An Optimal Resolution Sensitive Pyramid Representation for Hierarchical Memory Models

In this paper we propose and analyze a new hierarchical direct access data structure, namely the up-down pyramid, for storing and processing efficiently large sets of 2-dimensional data. We use as performance measure the cost of storage accesses on a hierrchical memory model with different access cost functions of theoretical and practical significance. We analyze, for our structure, the time complexity of the operation of retrieving the whole information associated to a datum, and prove that it is dependent only on the accessed resolution level without any overhead costs. Therefore, the up-down pyramid is an optmal representation for the considered model, with respect to other proposed direct access hierarchical structures. Namely we prove that, given a set of 2-dimensional data of size T×T, the up-down pyramid guarantees retrieving of information associated to location x in optimal time O(f(x)), where f(x) is the considered cost function in the hierarchical memory model, while in the pyramid and in the incomplete pyramid the time is Ω(f(T2)).