Directed acyclic graph - Knowledge (XXG)

889:. 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 509:

cycle would have to be oriented the wrong way. Therefore, every graph with a topological ordering is acyclic. Conversely, every directed acyclic graph has at least one topological ordering. The existence of a topological ordering can therefore be used as an equivalent definition of a directed acyclic graphs: they are exactly the graphs that have topological orderings. 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.

357: 438: 783: 1242:. 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. 458: 1194: 952: 416:. 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. 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 1425:, 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 167: 1353: 1182:

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

799: 136:. 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. 155: 22: 1563:

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

1209:

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

1181:

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

1074:

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

508:

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

112:

in a directed graph is 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

1453:

the vertices are documents with a single publication date. The edges represent the citations from the bibliography of one document to other necessarily earlier documents. The classic example comes from the citations between academic papers as pointed out in the 1965 article "Networks of Scientific

1375:

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

1291:

Graphs in which vertices represent events occurring at a definite time, and where the edges always point from the early time vertex to a late 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

900:

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

1591:

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,

1576:, 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 1252:, 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 73:, by arranging the vertices as a linear ordering that is consistent with all edge directions. DAGs have numerous scientific and computational applications, ranging from biology (evolution, family trees, epidemiology) to information science (citation networks) to computation (scheduling). 1308:. In the version history example below, each version of the software is associated with a unique time, typically the time the version was saved, committed or released. In the citation graph examples below, the documents are published at one time and can only refer to older documents. 1339:

The converse is also true. That is in any application 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

1101:. 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 1518:

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

1478:, earlier patents which are relevant to the current patent claim. By taking the special properties of directed acyclic graphs into account, one can analyse citation networks with techniques not available when analysing the general graphs considered in many studies using 1263:, 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 1587:, 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 1399:, 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. 707: 1226:

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.

1125:, 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 1319:

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

1486:

gives new insights into the citation distributions found in different applications highlighting clear differences in the mechanisms creating citations networks in different contexts. Another technique is

1296:

in the graph so you can never return to a vertex on a path. This reflects our natural intuition that causality means events can only affect the future, they never affect the past, and thus we have no

1474:

provide another example as judges support their conclusions in one case by recalling other earlier decisions made in previous cases. A final example is provided by patents which must refer to earlier

1580:. 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. 424:

of a partial order is a drawing of the transitive reduction in which the orientation of every edge is shown by placing the starting vertex of the edge in a lower position than its ending vertex.

1572:

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

1170:

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

3085: 1218:

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.

1174:, 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. 69:), with each edge directed from one vertex to another, such that following those directions will never form a closed loop. A directed graph is a DAG if and only if it can be 1506:

but it is simple enough to allow for analytic solutions for some of its properties. Many of these can be found by using results derived from the undirected version of the

1154:

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

1053: 1332:, 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 829:) is a DAG in which there is at most one directed path between any two vertices. Equivalently, it is a DAG in which the subgraph reachable from any vertex induces an 1549: 1113:

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

1075:

the only paths connecting their endpoints. Therefore, the transitive reduction can be constructed in the same asymptotic time bounds as the transitive closure.

451:

goes from earlier in the ordering (upper left) to later in the ordering (lower right). A directed graph is acyclic if and only if it has a topological ordering.

1565: 1311:

Sometimes events are not associated with a specific physical time. Provided that pairs of events have a purely causal relationship, that is edges represent

1205:(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 554: 570: 1202: 329:

into graph-theoretic terms. The same method of translating partial orders into DAGs works more generally: for every finite partially ordered set

3387: 3254: 3015: 2981: 2946: 2909: 2791: 2764: 2737: 2700: 2673: 2612: 2585: 2552: 2525: 2498: 2471: 2429: 2173: 1923: 1876: 1849: 1835: 1819: 1792: 1765: 1730: 1654: 1624: 516:

of the reachability relation for the DAG, so any two graphs representing the same partial order have the same set of topological orders.

232:). However, different DAGs may give rise to the same reachability relation and the same partial order. For example, a DAG with two edges 1388:

descent (father-son relationships) are trees within this graph. Because no one can become their own ancestor, family trees are acyclic.

1417:

representing the version history of a geometric structure over the course of a sequence of changes to the structure. For instance in a

2048: 3292: 2639: 2248: 2160:

based topological sorting algorithm, this validity check can be interleaved with the topological sorting algorithm itself; see e.g.

2126: 1700: 1197:

PERT chart for a project with five milestones (labeled 10–50) and six tasks (labeled A–F). There are two critical paths, ADF and BC.

1344:, i.e. there is at least one way to put the vertices in an order such that all edges point in the same direction along that order. 1093:

takes as input a vertex-weighted directed acyclic graph and seeks the minimum (or maximum) weight of a closure – a set of vertices

3453: 1324:

of a DAG is the undirected graph created by adding an (undirected) edge between all parents of the same vertex (sometimes called

1264: 104:

connecting pairs of vertices, where the vertices can be any kind of object that is connected in pairs by edges. In the case of a

1433:

in the Delaunay triangulation, follow a path in the history DAG, at each step moving to the replacement triangle that contains

2412:

Al-Mutawa, H. A.; Dietrich, J.; Marsland, S.; McCartin, C. (2014), "On the shape of circular dependencies in Java programs",

1268: 1206: 1271:

states that the dependencies between modules or components of a large software system should form a directed acyclic graph.

1964: 1328:), and then replacing all directed edges by undirected edges. Another type of graph with a similar causal structure is an 995:

into a single supervertex. When the graph is already acyclic, its smallest feedback vertex sets and feedback arc sets are

1511: 2381:

Cormen et al. 2001, Sections 24.1, The Bellman–Ford algorithm, pp. 588–592, and 24.3, Dijkstra's algorithm, pp. 595–601.

