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directed acyclic graph
In mathematics and computer science, a directed acyclic graph (DAG ), is a directed graph with no directed cycles. That is, it is formed by a collection of vertices and directed edges, each edge connecting one vertex to another, such that there is no way to start at some vertex ''v'' and follow a sequence of edges that eventually loops back to ''v'' again.〔.〕〔.〕〔.〕
DAGs may be used to model many different kinds of information. The reachability relation in a DAG forms a partial order, and any finite partial order may be represented by a DAG using reachability. A collection of tasks that must be ordered into a sequence, subject to constraints that certain tasks must be performed earlier than others, may be represented as a DAG with a vertex for each task and an edge for each constraint; algorithms for topological ordering may be used to generate a valid sequence. Additionally, DAGs may be used as a space-efficient representation of a collection of sequences with overlapping subsequences. DAGs are also used to represent systems of events or potential events and the causal relationships between them. DAGs may also be used to model processes in which data flows in a consistent direction through a network of processors, or states of a repository in a version-control system.
The corresponding concept for undirected graphs is a forest, an undirected graph without cycles. Choosing an orientation for a forest produces a special kind of directed acyclic graph called a polytree. However there are many other kinds of directed acyclic graph that are not formed by orienting the edges of an undirected acyclic graph. Moreover, every undirected graph has an acyclic orientation, an assignment of a direction for its edges that makes it into a directed acyclic graph. For these reasons it would be more accurate to call directed acyclic graphs acyclic directed graphs or acyclic digraphs.
== Mathematical properties ==
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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