{"title":"Slowing down top trees for better worst-case compression","authors":"","doi":"10.1016/j.tcs.2024.114764","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the top tree compression scheme introduced by Bille et al. [ICALP 2013] and construct an infinite family of trees on <em>n</em> nodes labeled from an alphabet of size <em>σ</em>, for which the size of the top DAG is <span><math><mi>Θ</mi><mo>(</mo><mfrac><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi>log</mi></mrow><mrow><mi>σ</mi></mrow></msub><mo></mo><mi>n</mi></mrow></mfrac><mi>log</mi><mo></mo><msub><mrow><mi>log</mi></mrow><mrow><mi>σ</mi></mrow></msub><mo></mo><mi>n</mi><mo>)</mo></math></span>. Our construction matches a previously known upper bound and exhibits a weakness of this scheme, as the information-theoretic lower bound is <span><math><mi>Ω</mi><mo>(</mo><mfrac><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi>log</mi></mrow><mrow><mi>σ</mi></mrow></msub><mo></mo><mi>n</mi></mrow></mfrac><mo>)</mo></math></span>. Lohrey et al. [IPL 2019] designed a more involved version of the original algorithm achieving the lower bound. We show that this can be also guaranteed by a very minor modification of the original scheme: informally, one only needs to ensure that different parts of the tree are not compressed too quickly. Arguably, our version is more uniform, and in particular, the compression procedure is oblivious to the value of <em>σ</em>.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397524003815","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the top tree compression scheme introduced by Bille et al. [ICALP 2013] and construct an infinite family of trees on n nodes labeled from an alphabet of size σ, for which the size of the top DAG is . Our construction matches a previously known upper bound and exhibits a weakness of this scheme, as the information-theoretic lower bound is . Lohrey et al. [IPL 2019] designed a more involved version of the original algorithm achieving the lower bound. We show that this can be also guaranteed by a very minor modification of the original scheme: informally, one only needs to ensure that different parts of the tree are not compressed too quickly. Arguably, our version is more uniform, and in particular, the compression procedure is oblivious to the value of σ.
期刊介绍:
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.