{"title":"Family Trees for Enumeration","authors":"Shin-Ichi Nakano","doi":"10.1142/s0129054123420078","DOIUrl":null,"url":null,"abstract":"In this paper we design efficient algorithms for enumerating (1) the rooted trees with exactly [Formula: see text] vertices, (2) the maximal planar graphs with exactly [Formula: see text] vertices and (3) the linear extensions of a given poset. Those algorithms are based on tree structures of objects, called the family trees. The first algorithm enumerates each ordered tree with exactly [Formula: see text] vertices in [Formula: see text] time for each after [Formula: see text] time preprocessing. The second algorithm enumerates each maximal planar graph with exactly [Formula: see text] vertices in [Formula: see text] time for each on average. The third algorithm enumerates each linear extension of a given poset in [Formula: see text] time for each after [Formula: see text] time preprocessing, where [Formula: see text] is the number of the element in the set of the poset.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Foundations of Computer Science","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1142/s0129054123420078","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we design efficient algorithms for enumerating (1) the rooted trees with exactly [Formula: see text] vertices, (2) the maximal planar graphs with exactly [Formula: see text] vertices and (3) the linear extensions of a given poset. Those algorithms are based on tree structures of objects, called the family trees. The first algorithm enumerates each ordered tree with exactly [Formula: see text] vertices in [Formula: see text] time for each after [Formula: see text] time preprocessing. The second algorithm enumerates each maximal planar graph with exactly [Formula: see text] vertices in [Formula: see text] time for each on average. The third algorithm enumerates each linear extension of a given poset in [Formula: see text] time for each after [Formula: see text] time preprocessing, where [Formula: see text] is the number of the element in the set of the poset.
在本文中，我们设计了有效的算法来枚举(1)具有完全[公式:见文]顶点的根树，(2)具有完全[公式:见文]顶点的最大平面图，以及(3)给定序集的线性扩展。这些算法基于对象的树状结构，称为家谱。第一种算法在[公式:见文本]时间内，为每一个经过[公式:见文本]时间预处理的有序树精确地枚举[公式:见文本]顶点。第二种算法在[公式:见文本]的平均时间内，精确地枚举每个顶点的每个最大平面图。第三种算法在[Formula: see text]时间内枚举给定偏序集的每个线性扩展，在[Formula: see text]时间预处理后的每个扩展，其中[Formula: see text]是偏序集集合中元素的编号。
期刊介绍：
The International Journal of Foundations of Computer Science is a bimonthly journal that publishes articles which contribute new theoretical results in all areas of the foundations of computer science. The theoretical and mathematical aspects covered include:
- Algebraic theory of computing and formal systems
- Algorithm and system implementation issues
- Approximation, probabilistic, and randomized algorithms
- Automata and formal languages
- Automated deduction
- Combinatorics and graph theory
- Complexity theory
- Computational biology and bioinformatics
- Cryptography
- Database theory
- Data structures
- Design and analysis of algorithms
- DNA computing
- Foundations of computer security
- Foundations of high-performance computing
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