 = Directed acyclic graph = 
 
 In mathematics and computer science , a directed acyclic graph ( DAG / ˈdæɡ / ) , is a finite directed graph with no directed cycles . That is , it consists of finitely many vertices and edges , with each edge directed from one vertex to another , such 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 . Equivalently , a DAG is a directed graph that has a topological ordering , a sequence of the vertices such that every edge is directed from earlier to later in the sequence . 
 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 . The program evaluation and review technique uses DAGs to model the milestones and activities of large human projects , and schedule these projects to use as little total time as possible . Scheduling of computer operations such as the updates to a spreadsheet or the compilation operations in a makefile can also be performed using topological orderings of DAGs . Combinational logic blocks in electronic circuit design , and the operations in dataflow programming languages , involve acyclic networks of processing elements . DAGs can also represent collections of events and their influence on each other , either in a probabilistic structure such as a Bayesian network or as a record of historical data such as family trees or the version histories of distributed revision control systems . DAGs can also be used as a compact representation of sequence data , such as the directed acyclic word graph representation of a collection of strings , or the binary decision diagram representation of sequences of binary choices . 
 Important polynomial time computational problems on DAGs include topological sorting ( finding a topological ordering ) , construction of the transitive closure and transitive reduction ( the largest and smallest DAGs with the same reachability relation , respectively ) , and the closure problem , in which the goal is to find a minimum @-@ weight subset of vertices with no edges connecting them to the rest of the graph . Transforming a directed graph with cycles into a DAG by deleting as few vertices or edges as possible ( the feedback vertex set and feedback edge set problem , respectively ) is NP @-@ hard , but any directed graph can be made into a DAG ( its condensation ) by contracting each strongly connected component into a single supervertex . The problems of finding shortest paths and longest paths can be solved on DAGs in linear time , in contrast to arbitrary graphs for which shortest path algorithms are slower and longest path problems are NP @-@ hard . 
 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 . To emphasize that DAGs are not the same thing as directed versions of undirected acyclic graphs , some authors call them acyclic directed graphs or acyclic digraphs . 
 
 = = Definitions = = 
 
 A graph is formed by a collection of vertices and edges , where the vertices are structureless objects that are connected in pairs by edges . In the case of a directed graph , each edge has an orientation , from one vertex to another vertex . A path in a directed graph can be described by a sequence of edges having the property that the ending vertex of each edge in the sequence is the same as the starting vertex of the next edge in the sequence ; a path forms a cycle if the starting vertex of its first edge equals the ending vertex of its last edge . A directed acyclic graph is a directed graph that has no cycles . 
 A vertex v of a directed graph is said to be reachable from another vertex u when there exists a path that starts at u and ends at v. As a special case , every vertex is considered to be reachable from itself ( by a path with zero edges ) . If a vertex can reach itself via a nontrivial path ( a path with one or more edges ) , then that path is a cycle , so another way to define directed acyclic graphs is that they are the graphs in which no vertex can reach itself via a nontrivial path . 
 A topological ordering of a directed graph is an ordering of its vertices into a sequence , such that for every edge the start vertex of the edge occurs earlier in the sequence than the ending vertex of the edge . A graph that has a topological ordering cannot have any cycles , because the edge into the earliest vertex of a cycle would have to be oriented the wrong way . Therefore , every graph with a topological ordering is acyclic . Conversely , every directed acyclic graph has a topological ordering . Therefore , this property can be used as an alternative definition of the directed acyclic graphs : they are exactly the graphs that have topological orderings . 
 
 = = Mathematical properties = = 
 
 
 = = = Reachability , transitive closure , and transitive reduction = = = 
 
 The reachability relationship in any directed acyclic graph can be formalized as a partial order ≤ on the vertices of the DAG . In this partial order , two vertices u and v are ordered as u ≤ v exactly when there exists a directed path from u to v in the DAG ; that is , when v is reachable from u . However , different DAGs may give rise to the same reachability relation and the same partial order . For example , the DAG with two edges a → b and b → c has the same reachability relation as the graph with three edges a → b , b → c , and a → c . Both of these DAGS produce the same partial order , in which the vertices are ordered as a ≤ b ≤ c . 
 If G is a DAG , its transitive closure is the graph with the most edges that represents the same reachability relation . It has an edge u → v whenever u can reach v. That is , it has an edge for every related pair u ≤ v of distinct elements in the reachability relation of G , and may therefore be thought of as a direct translation of the reachability relation ≤ into graph @-@ theoretic terms . The same method of translating partial orders into DAGs works more generally : for every finite partially ordered set ( S , ≤ ) , the graph that has a vertex for each member of S and an edge for each pair of elements related by u ≤ v is automatically a transitively closed DAG , and has ( S , ≤ ) as its reachability relation . In this way , every finite partially ordered set can be represented as the reachability relation of a DAG . 
 The transitive reduction of a DAG G is the graph with the fewest edges that represents the same reachability relation as G. It is a subgraph of G , formed by discarding the edges u → v for which G also contains a longer path connecting the same two vertices . Like the transitive closure , the transitive reduction is uniquely defined for DAGs . In contrast , for a directed graph that is not acyclic , there can be more than one minimal subgraph with the same reachability relation . 
 If a DAG G has a reachability relation described by the partial order ≤ , then the transitive reduction of G is a subgraph of G that has an edge u → v for every pair in the covering relation of ≤ . Transitive reductions are useful in visualizing the partial orders they represent , because they have fewer edges than other graphs representing the same orders and therefore lead to simpler graph drawings . A Hasse diagram of a partial order is a drawing of the transitive reduction in which the orientation of each edge is shown by placing the starting vertex of the edge in a lower position than its ending vertex . 
 
