arXiv:2611.16164 · v1

Exact and Approximation Algorithms for Computing Smallest SESs and BMSs

首次提交 2026-11-12 09:51

版本更新 2026-11-12 09:51

雷达观测 2026-11-13 14:16

摘要

Compression schemes represent repetitive strings compactly by using repeated substrings. Computing the minimum size of a representation of a given input string is important for compression and for understanding relationships between compression schemes, although this task is NP-hard for several schemes. In this paper, we introduce exact and approximation algorithms for finding smallest substring equation systems (SESs) and bidirectional macro schemes (BMSs). Specifically, for an input string of length $n$, we give deterministic and randomized algorithms for computing a smallest SES using $O^*(4^n)$ and $O^*(2^n)$ time and space, respectively. We also give a deterministic algorithm for computing a smallest BMS using $O^*(2^n)$ time and space. We further give deterministic and randomized $(1+α)$-approximation algorithms for smallest SESs using $O^*(4^{e^{-α}n})$ and $O^*(2^{e^{-α}n})$ time and space, respectively, for every $α\ge 0$. A conversion from SESs to BMSs gives $4(1+α)$-approximation algorithms for smallest BMSs with the same time and space bounds. All time bounds hold in the worst case, and the randomized algorithms return correct answers with high probability. Our algorithms adapt several approaches to exact and approximation algorithms for combinatorial problems, such as dynamic programming over determinant expansions and a greedy algorithm for monotone submodular functions.