Color Distance Oracles and Snippets: Separation Between Exact and Approximate Solutions
Noam Horowicz, Tsvi Kopelowitz

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 so that given two patterns and , one can locate the occurrences of the two patterns in 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 and one can quickly find the (distance between the) closest pair of points with colors and . 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…
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