2338: 2235: 1989: 1392: 992: 964: 93: 3120:

Clough, James R.; Gollings, Jamie; Loach, Tamar V.; Evans, Tim S. (2015), "Transitive reduction of citation networks",

2457: 1466:

of a paper is just the in-degree of the corresponding vertex of the citation network. This is an important measure in

984: 956: 913:

for its vertices and directing every edge from the earlier endpoint in the order to the later endpoint. The resulting

851: 1256:

in which each operation is performed by a parallel process as soon as another set of inputs becomes available to it.

1130: 1336:, for instance, these diagrams are often used to estimate the expected value of different choices for intervention. 2116: 1274: 914: 856: 2240: 1573: 2079:

in Proc. 3rd Annual Conference on Uncertainty in Artificial Intelligence (UAI 1987), Seattle, WA, USA, July 1987

3448: 1238:

that computes a function of an input, where the input and output of the function are represented as individual

372:

of a DAG is the graph with the fewest edges that has the same reachability relation as the DAG. It has an edge

3370:

Friedman, S. J.; Supowit, K. J. (1987), "Finding the optimal variable ordering for binary decision diagrams",

3083:

Price, Derek J. de Solla (1976), "A general theory of bibliometric and other cumulative advantage processes",

1781:

Bang-Jensen, Jørgen; Gutin, Gregory Z. (2008), "2.3 Transitive Digraphs, Transitive Closures and Reductions",

303:

of a DAG is the graph with the most edges that has the same reachability relation as the DAG. It has an edge

1972: 1584: 1455: 1126: 3287:, Encyclopedia of Mathematics and its Applications, vol. 105, Cambridge University Press, p. 18, 353:

as its reachability relation. In this way, every finite partially ordered set can be represented as a DAG.

3232: 2886: 1422: 1410: 1187: 543:(without restrictions on the order in which these numbers appear in a topological ordering of the DAG) is 3227:

Crochemore, Maxime; Vérin, Renaud (1997), "Direct construction of compact directed acyclic word graphs",

97: 58: 2372:

Cormen et al. 2001, Section 24.2, Single-source shortest paths in directed acyclic graphs, pp. 592–595.

1596:

that save space by allowing paths to rejoin when they agree on the results of all remaining decisions.

979:, a set of vertices or edges (respectively) that touches all cycles. However, the smallest such set is 3184: 3049: 3010:, Mathematical surveys and monographs, vol. 152, American Mathematical Society, pp. 93–94, 2969: 2336:

Picard, Jean-Claude (1976), "Maximal closure of a graph and applications to combinatorial problems",

2104: 2008: 1865:

Bender, Edward A.; Williamson, S. Gill (2005), "Example 26 (Linear extensions – topological sorts)",

1753: 1418: 1245: 1215: 1211: 1118: 1102: 1027: 945: 505: 369: 70: 1984: 3237: 2725: 1293: 1231: 1171: 1122: 972: 918: 882: 877: 830: 762: 562: 448: 444: 109: 101: 62: 2488: 356: 3393: 3352: 3260: 3174: 3147: 3129: 3102: 3032: 2915: 2435: 2189: 2157: 2054: 1998: 1670: 1488: 1253: 1031: 894: 885:

is the algorithmic problem of finding a topological ordering of a given DAG. It can be solved in

464: 325:

of the DAG, and may therefore be thought of as a direct translation of the reachability relation

300: 2754: 792:, a DAG in which the subgraph reachable from any vertex induces an undirected tree (e.g. in red) 2461: 1491:, which traces the citation links and suggests the most significant citation chains in a given 3415: 3383: 3288: 3250: 3210: 3065: 3040: 3011: 2977: 2942: 2905: 2867: 2787: 2781: 2760: 2733: 2721: 2696: 2669: 2635: 2629: 2608: 2602: 2581: 2548: 2542: 2521: 2515: 2494: 2467: 2425: 2262: 2244: 2169: 2163: 2122: 2044: 1940: 1919: 1872: 1866: 1845: 1839: 1815: 1788: 1761: 1726: 1696: 1650: 1620: 1467: 1377: 1365: 1357: 1329: 712: 525: 391: 3282: 3005: 2963: 2690: 2077: 1809: 1782: 1747: 1690: 3375: 3344: 3317: 3242: 3200: 3192: 3139: 3094: 3057: 2897: 2857: 2847: 2820: 2663: 2573: 2417: 2347: 2230: 2208: 2100: 2034: 1720: 1588: 1560: 1522: 1515: 1507: 1499: 1459: 1316: 1286: 1167: 988: 976: 960: 740: 513: 364:

representing the partial order of set inclusion (⊆) among the subsets of a three-element set

38: 3165:

Evans, T.S.; Calmon, L.; Vasiliauskaite, V. (2020), "The Longest Path in the Price Model",

2941:, Philadelphia, PA, USA: Society for Industrial and Applied Mathematics, pp. 845–854, 2359: 2258: 2355: 2254: 1944: 1695:, Cambridge Computer Science Texts, vol. 14, Cambridge University Press, p. 27, 1479: 1463: 1090: 1084: 758: 732: 282:. Both of these DAGs produce the same partial order, in which the vertices are ordered as 3188: 3053: 2973: 2012: 1757: 3321: 3308:

Lee, C. Y. (1959), "Representation of switching circuits by binary-decision programs",

3205: 2862: 2112: 1915: 1569: 1503: 1492: 1471: 1450: 1426: 720: 105: 50: 2939:

Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '01)

1649:, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, pp. 32–34, 1437:. The final triangle reached in this path must be the Delaunay triangle that contains 1150:

