{"title":"Reconstruction of hypermatrices from subhypermatrices","authors":"","doi":"10.1016/j.jcta.2024.105966","DOIUrl":null,"url":null,"abstract":"<div><div>For a given <em>n</em>, what is the smallest number <em>k</em> such that every sequence of length <em>n</em> is determined by the multiset of all its <em>k</em>-subsequences? This is called the <em>k</em>-deck problem for sequence reconstruction, and has been generalized to the two-dimensional case – reconstruction of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span>-matrices from submatrices. Previous works show that the smallest <em>k</em> is at most <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>)</mo></math></span> for sequences and at most <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>)</mo></math></span> for matrices. We study this <em>k</em>-deck problem for general dimension <em>d</em> and prove that, the smallest <em>k</em> is at most <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></msup><mo>)</mo></math></span> for reconstructing any <em>d</em> dimensional hypermatrix of order <em>n</em> from the multiset of all its subhypermatrices of order <em>k</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316524001055","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
For a given n, what is the smallest number k such that every sequence of length n is determined by the multiset of all its k-subsequences? This is called the k-deck problem for sequence reconstruction, and has been generalized to the two-dimensional case – reconstruction of -matrices from submatrices. Previous works show that the smallest k is at most for sequences and at most for matrices. We study this k-deck problem for general dimension d and prove that, the smallest k is at most for reconstructing any d dimensional hypermatrix of order n from the multiset of all its subhypermatrices of order k.
对于给定的 n,使得长度为 n 的每个序列都由其所有 k 个子序列的多集决定的最小数 k 是多少?这被称为序列重构的 k 层问题,并已被推广到二维情况--从子矩阵重构 n×n 矩阵。之前的研究表明,对于序列,最小的 k 至多为 O(n12),而对于矩阵,则至多为 O(n23)。我们研究了一般维数为 d 的 k 层问题,并证明了从其所有阶数为 k 的子超矩阵的多集重构任何阶数为 n 的 d 维超矩阵时,最小 k 至多为 O(ndd+1)。
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.