Online Searching for 3-Dimensional Data in Sublinear Time

We give a dynamic data structure that provides a sublinear time for online searching, and online maintaining it upon insertion and deletion. Each element in the data structure is a three-dimensional data item. The searching operation is to find a target element (x,y,z) that satisfies the constraints x < X, y < Y, and z < Z, for some input of positive real numbers X, Y, and Z.