Directed acyclic graph representations of partial orderings have many applications in

3442: 2997: 2655: 2226: 2212: 2026: 1593: 1403: 1361: 1114: 421: 417: 361: 184: 3419: 3397: 3356: 3264: 3151: 2919: 2692:

Probabilistic Boolean Networks: The Modeling and Control of Gene Regulatory Networks

2439: 2058: 1814:, Algorithms and Computation in Mathematics, vol. 5, Springer, pp. 92–93, 1201:

A somewhat different DAG-based formulation of scheduling constraints is used by the

983:

to find. An arbitrary directed graph may also be transformed into a DAG, called its

437: 3106: 3001: 2717: 2659: 2281: 1980: 1968: 1907: 1896: 1716: 1333: 782: 482: 180: 121: 34: 2934: 2193: 2029:; Zacks, Jeff (1994), "Multitrees: enriching and reusing hierarchical structure", 1352: 1193: 951: 463:

Adding the red edges to the blue directed acyclic graph produces another DAG, the

457: 3231:, Lecture Notes in Computer Science, vol. 1264, Springer, pp. 116–129, 3061: 2838:

Kirkpatrick, Bonnie B. (April 2011), "Haplotypes versus genotypes on pedigrees",

1300:. An example of this type of directed acyclic graph are those encountered in the 3278: 2965:

Architecture and Methods for Flexible Content Management in Peer-to-Peer Systems

2933:

Bender, Michael A.; Pemmasani, Giridhar; Skiena, Steven; Sumazin, Pavel (2001),

2901: 2073: 1976: 1385: 1381: 1372: 1321: 1301: 1297: 1249: 1178: 1163: 1159: 1155: 1068: 910: 886: 728: 54: 30: 3196: 1034:

to test reachability from each vertex. Alternatively, it can be solved in time

2824: 2577: 2108: 1235: 724: 702:{\displaystyle a_{n}=\sum _{k=1}^{n}(-1)^{k-1}{n \choose k}2^{k(n-k)}a_{n-k}.} 3246: 2266: 3424: 3348: 3143: 3007:

Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures

2808: 2351: 1949: 1871:, Dover Books on Computer Science, Courier Dover Publications, p. 142, 1564:

it would take to list out all of the sequences separately. For example, the

1475: 1406: 1312: 996: 890: 817: 789: 3214: 3098: 3069: 2871: 2852: 2568:

Dennis, Jack B. (1974), "First version of a data flow procedure language",

1315:

between the events, we will have a directed acyclic graph. For instance, a

2421: 2039: 1583:

The same idea of using a DAG to represent a family of paths occurs in the

1429:

queries to be answered efficiently: to find the location of a query point

859:

the edges of an undirected tree away from a particular vertex, called the

512:

The family of topological orderings of a DAG is the same as the family of

2239:. Series of Books in the Mathematical Sciences (1st ed.). New York: 1615:

Thulasiraman, K.; Swamy, M. N. S. (1992), "5.7 Acyclic Directed Graphs",

1260: 1183: 1151: 921:. Different total orders may lead to the same acyclic orientation, so an 901:

successfully orders all the vertices without meeting an error condition.

838: 805: 3379: 398:

of the DAG. It is a subgraph of the DAG, formed by discarding the edges

166: 2517:

What Every Engineer Should Know About Decision Making Under Uncertainty

2490:

The Compiler Design Handbook: Optimizations and Machine Code Generation

2031:

Proc. SIGCHI conference on Human Factors in Computing Systems (CHI '94)

1134: 980: 2236:

Computers and Intractability: A Guide to the Theory of NP-Completeness

846:) is a multitree formed by orienting the edges of an undirected tree. 2572:, Lecture Notes in Computer Science, vol. 19, pp. 362–376, 2003: 932:

acyclic orientations. The number of acyclic orientations is equal to

108:, each edge has an orientation, from one vertex to another vertex. A 798: 3179: 2286:

Structural Models: An Introduction to the Theory of Directed Graphs

2076:(1987), "The recovery of causal poly-trees from statistical data", 145:

Reachability relation, transitive closure, and transitive reduction

3134: 1458:

who went on to produce the first model of a citation network, the

1351: 1192: 950: 154: 21: 20: 1749:

Loop Transformations for Restructuring Compilers: The Foundations

1895:

Robinson, R. W. (1973), "Counting labeled acyclic digraphs", in

1787:, Springer Monographs in Mathematics, Springer, pp. 36–39, 1577: 1483: 1305: 769:

preserves the property that all matrix coefficients are 0 or 1.

190:

on the vertices of the DAG. In this partial order, two vertices

252:

has the same reachability relation as the DAG with three edges

2695:, Society for Industrial and Applied Mathematics, p. 58, 2631:

Large-Scale Software Architecture: A Practical Guide using UML

1592:

binary decision diagrams can be viewed as compressed forms of

1396: 1239: 1838:; Wayne, Kevin (2011), "4,2,25 Unique topological ordering", 765:, its adjacency matrix must have a zero diagonal, so adding 408:

for which the DAG also contains a longer directed path from

2935:"Finding least common ancestors in directed acyclic graphs" 808:, a DAG formed by orienting the edges of an undirected tree 549: 528:

problem of counting directed acyclic graphs was studied by

3432: 3372:

Proc. 24th ACM/IEEE Design Automation Conference (DAC '87)

2894:

IEEE Symposium on Information Visualization (INFOVIS 2005)

909:

Any undirected graph may be made into a DAG by choosing a

1267:

efficiently. At a higher level of code organization, the

971:

Any directed graph may be made into a DAG by removing a

3086:

Journal of the American Society for Information Science

1725:, Monographs in Computer Science, Springer, p. 9, 1121:

from a given starting vertex in DAGs in linear time by

753:

is a (0,1) matrix with all eigenvalues positive, where

716: 3335:

Akers, Sheldon B. (1978), "Binary decision diagrams",

113:

acyclic graph is a directed graph that has no cycles.

