An Encoding for Order-Preserving Matching

Encoding data structures store enough information to answer the queries they are meant to support but not enough to recover their underlying datasets. In this paper we give the first encoding data structure for the challenging problem of order-preserving pattern matching. This problem was introduced only a few years ago but has already attracted significant attention because of its applications in data analysis. Two strings are said to be an order-preserving match if the {\em relative order} of their characters is the same: e.g., $4, 1, 3, 2$ and $10, 3, 7, 5$ are an order-preserving match. We show how, given a string $S [1..n]$ over an arbitrary alphabet and a constant $c \geq 1$, we can build an $O (n \log \log n)$-bit encoding such that later, given a pattern $P [1..m]$ with $m \leq \lg^c n$, we can return the number of order-preserving occurrences of $P$ in $S$ in $O (m)$ time. Within the same time bound we can also return the starting position of some order-preserving match for $P$ in $S$ (if such a match exists). We prove that our space bound is within a constant factor of optimal; our query time is optimal if $\log \sigma = \Omega(\log n)$. Our space bound contrasts with the $\Omega (n \log n)$ bits needed in the worst case to store $S$ itself, an index for order-preserving pattern matching with no restrictions on the pattern length, or an index for standard pattern matching even with restrictions on the pattern length. Moreover, we can build our encoding knowing only how each character compares to $O (\lg^c n)$ neighbouring characters.