Color Distance Oracles and Snippets: Separation Between Exact and Approximate Solutions

TL;DR

This paper introduces new conditionally optimal algorithms for approximate color distance oracles and snippets problems, demonstrating the strict hardness of exact solutions and improving understanding of their computational tradeoffs.

Contribution

It presents new algorithms for approximate solutions to color distance oracles and snippets problems using fast matrix multiplication, and establishes the optimality of exact solutions under certain hypotheses.

Findings

Approximate CDO can be solved with specific preprocessing and query time tradeoffs.

Exact CDO is shown to be essentially optimal under the strong APSP hypothesis.

Approximate solutions are strictly easier than exact solutions for CDO.

Abstract

In the snippets problem, the goal is to preprocess text $T$ so that given two patterns $P_1$ and $P_2$, one can locate the occurrences of the two patterns in $T$ that are closest to each other, or report their distance. Kopelowitz and Krauthgamer [CPM2016] showed upper bound tradeoffs and conditional lower bounds tradeoffs for the snippets problem, by utilizing connections between the snippets problem and the problem of constructing a color distance oracle (CDO), which is a data structure that preprocess a set of points with associated colors so that given two colors $c$ and $c'$ one can quickly find the (distance between the) closest pair of points with colors $c$ and $c'$. However, the existing upper bound and lower bound curves are not tight. Inspired by recent advances by Kopelowitz and Vassilevska-Williams [ICALP2020] regarding Set-disjointness data structures, we introduce new conditionally optimal algorithms for $(1+\varepsilon)$ approximation versions of the snippets problem and the CDO problem, by applying fast matrix multiplication. For example, for CDO on $n$ points in an array with preprocessing time $\tilde{O}(n^a)$ and query time $\tilde{O}(n^b)$, assuming that $\omega=2$ (where $\omega$ is the exponent of $n$ in the runtime of the fastest matrix multiplication algorithm on two squared matrices of size $n\times n$), we show that approximate CDO can be solved with the following tradeoff $$ a + 2b = 2 \text{ if } 0 \leq b \leq \frac1 3$$ $$ 2a + b = 3 \text{ if } \frac13\leq b \leq 1.$$ Moreover, we prove that for exact CDO on points in an array, the algorithm of Kopelowitz and Krauthgamer [CPM2016], is essentially optimal assuming that the strong APSP hypothesis holds for randomized algorithms. Thus, the exact version of CDO is strictly harder than the approximate version.