1525: 573: 76:

Directed acyclic graphs are sometimes instead called

1985:"Acyclic digraphs and eigenvalues of (0,1)-matrices" 1645:

Bang-Jensen, Jørgen (2008), "2.1 Acyclic Digraphs",

345:

is automatically a transitively closed DAG, and has

1230:For instance, in electronic circuit design, static 337:, the graph that has a vertex for every element of 2887:"Interactive visualization of genealogical graphs" 1844:(4th ed.), Addison-Wesley, pp. 598–599, 1543: 1234:blocks can be represented as an acyclic system of 701: 649: 636: 2323: 2284:; Norman, Robert Z.; Cartwright, Dorwin (1965), 1692:Simulation Techniques for Discrete Event Systems 1158:after one of the cells has been changed, or the 2689:Shmulevich, Ilya; Dougherty, Edward R. (2010), 2414:23rd Australian Software Engineering Conference 208:exactly when there exists a directed path from 1559:Directed acyclic graphs may also be used as a 1304:though in this case the graphs considered are 1123:processing the vertices in a topological order 967:of the blue graph into a single yellow vertex. 547:1, 1, 3, 25, 543, 29281, 3781503, … (sequence 1784:Digraphs: Theory, Algorithms and Applications 1647:Digraphs: Theory, Algorithms and Applications 1190:for low-level computer program optimization. 1056:; this is a theoretical improvement over the 1054:exponent for matrix multiplication algorithms 467:of the blue graph. For each red or blue edge 8: 3374:, New York, NY, USA: ACM, pp. 348–356, 2132:Section 22.4, Topological sort, pp. 549–552. 1384:descent (mother-daughter relationships) and 1248:languages describe systems of operations on 1133:, and longest paths in arbitrary graphs are 1007:The transitive closure of a given DAG, with 999:, and its condensation is the graph itself. 959:of the blue directed graph. It is formed by 2493:(2nd ed.), CRC Press, pp. 19–39, 2121:(2nd ed.), MIT Press and McGraw-Hill, 1502:is too simple to be a realistic model of a 1003:Transitive closure and transitive reduction 3031:Price, Derek J. de Solla (July 30, 1965), 2885:McGuffin, M. J.; Balakrishnan, R. (2005), 2451: 2449: 2144: 1162:of a piece of computer software after its 341:and an edge for every pair of elements in 3236: 3204: 3178: 3133: 2861: 2851: 2730:Probabilistic Networks and Expert Systems 2601:Touati, Sid; de Dinechin, Benoit (2014), 2466:(2nd ed.), CRC Press, p. 1181, 2271:, Problems GT7 and GT8, pp. 191–192. 2038: 2002: 1746:Banerjee, Utpal (1993), "Exercise 2(c)", 1524: 1341: 955:The yellow directed acyclic graph is the 684: 659: 648: 635: 633: 621: 602: 591: 578: 572: 2809:"Causal diagrams for empirical research" 2628:Garland, Jeff; Anthony, Richard (2003), 719:proved, that the same numbers count the 529: 355: 128:when there exists a path that starts at 2487:Srikant, Y. N.; Shankar, Priti (2007), 2095: 2093: 1890: 1888: 1604: 1203:program evaluation and review technique 489:: there exists a blue path starting at 2783:Encyclopedia of Epidemiology, Volume 1 2668:, Oxford University Press, p. 4, 2634:, John Wiley & Sons, p. 215, 2607:, John Wiley & Sons, p. 123, 2399: 2311: 2299: 2140: 2138: 1901:New Directions in the Theory of Graphs 1868:A Short Course in Discrete Mathematics 1640: 1638: 1636: 1610: 1608: 1302:causal set approach to quantum gravity 16:Directed graph with no directed cycles 1722:The Design and Analysis of Algorithms 1675:Graph theory: an algorithmic approach 1166:has been changed. In this context, a 561:These numbers may be computed by the 7: 2288:, John Wiley & Sons, p. 63 1619:, John Wiley and Son, p. 118, 872:Topological sorting and recognition 447:of a directed acyclic graph: every 25:Example of a directed acyclic graph 3435:– an online tool for creating DAGs 3322:10.1002/j.1538-7305.1959.tb01585.x 2756:The Technology Management Handbook 1903:, Academic Press, pp. 239–273 1677:, Academic Press, pp. 170–174 1015:edges, may be constructed in time 925:-vertex graph can have fewer than 640: 120:of a directed graph is said to be 14: 2456:Gross, Jonathan L.; Yellen, Jay; 1177:For instance, when one cell of a 2840:Algorithms for Molecular Biology 