 = = = Topological ordering = = = 
 
 Every directed acyclic graph has a topological ordering , an ordering of the vertices such that the starting endpoint of every edge occurs earlier in the ordering than the ending endpoint of the edge . The existence of such an ordering can be used to characterize DAGs : a directed graph is a DAG if and only if it has a topological ordering . In general , this ordering is not unique ; a DAG has a unique topological ordering if and only if it has a directed path containing all the vertices , in which case the ordering is the same as the order in which the vertices appear in the path . 
 The family of topological orderings of a DAG is the same as the family of linear extensions of the reachability relation for the DAG , so any two graphs representing the same partial order have the same set of topological orders . 
 
 = = = Combinatorial enumeration = = = 
 
 The graph enumeration problem of counting directed acyclic graphs was studied by Robinson ( 1973 ) . The number of DAGs on n labeled vertices , for n = 0 , 1 , 2 , 3 , … ( without restrictions on the order in which these numbers appear in a topological ordering of the DAG ) is 
 1 , 1 , 3 , 25 , 543 , 29281 , 3781503 , … ( sequence A003024 in the OEIS ) . 
 These numbers may be computed by the recurrence relation 
 <formula> 
 Eric W. Weisstein conjectured , and McKay et al . ( 2004 ) proved , that the same numbers count the ( 0 @,@ 1 ) matrices for which all eigenvalues are positive real numbers . The proof is bijective : a matrix A is an adjacency matrix of a DAG if and only if A + I is a ( 0 @,@ 1 ) matrix with all eigenvalues positive , where I denotes the identity matrix . Because a DAG cannot have self @-@ loops , its adjacency matrix must have a zero diagonal , so adding I preserves the property that all matrix coefficients are 0 or 1 . 
 
 = = = Related families of graphs = = = 
 
 A polytree is a directed graph formed by orienting the edges of a free tree . Every polytree is a DAG . In particular , this is true of the arborescences formed by directing all edges outwards from the roots of a tree . 
 A multitree ( also called a strongly unambiguous graph or a mangrove ) is a directed graph in which there is at most one directed path ( in either direction ) between any two vertices ; equivalently , it is a DAG in which , for every vertex v , the subgraph reachable from v forms a tree . 
 
 = = Computational problems = = 
 
 
 = = = Topological sorting and recognition = = = 
 
 Topological sorting is the algorithmic problem of finding a topological ordering of a given DAG . It can be solved in linear time . Kahn 's algorithm for topological sorting builds the vertex ordering directly . It maintains a list of vertices that have no incoming edges from other vertices that have not already been included in the partially constructed topological ordering ; initially this list consists of the vertices with no incoming edges at all . Then , it repeatedly adds one vertex from this list to the end of the partially constructed topological ordering , and checks whether its neighbors should be added to the list . The algorithm terminates when all vertices have been processed in this way . Alternatively , a topological ordering may be constructed by reversing a postorder numbering of a depth @-@ first search graph traversal . 
 It is also possible to check whether a given directed graph is a DAG in linear time , either by attempting to find a topological ordering and then testing for each edge whether the resulting ordering is valid or alternatively , for some topological sorting algorithms , by verifying that the algorithm successfully orders all the vertices without meeting an error condition . 
 
 = = = Construction from cyclic graphs = = = 
 
 Any undirected graph may be made into a DAG by choosing a total order for its vertices and directing every edge from the earlier endpoint in the order to the later endpoint . The resulting orientation of the edges is called an acyclic orientation . Different total orders may lead to the same acyclic orientation , so an n @-@ vertex graph can have fewer than n ! acyclic orientations . The number of acyclic orientations is equal to | χ ( − 1 ) | , where χ is the chromatic polynomial of the given graph . 
 Any directed graph may be made into a DAG by removing a feedback vertex set or a feedback arc set , a set of vertices or edges ( respectively ) that touches all cycles . However , the smallest such set is NP @-@ hard to find . An arbitrary directed graph may also be transformed into a DAG , called its condensation , by contracting each of its strongly connected components into a single supervertex . When the graph is already acyclic , its smallest feedback vertex sets and feedback arc sets are empty , and its condensation is the graph itself . 
 
