Space-Efficient Indexes for Uncertain Strings

TL;DR

This paper introduces a space-efficient index for uncertain strings that significantly reduces storage requirements while maintaining fast pattern matching capabilities, addressing the impracticality of existing large indexes.

Contribution

The authors propose a novel index with expected size $rac{nz}{ ext{ell}} imes ext{log} z$, which is much smaller than previous methods, enabling efficient pattern searches in uncertain strings.

Findings

Index is up to two orders of magnitude smaller than state-of-the-art.

Supports fast pattern matching for patterns of length at least $ ext{ell}$.

Offers competitive or faster query and construction times.

Abstract

Strings in the real world are often encoded with some level of uncertainty. In the character-level uncertainty model, an uncertain string $X$ of length $n$ on an alphabet $\Sigma$ is a sequence of $n$ probability distributions over $\Sigma$. Given an uncertain string $X$ and a weight threshold $\frac{1}{z}\in(0,1]$, we say that pattern $P$ occurs in $X$ at position $i$, if the product of probabilities of the letters of $P$ at positions $i,\ldots,i+|P|-1$ is at least $\frac{1}{z}$. While indexing standard strings for online pattern searches can be performed in linear time and space, indexing uncertain strings is much more challenging. Specifically, the state-of-the-art index for uncertain strings has $\mathcal{O}(nz)$ size, requires $\mathcal{O}(nz)$ time and $\mathcal{O}(nz)$ space to be constructed, and answers pattern matching queries in the optimal $\mathcal{O}(m+|\text{Occ}|)$ time, where $m$ is the length of $P$ and $|\text{Occ}|$ is the total number of occurrences of $P$ in $X$. For large $n$ and (moderate) $z$ values, this index is completely impractical to construct, which outweighs the benefit of the supported optimal pattern matching queries. We were thus motivated to design a space-efficient index at the expense of slower yet competitive pattern matching queries. We propose an index of $\mathcal{O}(\frac{nz}{\ell}\log z)$ expected size, which can be constructed using $\mathcal{O}(\frac{nz}{\ell}\log z)$ expected space, and supports very fast pattern matching queries in expectation, for patterns of length $m\geq \ell$. We have implemented and evaluated several versions of our index. The best-performing version of our index is up to two orders of magnitude smaller than the state of the art in terms of both index size and construction space, while offering faster or very competitive query and construction times.