2194:"Acyclic orientations of graphs" 1265:common subexpression elimination 797: 781: 456: 436: 183:of a DAG can be formalized as a 165: 153: 3033:"Networks of Scientific Papers" 1811:Graphs, Networks and Algorithms 905:Construction from cyclic graphs 321:) in the reachability relation 3337:IEEE Transactions on Computers 3284:Applied Combinatorics on Words 3229:Combinatorial Pattern Matching 2759:, CRC Press, p. 9-7, 2728:(1999), "3.2.1 Moralization", 2324:Bang-Jensen & Gutin (2008) 2168:, Springer, pp. 179–181, 1538: 1532: 1360:, with many marriages between 1269:acyclic dependencies principle 675: 663: 618: 608: 1: 3310:Bell System Technical Journal 2604:Advanced Backend Optimization 1617:Graphs: Theory and Algorithms 1348:Genealogy and version history 993:strongly connected components 394:of the reachability relation 3062:10.1126/science.149.3683.510 2732:, Springer, pp. 31–33, 2213:10.1016/0012-365X(73)90108-8 1990:Journal of Integer Sequences 1413:, the algorithm maintains a 1393:distributed revision control 1186:for program compilation and 965:strongly connected component 761:. Because a DAG cannot have 382:for every pair of vertices ( 313:for every pair of vertices ( 3122:Journal of Complex Networks 2902:10.1109/INFVIS.2005.1532124 2541:Sapatnekar, Sachin (2004), 2390:Cormen et al. 2001, p. 966. 2165:The Algorithm Design Manual 1910:; Palmer, Edgar M. (1973), 1808:Jungnickel, Dieter (2012), 1566:directed acyclic word graph 1275:Feedforward neural networks 1214:in this DAG represents the 1097:, such that no edges leave 3470: 3197:10.1038/s41598-020-67421-8 2520:, CRC Press, p. 160, 2162:Skiena, Steven S. (2009), 2118:Introduction to Algorithms 1284: 1082: 917:of the edges is called an 875: 823:strongly unambiguous graph 773:Related families of graphs 541: = 0, 1, 2, 3, … 216:in the DAG; that is, when 57:. That is, it consists of 2753:Dorf, Richard C. (1998), 2578:10.1007/3-540-06859-7_145 2547:, Springer, p. 133, 2241:W. H. Freeman and Company 1391:The version history of a 1210:to perform the task. The 520:Combinatorial enumeration 3247:10.1007/3-540-63220-4_55 2968:, Springer, p. 59, 2780:Boslaugh, Sarah (2008), 2463:Handbook of Graph Theory 2416:, IEEE, pp. 48–57, 1945:"Weisstein's Conjecture" 1752:, Springer, p. 19, 1222:Data processing networks 855:is a polytree formed by 743:of a DAG if and only if 532:. The number of DAGs on 172:Its transitive reduction 3454:Directed acyclic graphs 3349:10.1109/TC.1978.1675141 2825:10.1093/biomet/82.4.669 2726:Spiegelhalter, David J. 2352:10.1287/mnsc.22.11.1268 1585:binary decision diagram 1456:Derek J. de Solla Price 1376:father's side) causing 140:Mathematical properties 78:acyclic directed graphs 3099:10.1002/asi.4630270505 2962:Bartlang, Udo (2010), 2853:10.1186/1748-7188-6-10 2514:Wang, John X. (2002), 1561:compact representation 1545: 1544:{\displaystyle \ln(n)} 1423:Delaunay triangulation 1419:randomized incremental 1411:computational geometry 1369: 1198: 1188:instruction scheduling 1131:Bellman–Ford algorithm 968: 867:Computational problems 703: 607: 536:labeled vertices, for 365: 43:directed acyclic graph 26: 3144:10.1093/comnet/cnu039 2807:Pearl, Judea (1995), 2786:, SAGE, p. 255, 2722:Lauritzen, Steffen L. 2570:Programming Symposium 2422:10.1109/ASWEC.2014.15 2105:Leiserson, Charles E. 2040:10.1145/191666.191778 1912:Graphical Enumeration 1546: 1512:Barabási–Albert model 1355: 1306:transitively complete 1277:are another example. 1196: 1050: < 2.373 954: 863:of the arborescence. 704: 587: 359: 181:reachability relation 71:topologically ordered 24: 2201:Discrete Mathematics 2033:, pp. 330–336, 1689:Mitrani, I. (1982), 1523: 1484:transitive reduction 1342:topological ordering 1246:Dataflow programming 1127:Dijkstra's algorithm 1103:maximum flow problem 1028:breadth-first search 948:of the given graph. 