# Fine-Grained Complexity of Analyzing Compressed Data: Quantifying Improvements over Decompress-And-Solve

## TL;DR

This paper investigates the complexity of analyzing compressed data directly, establishing bounds and optimality results for various problems under common compression schemes, and introduces a framework for conditional lower bounds.

## Contribution

It provides a unified framework for proving lower bounds on analyzing compressed data and determines the optimality of decompress-and-solve for several fundamental problems.

## Key findings

- LCS and Pattern Matching with Wildcards bounds are optimal under ETH.
- Decompress-and-solve is optimal for Grammar Parsing and RNA Folding under the k-Clique conjecture.
- Decompress-and-solve is not optimal for Disjointness, with a new algorithm.

## Abstract

Can we analyze data without decompressing it? As our data keeps growing, understanding the time complexity of problems on compressed inputs, rather than in convenient uncompressed forms, becomes more and more relevant. Suppose we are given a compression of size $n$ of data that originally has size $N$, and we want to solve a problem with time complexity $T(\cdot)$. The naive strategy of "decompress-and-solve" gives time $T(N)$, whereas "the gold standard" is time $T(n)$: to analyze the compression as efficiently as if the original data was small. We restrict our attention to data in the form of a string (text, files, genomes, etc.) and study the most ubiquitous tasks. While the challenge might seem to depend heavily on the specific compression scheme, most methods of practical relevance (Lempel-Ziv-family, dictionary methods, and others) can be unified under the elegant notion of Grammar Compressions. A vast literature, across many disciplines, established this as an influential notion for Algorithm design. We introduce a framework for proving (conditional) lower bounds in this field, allowing us to assess whether decompress-and-solve can be improved, and by how much. Our main results are: - The $O(nN\sqrt{\log{N/n}})$ bound for LCS and the $O(\min\{N \log N, nM\})$ bound for Pattern Matching with Wildcards are optimal up to $N^{o(1)}$ factors, under the Strong Exponential Time Hypothesis. (Here, $M$ denotes the uncompressed length of the compressed pattern.) - Decompress-and-solve is essentially optimal for Context-Free Grammar Parsing and RNA Folding, under the $k$-Clique conjecture. - We give an algorithm showing that decompress-and-solve is not optimal for Disjointness.