How is a directed acyclic graph (DAG) represented?

A directed acyclic graph (DAG) is represented using vertices and edges, with each edge having a direction and no cycles.

A directed acyclic graph, or DAG, is a concept in mathematics and computer science. It is a directed graph with no directed cycles. That is, it consists of vertices and edges, with each edge directed from one vertex to another, such that following those directions will never form a closed loop.

The representation of a DAG is similar to any other graph. It consists of a set of vertices and a set of directed edges. Each edge has an initial vertex, or 'tail', and a terminal vertex, or 'head'. The edge is said to be directed from the tail to the head. The vertices are usually represented as circles or dots, while the edges are represented as arrows pointing from the tail to the head.

In a DAG, no directed cycles are allowed. This means that there is no way to start at any vertex v and follow a consistently-directed sequence of edges that eventually loops back to v again. This is what makes it 'acyclic'.

DAGs can be represented in several ways in computer memory. The two most common representations are adjacency lists and adjacency matrices. An adjacency list for a vertex v is a list of all the vertices that v has a direct edge to. An adjacency matrix is a 2D array where the cell at the i-th row and j-th column is true if there is an edge from the i-th vertex to the j-th vertex, and false otherwise.

In computer science, DAGs are used in many different algorithms and data structures. For example, they are used in scheduling problems, data compression algorithms, and to represent expressions in compilers. They are also used in network theory, bioinformatics, and many other fields.

In summary, a directed acyclic graph (DAG) is represented using vertices and edges, with each edge having a direction and no cycles. The representation can be visual with circles and arrows, or in computer memory with adjacency lists or matrices.