946:chromatic polynomial 571: 506:topological ordering 445:topological ordering 428:Topological ordering 370:transitive reduction 124:from another vertex 3380:10.1145/37888.37941 3189:2020NatSR..1010503E 3054:1965Sci...149..510D 2974:2010aamf.book.....B 2716:Cowell, Robert G.; 2190:Stanley, Richard P. 2013:2004JIntS...7...33M 1758:1993ltfr.book.....B 1671:Christofides, Nicos 1462:. In this case the 1356:Family tree of the 1232:combinational logic 1172:circular dependency 973:feedback vertex set 919:acyclic orientation 883:Topological sorting 878:Topological sorting 717:McKay et al. (2004) 563:recurrence relation 3416:Weisstein, Eric W. 3167:Scientific Reports 2896:, pp. 16–23, 2339:Management Science 2158:depth-first search 2085:, pp. 222–228 1971:; Wanless, I. M.; 1941:Weisstein, Eric W. 1541: 1489:main path analysis 1370: 1254:parallel algorithm 1199: 1032:depth-first search 969: 895:depth-first search 699: 465:transitive closure 366: 301:transitive closure 228:is reachable from 27: 3420:"Acyclic Digraph" 3389:978-0-8186-0781-3 3256:978-3-540-63220-7 3048:(3683): 510–515, 3017:978-0-8218-7533-9 2983:978-3-8348-9645-2 2948:978-0-89871-490-6 2911:978-0-7803-9464-3 2793:978-1-4129-2816-8 2766:978-0-8493-8577-3 2739:978-0-387-98767-5 2702:978-0-89871-692-4 2675:978-0-19-803928-0 2614:978-1-118-64894-0 2587:978-3-540-06859-4 2554:978-1-4020-7671-8 2527:978-0-8247-4373-4 2500:978-1-4200-4383-9 2473:978-1-4398-8018-0 2431:978-1-4799-3149-1 2346:(11): 1268–1272, 2231:Johnson, David S. 2227:Garey, Michael R. 2175:978-1-84800-070-4 2145:Jungnickel (2012) 2109:Rivest, Ronald L. 2101:Cormen, Thomas H. 2027:Furnas, George W. 2016:, Article 04.3.3. 1925:978-0-12-324245-7 1878:978-0-486-43946-4 1851:978-0-13-276256-4 1836:Sedgewick, Robert 1821:978-3-642-32278-5 1794:978-1-84800-998-1 1767:978-0-7923-9318-4 1732:978-0-387-97687-7 1656:978-1-84800-997-4 1626:978-0-471-51356-8 1514:. However, since 1468:citation analysis 1378:pedigree collapse 1366:pedigree collapse 1358:Ptolemaic dynasty 1330:influence diagram 1281:Causal structures 897:graph traversal. 715:conjectured, and 713:Eric W. Weisstein 647: 526:graph enumeration 514:linear extensions 392:covering relation 3461: 3429: 3428: 3402: 3400: 3367: 3361: 3359: 3332: 3326: 3324: 3305: 3299: 3297: 3275: 3269: 3267: 3240: 3224: 3218: 3217: 3208: 3182: 3162: 3156: 3154: 3137: 3117: 3111: 3109: 3080: 3074: 3072: 3037: 3028: 3022: 3020: 2994: 2988: 2986: 2959: 2953: 2951: 2930: 2924: 2922: 2891: 2882: 2876: 2874: 2865: 2855: 2835: 2829: 2827: 2804: 2798: 2796: 2777: 2771: 2769: 2750: 2744: 2742: 2718:Dawid, A. Philip 2713: 2707: 2705: 2686: 2680: 2678: 2652: 2646: 2644: 2625: 2619: 2617: 2598: 2592: 2590: 2565: 2559: 2557: 2538: 2532: 2530: 2511: 2505: 2503: 2484: 2478: 2476: 2453: 2444: 2442: 2409: 2403: 2397: 2391: 2388: 2382: 2379: 2373: 2370: 2364: 2362: 2333: 2327: 2321: 2315: 2309: 2303: 2297: 2291: 2289: 2278: 2272: 2270: 2223: 2217: 2215: 2198: 2186: 2180: 2178: 2154: 2148: 2142: 2133: 2131: 2097: 2088: 2086: 2084: 2072:Rebane, George; 2069: 2063: 2061: 2042: 2023: 2017: 2015: 2006: 1977:Sloane, N. J. A. 1961: 1955: 1954: 1953: 1936: 1930: 1928: 1904: 1892: 1883: 1881: 1862: 1856: 1854: 1832: 1826: 1824: 1805: 1799: 1797: 1778: 1772: 1770: 1743: 1737: 1735: 1713: 1707: 1705: 1686: 1680: 1678: 1667: 1661: 1659: 1642: 1631: 1629: 1612: 1589:truth assignment 1555:Data compression 1550: 1548: 1547: 1542: 1504:citation network 1480:network analysis 1472:Court judgements 1440: 1436: 1432: 1395:system, such as 1380:. The graphs of 1317:Bayesian network 1313:causal relations 1287:Bayesian network 1168:dependency graph 1066: 1051: 1044: 1026:by using either 1025: 1014: 1010: 977:feedback arc set 943: 939: 931: 924: 801: 785: 768: 756: 752: 741:adjacency matrix 738: 708: 706: 705: 700: 695: 694: 679: 678: 654: 653: 652: 639: 632: 631: 606: 601: 583: 582: 552: 542: 535: 496: 492: 488: 480: 476: 460: 440: 415: 411: 407: 397: 389: 385: 381: 352: 344: 340: 336: 328: 324: 320: 316: 312: 295: 281: 271: 261: 251: 241: 231: 227: 223: 219: 215: 211: 207: 197: 193: 189: 169: 157: 135: 131: 127: 119: 82:acyclic digraphs 39:computer science 3469: 3468: 3464: 3463: 3462: 3460: 3459: 3458: 3449:Directed graphs 3439: 3438: 3414: 3413: 3410: 3405: 3390: 3369: 3368: 3364: 3334: 3333: 3329: 3307: 3306: 3302: 3295: 3277: 3276: 3272: 3257: 3226: 3225: 