961
个编辑
更改
跳到导航
跳到搜索
第257行：
第257行： − + − + − + − 二部图在现代编码理论中得到了广泛的应用，特别是在信道码字的译码方面。因子图和坦纳图就是这方面的例子。图是一个二部图，其中一边的顶点代表一个码字的数字，另一边的顶点代表一个码字中期望和为零的数字组合。< ref > { citation+ − | last = Moon | first = Todd K. − | last = Moon | first = Todd K. − | last = Moon | first = Todd k.+ − | isbn = 9780471648000 + − | isbn = 9780471648000+ − 9780471648000 − | page = 638 − | page = 638+ − − 638 − − | publisher = John Wiley & Sons − − | publisher = John Wiley & Sons − − 2012年3月24日 | publisher = 约翰威立 − − | title = Error Correction Coding: Mathematical Methods and Algorithms − − | title = Error Correction Coding: Mathematical Methods and Algorithms − − | title = 纠错编码: 数学方法和算法 − − | url = https://books.google.com/books?id=adxb8CRx5vQC&pg=PA638 − − | url = https://books.google.com/books?id=adxb8CRx5vQC&pg=PA638 − − Https://books.google.com/books?id=adxb8crx5vqc&pg=pa638 − − | year = 2005}}.</ref> A factor graph is a closely related [[belief network]] used for probabilistic decoding of [[LDPC]] and [[turbo codes]].<ref>{{harvtxt|Moon|2005}}, p. 686.</ref> − − | year = 2005}}.</ref> A factor graph is a closely related belief network used for probabilistic decoding of LDPC and turbo codes. − − 2005}}.因子图是一种密切相关的置信网络，用于 LDPC 码和 turbo 码的概率译码。 − − − − In computer science, a [[Petri net]] is a mathematical modeling tool used in analysis and simulations of concurrent systems. A system is modeled as a bipartite directed graph with two sets of nodes: A set of "place" nodes that contain resources, and a set of "event" nodes which generate and/or consume resources. There are additional constraints on the nodes and edges that constrain the behavior of the system. Petri nets utilize the properties of bipartite directed graphs and other properties to allow mathematical proofs of the behavior of systems while also allowing easy implementation of simulations of the system.<ref>{{citation − − In computer science, a Petri net is a mathematical modeling tool used in analysis and simulations of concurrent systems. A system is modeled as a bipartite directed graph with two sets of nodes: A set of "place" nodes that contain resources, and a set of "event" nodes which generate and/or consume resources. There are additional constraints on the nodes and edges that constrain the behavior of the system. Petri nets utilize the properties of bipartite directed graphs and other properties to allow mathematical proofs of the behavior of systems while also allowing easy implementation of simulations of the system.<ref>{{citation − − 在计算机科学中，Petri 网是一种用于并发系统分析和仿真的数学建模工具。一个系统被建模为由两组节点组成的二分有向图: 一组包含资源的“位置”节点和一组生成和/或消耗资源的“事件”节点。在节点和边上有额外的约束，约束系统的行为。Petri 网利用二部有向图的性质和其他性质，允许对系统的行为进行数学证明，同时也允许容易地实现系统的模拟。< ref > { citation − − | last1 = Cassandras | first1 = Christos G. − − | last1 = Cassandras | first1 = Christos G. − − 1 = Cassandras | first1 = Christos g. − − | last2 = Lafortune | first2 = Stephane − − | last2 = Lafortune | first2 = Stephane − − 2 = Lafortune | first2 = Stephane − − | edition = 2nd − − | edition = 2nd − − 2nd − − | isbn = 9780387333328 − − | isbn = 9780387333328 − − 9780387333328 − − | page = 224 − − | page = 224 − − 224 − − | publisher = Springer − − | publisher = Springer − − | publisher = Springer − − | title = Introduction to Discrete Event Systems − − | title = Introduction to Discrete Event Systems − − | title = 离散事件系统导论 − − | url = https://books.google.com/books?id=AxguNHDtO7MC&pg=PA224 − − | url = https://books.google.com/books?id=AxguNHDtO7MC&pg=PA224 − − Https://books.google.com/books?id=axgunhdto7mc&pg=pa224 − − | year = 2007}}.</ref> − − | year = 2007}}.</ref> − − | year = 2007} . </ref > − − − − − − In projective geometry, Levi graphs are a form of bipartite graph used to model the incidences between points and lines in a configuration. Corresponding to the geometric property of points and lines that every two lines meet in at most one point and every two points be connected with a single line, Levi graphs necessarily do not contain any cycles of length four, so their girth must be six or more.<ref>{{citation − − 在射影几何中，Levi 图是二部图的一种形式，用来模拟配置中点和线之间的冲突。对应于点和线的几何性质，每两条直线最多相交一点，每两点与一条直线相连，Levi 图不一定包含任何长度为4的圈，所以它们的周长必须是6或更多。< ref > { citation − − | last = Grünbaum | first = Branko | author-link = Branko Grünbaum − − | last = Grünbaum | first = Branko | author-link = Branko Grünbaum − − | last = Grünbaum | first = Branko | author-link = Branko Grünbaum − − | isbn = 9780821843086 − − | isbn = 9780821843086 − − 9780821843086 − − | page = 28 − − | page = 28 − − 28 − − | publisher = [[American Mathematical Society]] − − | publisher = American Mathematical Society − − 美国数学学会 − − | series = [[Graduate Studies in Mathematics]] − − | series = Graduate Studies in Mathematics − − 数学研究生课程 − − | title = Configurations of Points and Lines − − | title = Configurations of Points and Lines − − | title = 点和线的配置 − − | url = https://books.google.com/books?id=mRw571GNa5UC&pg=PA28 − − | url = https://books.google.com/books?id=mRw571GNa5UC&pg=PA28 − − Https://books.google.com/books?id=mrw571gna5uc&pg=pa28 − − | volume = 103 − − | volume = 103 − − 103 − − | year = 2009}}.</ref> − − | year = 2009}}.</ref> − − 2009} . </ref > + +
