Computation over APT compressed data

Abstract

The Arithmetic Progressions Tree (APT) is a data structure storing an encoding of a monotonic sequence $L$ in $[1..n]$. Previous work on APTs focused on its theoretical and experimental compression guarantees. This paper is the first to consider computations over APT compressed data. In particular:

1. We show how to perform a search for any sub-sequence/a set of the monotone sequence $L$ in time proportional to the query sub-sequence length/set size multiplied by the size of the APT compressed representation of $L$.

2. We show how, given the APT compressed representation of the monotone sequence $L$, we can find a minimum run-length of $L$ in constant time, a maximum run-length of $L$ in $O(logn)$ time, and all runs of $L$ in constant time plus the output size.

3. We show how, given the APT compressed representation of the monotone sequence $L$, we can answer whether a consecutive periodic pattern $P$ is represented by an APT-node in $O(logn)$ time and report occurrences of $P$ in $L$ within the processing time of the output size.

4. In addition, we improve the APT construction algorithm time and space complexity.