3221: 3164: 3163: 3159: 3119: 3118: 3114: 3082: 3081: 3077: 3035: 3030: 3029: 3025: 3018: 2996: 2995: 2991: 2984: 2961: 2960: 2956: 2949: 2932: 2931: 2927: 2912: 2889: 2884: 2883: 2879: 2837: 2836: 2832: 2806: 2805: 2801: 2794: 2779: 2778: 2774: 2767: 2752: 2751: 2747: 2740: 2715: 2714: 2710: 2703: 2688: 2687: 2683: 2676: 2665:Causal Learning 2654: 2653: 2649: 2642: 2627: 2626: 2622: 2615: 2600: 2599: 2595: 2588: 2567: 2566: 2562: 2555: 2540: 2539: 2535: 2528: 2513: 2512: 2508: 2501: 2486: 2485: 2481: 2474: 2455: 2454: 2447: 2432: 2411: 2410: 2406: 2398: 2394: 2389: 2385: 2380: 2376: 2371: 2367: 2335: 2334: 2330: 2322: 2318: 2310: 2306: 2298: 2294: 2280: 2279: 2275: 2251: 2225: 2224: 2220: 2196: 2188: 2187: 2183: 2176: 2161: 2155: 2151: 2143: 2136: 2129: 2113:Stein, Clifford 2099: 2098: 2091: 2082: 2071: 2070: 2066: 2051: 2025: 2024: 2020: 1963: 1962: 1958: 1939: 1938: 1937: 1933: 1926: 1906: 1894: 1893: 1886: 1879: 1864: 1863: 1859: 1852: 1834: 1833: 1829: 1822: 1807: 1806: 1802: 1795: 1780: 1779: 1775: 1768: 1745: 1744: 1740: 1733: 1715: 1714: 1710: 1703: 1688: 1687: 1683: 1669: 1668: 1664: 1657: 1644: 1643: 1634: 1627: 1614: 1613: 1606: 1602: 1557: 1521: 1520: 1482:. For instance 1447: 1445:Citation graphs 1438: 1434: 1430: 1362:close relatives 1350: 1289: 1283: 1224: 1148: 1143: 1111: 1109:Path algorithms 1091:closure problem 1087: 1085:Closure problem 1081: 1079:Closure problem 1057: 1046: 1035: 1016: 1012: 1008: 1005: 941: 933: 926: 922: 907: 893:numbering of a 880: 874: 869: 842:(also called a 831:undirected tree 821:(also called a 813: 812: 811: 810: 809: 802: 794: 793: 786: 775: 766: 759:identity matrix 754: 744: 736: 731:. The proof is 680: 655: 634: 617: 574: 569: 568: 548: 537: 533: 530:Robinson (1973) 522: 502: 501: 500: 499: 498: 494: 490: 486: 478: 468: 461: 453: 452: 441: 430: 413: 409: 399: 395: 387: 383: 373: 346: 342: 338: 330: 326: 322: 318: 314: 304: 283: 273: 263: 253: 243: 233: 229: 225: 221: 217: 213: 209: 199: 198:are ordered as 195: 191: 187: 177: 176: 175: 174: 173: 170: 162: 161: 158: 147: 142: 133: 129: 125: 117: 90: 55:directed cycles 33:, particularly 17: 12: 11: 5: 3467: 3465: 3457: 3456: 3451: 3441: 3440: 3437: 3436: 3430: 3409: 3408:External links 3406: 3404: 3403: 3388: 3362: 3343:(6): 509–516, 3327: 3316:(4): 985–999, 3300: 3293: 3270: 3255: 3238:10.1.1.53.6273 3219: 3157: 3128:(2): 189–203, 3112: 3093:(5): 292–306, 3075: 3023: 3016: 2989: 2982: 2954: 2947: 2925: 2910: 2877: 2830: 2819:(4): 669–709, 2799: 2792: 2772: 2765: 2745: 2738: 2708: 2701: 2681: 2674: 2656:Gopnik, Alison 2647: 2640: 2620: 2613: 2593: 2586: 2560: 2553: 2533: 2526: 2506: 2499: 2479: 2472: 2445: 2430: 2404: 2392: 2383: 2374: 2365: 2328: 2316: 2304: 2292: 2273: 2249: 2218: 2207:(2): 171–178, 2181: 2174: 2149: 2134: 2127: 2089: 2064: 2050:978-0897916509 2049: 2018: 1956: 1931: 1924: 1918:, p. 19, 1916:Academic Press 1884: 1877: 1857: 1850: 1827: 1820: 1800: 1793: 1773: 1766: 1738: 1731: 1708: 1701: 1681: 1662: 1655: 1632: 1625: 1603: 1601: 1598: 1594:decision trees 1570:data structure 1556: 1553: 1540: 1537: 1534: 1531: 1528: 1493:citation graph 1464:citation count 1451:citation graph 1446: 1443: 1427:point location 1421:algorithm for 1349: 1346: 1285:Main article: 1282: 1279: 1223: 1220: 1147: 1144: 1142: 1139: 1115:shortest paths 1110: 1107: 1083:Main article: 1080: 1077: 1004: 1001: 906: 903: 876:Main article: 873: 870: 868: 865: 803: 796: 795: 787: 780: 779: 778: 777: 776: 774: 771: 723:for which all 721:(0,1) matrices 710: 709: 698: 693: 690: 687: 683: 677: 674: 671: 668: 665: 662: 658: 651: 646: 643: 638: 630: 627: 624: 620: 616: 613: 610: 605: 600: 597: 594: 590: 586: 581: 577: 559: 558: 521: 518: 493:and ending at 462: 455: 454: 442: 435: 434: 433: 432: 431: 429: 426: 418:graph drawings 171: 164: 163: 159: 152: 151: 150: 149: 148: 146: 143: 141: 138: 106:directed graph 89: 86: 51:directed graph 15: 13: 10: 9: 6: 4: 3: 2: 3466: 3455: 3452: 3450: 3447: 3446: 3444: 3434: 3431: 3427: 3426: 3421: 3417: 3412: 3411: 3407: 3399: 3395: 3391: 3385: 3381: 3377: 3373: 3366: 3363: 3358: 3354: 3350: 3346: 3342: 