→Additional applications
Dulmage–Mendelsohn 分解是二分图的结构分解，可用于寻找最大匹配。
Dulmage–Mendelsohn 分解是二分图的结构分解，可用于寻找最大匹配。
==Additional applications==
== Additional applications 其他应用 ==
Bipartite graphs are extensively used in modern [[coding theory]], especially to decode [[codeword]]s received from the channel. [[Factor graph]]s and [[Tanner graph]]s are examples of this. A Tanner graph is a bipartite graph in which the vertices on one side of the bipartition represent digits of a codeword, and the vertices on the other side represent combinations of digits that are expected to sum to zero in a codeword without errors.<ref>{{citation
Bipartite graphs are extensively used in modern [[coding theory]], especially to decode [[codeword]]s received from the channel. [[Factor graph]]s and [[Tanner graph]]s are examples of this. A Tanner graph is a bipartite graph in which the vertices on one side of the bipartition represent digits of a codeword, and the vertices on the other side represent combinations of digits that are expected to sum to zero in a codeword without errors.
Bipartite graphs are extensively used in modern coding theory, especially to decode codewords received from the channel. Factor graphs and Tanner graphs are examples of this. A Tanner graph is a bipartite graph in which the vertices on one side of the bipartition represent digits of a codeword, and the vertices on the other side represent combinations of digits that are expected to sum to zero in a codeword without errors.<ref>{{citation
Bipartite graphs are extensively used in modern coding theory, especially to decode codewords received from the channel. Factor graphs and Tanner graphs are examples of this. A Tanner graph is a bipartite graph in which the vertices on one side of the bipartition represent digits of a codeword, and the vertices on the other side represent combinations of digits that are expected to sum to zero in a codeword without errors.
二分图在现代编码理论中被广泛使用，尤其是对从通道接收到的密码进行解码。例如因子图Factor graphs和坦纳图Tanner graphs。Tanner图是二分图，其中二分的一侧顶点表示一个密码数字，另一侧顶点表示一个数字的组合，这些数字的总和期望为零而没有错误。因子图与置信网络密切相关，用于LDPC和Turbo码的概率解码。
In computer science, a [[Petri net]] is a mathematical modeling tool used in analysis and simulations of concurrent systems. A system is modeled as a bipartite directed graph with two sets of nodes: A set of "place" nodes that contain resources, and a set of "event" nodes which generate and/or consume resources. There are additional constraints on the nodes and edges that constrain the behavior of the system. Petri nets utilize the properties of bipartite directed graphs and other properties to allow mathematical proofs of the behavior of systems while also allowing easy implementation of simulations of the system.
In computer science, a Petri net is a mathematical modeling tool used in analysis and simulations of concurrent systems. A system is modeled as a bipartite directed graph with two sets of nodes: A set of "place" nodes that contain resources, and a set of "event" nodes which generate and/or consume resources. There are additional constraints on the nodes and edges that constrain the behavior of the system. Petri nets utilize the properties of bipartite directed graphs and other properties to allow mathematical proofs of the behavior of systems while also allowing easy implementation of simulations of the system.
在计算机科学中，Petri网是用于分析和模拟并发系统的数学建模工具。该系统被建模为具有两组点集的二分有向图：一组“库所”点集用于容纳，以及一组“事件”点集用于生成和（或）消耗资源。他们通过约束节点和连边的传递来进一步限制系统的行为。Petri网利用二分有向图的属性和其他属性来允许对系统行为进行数学证明，同时还可以轻松实现系统仿真。
In [[projective geometry]], [[Levi graph]]s are a form of bipartite graph used to model the incidences between points and lines in a [[configuration (geometry)|configuration]]. Corresponding to the geometric property of points and lines that every two lines meet in at most one point and every two points be connected with a single line, Levi graphs necessarily do not contain any cycles of length four, so their [[girth (graph theory)|girth]] must be six or more.
In [[projective geometry]], [[Levi graph]]s are a form of bipartite graph used to model the incidences between points and lines in a [[configuration (geometry)|configuration]]. Corresponding to the geometric property of points and lines that every two lines meet in at most one point and every two points be connected with a single line, Levi graphs necessarily do not contain any cycles of length four, so their [[girth (graph theory)|girth]] must be six or more.<ref>{{citation
In projective geometry, Levi graphs are a form of bipartite graph used to model the incidences between points and lines in a configuration. Corresponding to the geometric property of points and lines that every two lines meet in at most one point and every two points be connected with a single line, Levi graphs necessarily do not contain any cycles of length four, so their girth must be six or more.
在射影几何中，Levi图是二分图的一种形式，用于对配置点和线之间的入射进行建模。对应于点和线的几何特性，即每两条线最多相交一个点，并且每两个点都由一条线连接，因此Levi图不一定包含任何长度为4的环，进而其周长必须为6或更大。
==See also==
==See also==