 = = = Transitive closure and transitive reduction = = = 
 
 The transitive closure of a given DAG , with n vertices and m edges , may be constructed in time O ( mn ) by using either breadth @-@ first search or depth @-@ first search to test reachability from each vertex . Alternatively , it can be solved in time O ( nω ) where ω < 2 @.@ 373 is the exponent for fast matrix multiplication algorithms ; this is a theoretical improvement over the O ( mn ) bound for dense graphs . 
 In all of these transitive closure algorithms , it is possible to distinguish pairs of vertices that are reachable by at least one path of length two or more from pairs that can only be connected by a length @-@ one path . The transitive reduction consists of the edges that form length @-@ one paths that are the only paths connecting their endpoints . Therefore , the transitive reduction can be constructed in the same asymptotic time bounds as the transitive closure . 
 
 = = = Closure problem = = = 
 
 The closure problem takes as input a directed acyclic graph with weights on its vertices and seeks the minimum ( or maximum ) weight of a closure , a set of vertices with no outgoing edges . ( The problem may be formulated for directed graphs without the assumption of acyclicity , but with no greater generality , because in this case it is equivalent to the same problem on the condensation of the graph . ) It may be solved in polynomial time using a reduction to the maximum flow problem . 
 
 = = = Path algorithms = = = 
 
 Some algorithms become simpler when used on DAGs instead of general graphs , based on the principle of topological ordering . For example , it is possible to find shortest paths and longest paths from a given starting vertex in DAGs in linear time by processing the vertices in a topological order , and calculating the path length for each vertex to be the minimum or maximum length obtained via any of its incoming edges . In contrast , for arbitrary graphs the shortest path may require slower algorithms such as Dijkstra 's algorithm or the Bellman – Ford algorithm , and longest paths in arbitrary graphs are NP @-@ hard to find . 
 
 = = Applications = = 
 
 
 = = = Scheduling = = = 
 
 Directed acyclic graphs representations of partial orderings have many applications in scheduling for systems of tasks with ordering constraints . An important class of problems of this type concern collections of objects that need to be updated , such 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 each 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 generally 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 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 value from another cell . In such a case , the value that is used must be recalculated earlier than the expression that uses it . Topologically ordering the dependency graph , and using this topological order to schedule the cell updates , allows the whole spreadsheet to be updated with only a single evaluation per cell . Similar problems of task ordering arise in makefiles for program compilation and instruction scheduling for low @-@ level computer program optimization . 
 A somewhat different DAG @-@ based formulation of scheduling constraints is used by the program evaluation and review technique ( PERT ) , a method for management of large human projects that was one of the first applications of DAGs . In this method , the vertices of a DAG represent 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 mark 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 controls the total time for the project . Individual milestones can be scheduled according to the lengths of the longest paths ending at their vertices . 
 
 = = = Data processing networks = = = 
 
 A directed acyclic graph may be used to represent a network of processing elements . In this representation , data enters a processing element through its incoming edges and leaves the element 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 it is captured by a register or state element which maintains 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 reference 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 many 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 allows the compiler to perform common subexpression elimination efficiently . 
 
 = = = Causal structures = = = 
 
 Graphs that have vertices representing events , and edges representing causal relations between events , are often acyclic – arranging the vertices in linear order of time , all arrows point in the same direction as time , from parent to child ( due to causality affecting the future , not the past ) , and thus have no loops . 
 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 all 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 made 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 . 
 
 = = = Genealogy and version history = = = 
 
 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 possibility 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 " relationships between women ) and patrilineal descent ( " father " relationships between men ) are trees within this graph . Because no one can become their own ancestor , family trees are acyclic . 
 For the same reason , the version history of a distributed revision control system generally 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 many randomized algorithms in computational geometry , the algorithm maintains a history DAG representing the version history of a geometric structure over the course of a sequence of changes to the structure . For instance in a randomized incremental algorithm for Delaunay triangulation , the triangulation changes 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 . The final triangle reached in this path must be the Delaunay triangle that contains q . 
 
 = = = Data compression = = = 
 
 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 given sequences . When many of the sequences share the same subsequences , these shared subsequences can be represented by a shared part of the DAG , allowing the representation to use 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 source 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 . 
 The same idea of using a DAG to represent a family of paths occurs in the binary decision diagram , a DAG @-@ based data structure for representing binary functions . In a binary decision diagram , each non @-@ sink vertex is labeled by the name of a binary variable , and each sink and each edge is labeled by a 0 or 1 . The function value for any truth assignment to the variables is the value at the sink found by following a path , starting from the single source vertex , that at each non @-@ sink vertex follows the outgoing edge labeled with the value of that vertex 's variable . Just as directed acyclic word graphs can be viewed as a compressed form of tries , binary decision diagrams can be viewed as compressed forms of decision trees that save space by allowing paths to rejoin when they agree on the results of all remaining decisions . 
 
 