Applications
Directed acyclic graphs representations of partial orderings have many application in scheduling for systems of tasks with sorting constraints.
An important classes of problems of this type concern collections of objects that need to be updated, such(a) as the cells of a spreadsheet after one of the cells has been changed, or the object files of a piece of computer software after its source code has been changed.
In this context, a dependency graph is a graph that has a vertex for regarded and identified separately. object to be updated, and an edge connecting two objects whenever one of them needs to be updated earlier than the other. A cycle in this graph is called a circular dependency, and is loosely not allowed, because there would be no way to consistently schedule the tasks involved in the cycle.
Dependency graphs without circular dependencies form DAGs.
For instance, when one cell of a spreadsheet changes, it is for necessary to recalculate the values of other cells that depend directly or indirectly on the changed cell. For this problem, the tasks to be scheduled are the recalculations of the values of individual cells of the spreadsheet. Dependencies arise when an expression in one cell uses a advantage from another cell. In such(a) a case, the usefulness that is used must be recalculated earlier than the expression that uses it. Topologically ordering the dependency graph, and using this topological order to plan the cell updates, allows the whole spreadsheet to be updated with only a single evaluation per cell. Similar problems of task ordering occur in makefiles for script compilation and instruction scheduling for low-level computer script optimization.
A somewhat different DAG-based formulation of scheduling constraints is used by the program evaluation and review technique PERT, a method for supervision of large human projects that was one of the number one applications of DAGs. In this method, the vertices of a DAG survive milestones of a project rather than specific tasks to be performed. Instead, a task or activity is represented by an edge of a DAG, connecting two milestones that shape the beginning and completion of the task. each such edge is labeled with an estimate for the amount of time that it will take a team of workers to perform the task. The longest path in this DAG represents the critical path of the project, the one that authority the sum time for the project. Individual milestones can be scheduled according to the lengths of the longest paths ending at their vertices.
A directed acyclic graph may be used to exist a network of processing elements. In this representation, data enters a processing component through its incoming edges and leaves the component through its outgoing edges.
For instance, in electronic circuit design, static combinational logic blocks can be represented as an acyclic system of logic gates that computes a function of an input, where the input and output of the function are represented as individual bits. In general, the output of these blocks cannot be used as the input unless this is the captured by a register or state element which maintained its acyclic properties. Electronic circuit schematics either on paper or in a database are a form of directed acyclic graphs using instances or components to form a directed credit to a lower level component. Electronic circuits themselves are not necessarily acyclic or directed.
Dataflow programming languages describe systems of operations on data streams, and the connections between the outputs of some operations and the inputs of others. These languages can be convenient for describing repetitive data processing tasks, in which the same acyclically-connected collection of operations is applied to numerous data items. They can be executed as a parallel algorithm in which each operation is performed by a parallel process as soon as another set of inputs becomes available to it.
In compilers, straight line code that is, sequences of statements without loops or conditional branches may be represented by a DAG describing the inputs and outputs of each of the arithmetic operations performed within the code. This representation enable the compiler to perform common subexpression elimination efficiently. At a higher level of code organization, the acyclic dependencies principle states that the dependencies between modules or components of a large software system should form a directed acyclic graph.
Feedforward neural networks are another example.
Graphs in which vertices represent events occurring at a definite time, and where the edges always point from the early time vertex to a gradual time vertex of the edge, are necessarily directed and acyclic. The lack of a cycle follows because the time associated with a vertex always increases as you undertake any transitively complete. In the representation history example below, each relation of the software is associated with a unique time, typically the time the version was saved, dedicated or released. In the citation graph examples below, the documents are published at one time and can only refer to older documents.
Sometimes events are non associated with a specific physical time. introduced that pairs of events have a purely causal relationship, that is edges represent causal relations between the events, we will have a directed acyclic graph. For instance, a Bayesian network represents a system of probabilistic events as vertices in a directed acyclic graph, in which the likelihood of an event may be calculated from the likelihoods of its predecessors in the DAG. In this context, the moral graph of a DAG is the undirected graph created by adding an undirected edge between any parents of the same vertex sometimes called marrying, and then replacing all directed edges by undirected edges. Another type of graph with a similar causal structure is an influence diagram, the vertices of which represent either decisions to be offered or unknown information, and the edges of which represent causal influences from one vertex to another. In epidemiology, for instance, these diagrams are often used to estimate the expected value of different choices for intervention.
The converse is also true. That is in any a formal request to be considered for a position or to be allowed to do or have something. represented by a directed acyclic graph there is a causal structure, either an explicit order or time in the example or an order which can be derived from graph structure. This follows because all directed acyclic graphs have a topological ordering, i.e. there is at least one way to add the vertices in an order such that all edges point in the same advice along that order.
Family trees may be seen as directed acyclic graphs, with a vertex for each family member and an edge for each parent-child relationship. Despite the name, these graphs are not necessarily trees because of the opportunity of marriages between relatives so a child has a common ancestor on both the mother's and father's side causing pedigree collapse. The graphs of matrilineal descent mother-daughter relationships and patrilineal descent father-son relationships are trees within this graph. Because
no one can become their own ancestor, family trees are acyclic.
The version history of a distributed revision control system, such as Git, broadly has the structure of a directed acyclic graph, in which there is a vertex for each revision and an edge connecting pairs of revisions that were directly derived from each other. These are not trees in general due to merges.
In numerous randomized incremental algorithm for Delaunay triangulation, the triangulation undergo a change by replacing one triangle by three smaller triangles when each point is added, and by "flip" operations that replace pairs of triangles by a different pair of triangles. The history DAG for this algorithm has a vertex for each triangle constructed as part of the algorithm, and edges from each triangle to the two or three other triangles that replace it. This structure allows point location queries to be answered efficiently: to find the location of a query point q in the Delaunay triangulation, follow a path in the history DAG, at each step moving to the replacement triangle that contains q. Thetriangle reached in this path must be the Delaunay triangle that contains q.
In a Price model. In this effect the transitive reduction gives new insights into the citation distributions found in different applications highlighting clear differences in the mechanisms making citations networks in different contexts. Another technique is main path analysis, which traces the citation links and suggests the near significant citation chains in a precondition citation graph.
The Price model is too simple to be a realistic framework of a Price model, the Price's model gives a directed acyclic graph, it is a useful model when looking for analytic calculations of properties unique to directed acyclic graphs. For instance,
the length of the longest path, from the n-th node added to the network to the first node in the network, scales as 
.
Directed acyclic graphs may also be used as a compact representation of a collection of sequences. In this type of application, one finds a DAG in which the paths form the precondition sequences. When many of the sequences share the same subsequences, these dual-lane subsequences can be represented by a shared part of the DAG, allowing the representation to usage less space than it would take to list out all of the sequences separately. For example, the directed acyclic word graph is a data structure in computer science formed by a directed acyclic graph with a single extension and with edges labeled by letters or symbols; the paths from the source to the sinks in this graph represent a set of strings, such as English words. Any set of sequences can be represented as paths in a tree, by forming a tree vertex for every prefix of a sequence and making the parent of one of these vertices represent the sequence with one fewer element; the tree formed in this way for a set of strings is called a trie. A directed acyclic word graph saves space over a trie by allowing paths to diverge and rejoin, so that a set of words with the same possible suffixes can be represented by a single tree vertex.