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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク De Bruijn graph ： ウィキペディア英語版 De Bruijn graph In graph theory, an ''n''-dimensional De Bruijn graph of ''m'' symbols is a directed graph representing overlaps between sequences of symbols. It has ''m''''n'' vertices, consisting of all possible length-''n'' sequences of the given symbols; the same symbol may appear multiple times in a sequence. If we have the set of ''m'' symbols $S:=\backslash$ then the set of vertices is: : $V=S^n=\backslash .$ If one of the vertices can be expressed as another vertex by shifting all its symbols by one place to the left and adding a new symbol at the end of this vertex, then the latter has a directed edge to the former vertex. Thus the set of arcs (aka directed edges) is : $E=\backslash .$ Although De Bruijn graphs are named after Nicolaas Govert de Bruijn, they were discovered independently by both De Bruijn〔 and I. J. Good.〔 Much earlier, Camille Flye Sainte-Marie〔 implicitly used their properties. == Properties == * If $n=1$ then the condition for any two vertices forming an edge holds vacuously, and hence all the vertices are connected forming a total of $m^2$ edges. * Each vertex has exactly $m$ incoming and $m$ outgoing edges. * Each $n$-dimensional De Bruijn graph is the line digraph of the $(n\; -\; 1)$-dimensional De Bruijn graph with the same set of symbols.〔 * Each De Bruijn graph is Eulerian and Hamiltonian. The Euler cycles and Hamiltonian cycles of these graphs (equivalent to each other via the line graph construction) are De Bruijn sequences. The line graph construction of the three smallest binary De Bruijn graphs is depicted below. As can be seen in the illustration, each vertex of the $n$-dimensional De Bruijn graph corresponds to an edge of the $(n\; -\; 1)$-dimensional De Bruijn graph, and each edge in the $n$-dimensional De Bruijn graph corresponds to a two-edge path in the $(n\; -\; 1)$-dimensional De Bruijn graph. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「De Bruijn graph」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.