it is
s = sum(x, y) - sum(x, y0 - 1) - sum(x0 - 1, y) + sum(x0 - 1, y0 - 1)
Thanks in advance
http://www.comp.nus.edu.sg/~stevenha/ft.pdf
Here is 2D case:
378
Fenwick trees are faster and extremely simple to implement. The asymptotic bounds are equivalent, but the most basic query and update code is almost branchless, non-recursive, and uses very few operations. The segment tree versions of this can be made almost as fast, but this does take extra effort.
A Fenwick tree or binary indexed tree is a data structure that helps compute prefix sums efficiently. Computing prefix sums are often important in various other algorithms, not to mention several competitive programming problems. For example, they are used to implement the arithmetic coding algorithm.
A Fenwick tree or binary indexed tree is a data structure that can efficiently update elements and calculate prefix sums in a table of numbers. This structure was proposed by Boris Ryabko in 1989 with a further modification published in 1992.
A Fenwick tree, also called a binary indexed tree (BIT), is a data structure that can efficiently update elements and calculate range sums on a list of numbers.
formula involves total 8 computations
value1 = sum(x,y,z)- sum(x0-1,y,z) - sum(x,y0-1,z) + sum(x0-1,y0-1,z)
value2 = sum(x,y,z0-1) - sum(x0-1,y,z0-1) - sum(x,y0-1,z0-1) + sum(x0-1,y0-1,z0-1)
final answer = value1 - value2
Time complexity of code is 8 (logn)^3
How can you visualize.
1> assume it as a cube.
2> value 1 gives answer for upper square layer of cube, but it include sum upto 0 in z-axis.
You only want upto z0, so you have subtract that extra part(from z0 to 0, and that is value2).
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