Entropy as Computational Complexity

• Item type: Journal(1)
• 公開日: 2010-09-15
• タイトル: Entropy as Computational Complexity
• タイトル / 言語: en
• タイトル: Entropy as Computational Complexity
• 言語: eng
• キーワード / 主題Scheme: Other
• キーワード / 主題: 一般論文
• 資源タイプ / 資源タイプ識別子: http://purl.org/coar/resource_type/c_6501
• 資源タイプ: journal article
• 著者所属: Department of Computer Science and Software Engineering University of Canterbury
• 著者所属: School of Informatics Kansai University
• 著者所属(英): en
• 著者所属(英): Department of Computer Science and Software Engineering University of Canterbury
• 著者所属(英): en
• 著者所属(英): School of Informatics Kansai University
• 著者名: Tadao, Takaoka; Yuji, Nakagawa
• 著者名(英): Tadao, Takaoka; Yuji, Nakagawa
• 論文抄録 / 内容記述タイプ: Other
• 論文抄録 / 内容記述: If the given problem instance is partially solved, we want to minimize our effort to solve the problem using that information. In this paper we introduce the measure of entropy, H(S), for uncertainty in partially solved input data S(X)=(X1, ..., Xk), where X is the entire data set, and each Xi is already solved. We propose a generic algorithm that merges Xi's repeatedly, and finishes when k becomes 1. We use the entropy measure to analyze three example problems, sorting, shortest paths and minimum spanning trees. For sorting Xi is an ascending run, and for minimum spanning trees, Xi is interpreted as a partially obtained minimum spanning tree for a subgraph. For shortest paths, Xi is an acyclic part in the given graph. When k is small, the graph can be regarded as nearly acyclic. The entropy measure, H(S), is defined by regarding pi=|Xi|/|X| as a probability measure, that is, H(S)=-n(p1logp1+...+pklogpk), where n=|X1|+....+|Xk|. We show that we can sort the input data S(X) in O(H(S)) time, and that we can complete the minimum cost spanning tree in O(m+H(S)) time, where m in the number of edges. Then we solve the shortest path problem in O(m+H(S)) time. Finally we define dual entropy on the partitioning process, whereby we give the time bounds on a generic quicksort and the shortest path problem for another kind of nearly acyclic graphs.
• 論文抄録(英) / 内容記述タイプ: Other
• 論文抄録(英) / 内容記述: If the given problem instance is partially solved, we want to minimize our effort to solve the problem using that information. In this paper we introduce the measure of entropy, H(S), for uncertainty in partially solved input data S(X)=(X1, ..., Xk), where X is the entire data set, and each Xi is already solved. We propose a generic algorithm that merges Xi's repeatedly, and finishes when k becomes 1. We use the entropy measure to analyze three example problems, sorting, shortest paths and minimum spanning trees. For sorting Xi is an ascending run, and for minimum spanning trees, Xi is interpreted as a partially obtained minimum spanning tree for a subgraph. For shortest paths, Xi is an acyclic part in the given graph. When k is small, the graph can be regarded as nearly acyclic. The entropy measure, H(S), is defined by regarding pi=|Xi|/|X| as a probability measure, that is, H(S)=-n(p1logp1+...+pklogpk), where n=|X1|+....+|Xk|. We show that we can sort the input data S(X) in O(H(S)) time, and that we can complete the minimum cost spanning tree in O(m+H(S)) time, where m in the number of edges. Then we solve the shortest path problem in O(m+H(S)) time. Finally we define dual entropy on the partitioning process, whereby we give the time bounds on a generic quicksort and the shortest path problem for another kind of nearly acyclic graphs.
• 書誌レコードID / 収録物識別子タイプ: NCID
• 書誌レコードID / 収録物識別子: AN00116647
• 書誌情報: 情報処理学会論文誌; 巻 51, 号 9, p. 1832-1846, 発行日 2010-09-15
• ISSN / 収録物識別子タイプ: ISSN
• ISSN / 収録物識別子: 1882-7764