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 O(4d*logdn) 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 Omega(nd-1) per update/query.