A New Algorithm for Updating and Querying Sub-arrays of Multidimensional Arrays

Given a $d$-dimensional array $A$, an update operation adds a given constant $C$ to each element within a continuous sub-array of $A$. A query operation computes the sum of all the elements within a continuous sub-array of $A$. The one-dimensional update and query handling problem has been studied intensively and is usually solved using segment trees with lazy propagation technique. In this paper, we present a new algorithm incorporating Binary Indexed Trees and Inclusion-Exclusion Principle to accomplish the same task. We extend the algorithm to update and query sub-matrices of matrices (two-dimensional array). Finally, we propose a general form of the algorithm for $d$-dimensions which achieves $\mathcal{O}(4^d*\log^{d}n)$ time complexity for both updates and queries. This is an improvement over the previously known algorithms which utilize hierarchical data structures like quadtrees and octrees and have a worst-case time complexity of $Ω(n^{d-1})$ per update/query.

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