Sequences of radius k: how to fetch many huge objects into small memory for pairwise computations

• Authors: Jerzy W. Jaromczyk;Zbigniew Lonc
• Venue: ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
• Year: 2004

Let a1, a2, ..., am be a sequence over [n]={1,...n} We say that a sequence a1, a2, .. am has the k-radius property if every pair of different elements in [n] occurs at least once within distance at most k; the distanced(ai,aj)=|i−j | We demonstrate lower and (asymptotically) matching upper bounds for sequences with the k-radius property Such sequences are applicable, for example, in computations of two-argument functions for all $\binom{n}{2}$ pairs of large objects such as medical images, bitmaps or matrices, when processing occurs in a memory of size capable of storing k + 1 objects, k n We focus on the model when elements are read into the memory in a FIFO fashion that correspond to streaming the data or a special type of caching We present asymptotically optimal constructions; they are based on the Euler totient theorem and recursion.