3338: 3331: 3328: 3323: 3319: 3315: 3311: 3304: 3301: 3296: 3294:9780521848022 3290: 3286: 3285: 3280: 3274: 3271: 3266: 3262: 3258: 3252: 3248: 3244: 3239: 3234: 3230: 3223: 3220: 3216: 3212: 3207: 3202: 3198: 3194: 3190: 3186: 3181: 3176: 3172: 3168: 3161: 3158: 3153: 3149: 3145: 3141: 3136: 3131: 3127: 3123: 3116: 3113: 3108: 3104: 3100: 3096: 3092: 3088: 3087: 3079: 3076: 3071: 3067: 3063: 3059: 3055: 3051: 3047: 3043: 3042: 3034: 3027: 3024: 3019: 3013: 3009: 3008: 3003: 3002:Sharir, Micha 2999: 2993: 2990: 2985: 2979: 2975: 2971: 2967: 2966: 2958: 2955: 2950: 2944: 2940: 2936: 2929: 2926: 2921: 2917: 2913: 2907: 2903: 2899: 2895: 2888: 2881: 2878: 2873: 2869: 2864: 2859: 2854: 2849: 2845: 2841: 2834: 2831: 2826: 2822: 2818: 2814: 2810: 2803: 2800: 2795: 2789: 2785: 2784: 2776: 2773: 2768: 2762: 2758: 2757: 2749: 2746: 2741: 2735: 2731: 2727: 2723: 2719: 2712: 2709: 2704: 2698: 2694: 2693: 2685: 2682: 2677: 2671: 2667: 2666: 2661: 2660:Schulz, Laura 2657: 2651: 2648: 2643: 2641:9780470856383 2637: 2633: 2632: 2624: 2621: 2616: 2610: 2606: 2605: 2597: 2594: 2589: 2583: 2579: 2575: 2571: 2564: 2561: 2556: 2550: 2546: 2545: 2537: 2534: 2529: 2523: 2519: 2518: 2510: 2507: 2502: 2496: 2492: 2491: 2483: 2480: 2475: 2469: 2465: 2464: 2459: 2452: 2450: 2446: 2441: 2437: 2433: 2427: 2423: 2419: 2415: 2408: 2405: 2401: 2400:Skiena (2009) 2396: 2393: 2387: 2384: 2378: 2375: 2369: 2366: 2361: 2357: 2353: 2349: 2345: 2341: 2340: 2332: 2329: 2325: 2320: 2317: 2313: 2312:Skiena (2009) 2308: 2305: 2301: 2300:Skiena (2009) 2296: 2293: 2287: 2283: 2282:Harary, Frank 2277: 2274: 2268: 2264: 2260: 2256: 2252: 2250:9780716710455 2246: 2242: 2238: 2237: 2232: 2228: 2222: 2219: 2214: 2210: 2206: 2202: 2195: 2191: 2185: 2182: 2177: 2171: 2167: 2166: 2159: 2153: 2150: 2146: 2141: 2139: 2135: 2130: 2128:0-262-03293-7 2124: 2120: 2119: 2114: 2110: 2106: 2102: 2096: 2094: 2090: 2081: 2080: 2075: 2068: 2065: 2060: 2056: 2052: 2046: 2041: 2036: 2032: 2028: 2022: 2019: 2014: 2010: 2005: 2000: 1996: 1992: 1991: 1986: 1982: 1978: 1974: 1973:Oggier, F. E. 1970: 1966: 1960: 1957: 1952: 1951: 1946: 1942: 1935: 1932: 1927: 1921: 1917: 1913: 1909: 1908:Harary, Frank 1902: 1898: 1891: 1889: 1885: 1880: 1874: 1870: 1869: 1861: 1858: 1853: 1847: 1843: 1842: 1837: 1831: 1828: 1823: 1817: 1813: 1812: 1804: 1801: 1796: 1790: 1786: 1785: 1777: 1774: 1769: 1763: 1759: 1755: 1751: 1750: 1742: 1739: 1734: 1728: 1724: 1723: 1718: 1717:Kozen, Dexter 1712: 1709: 1704: 1702:9780521282826 1698: 1694: 1693: 1685: 1682: 1676: 1672: 1666: 1663: 1658: 1652: 1648: 1641: 1639: 1637: 1633: 1628: 1622: 1618: 1611: 1609: 1605: 1599: 1597: 1595: 1590: 1586: 1581: 1579: 1575: 1571: 1567: 1562: 1554: 1552: 1535: 1529: 1526: 1517: 1516:Price's model 1513: 1509: 1505: 1501: 1496: 1494: 1490: 1485: 1481: 1477: 1473: 1469: 1465: 1461: 1457: 1452: 1444: 1442: 1428: 1424: 1420: 1416: 1412: 1408: 1405: 1400: 1398: 1394: 1389: 1387: 1383: 1379: 1374: 1367: 1363: 1359: 1354: 1347: 1345: 1343: 1337: 1335: 1331: 1327: 1323: 1318: 1314: 1309: 1307: 1303: 1299: 1295: 1288: 1280: 1278: 1276: 1272: 1270: 1266: 1262: 1257: 1255: 1251: 1247: 1243: 1241: 1237: 1233: 1228: 1221: 1219: 1217: 1216:critical path 1213: 1208: 1204: 1195: 1191: 1189: 1185: 1180: 1175: 1173: 1169: 1165: 1161: 1157: 1153: 1145: 1140: 1138: 1136: 1132: 1128: 1124: 1120: 1119:longest paths 1116: 1108: 1106: 1104: 1100: 1096: 1092: 1086: 1078: 1076: 1072: 1070: 1064: 1060: 1055: 1049: 1042: 1038: 1033: 1029: 1023: 1019: 1011:vertices and 1002: 1000: 998: 994: 990: 986: 982: 978: 974: 966: 962: 958: 953: 949: 947: 937: 929: 920: 916: 912: 904: 902: 898: 896: 892: 888: 884: 879: 871: 866: 864: 862: 858: 854: 853: 847: 845: 844:directed tree 841: 840: 834: 832: 828: 824: 820: 819: 807: 800: 791: 784: 772: 770: 764: 760: 751: 748: + 747: 742: 734: 730: 727:are positive 726: 722: 718: 714: 696: 691: 688: 685: 681: 672: 669: 666: 660: 656: 644: 641: 628: 625: 622: 614: 611: 603: 598: 595: 592: 588: 584: 579: 575: 567: 566: 565: 564: 556: 551: 546: 545: 544: 540: 531: 527: 519: 517: 515: 510: 507: 484: 475: 471: 466: 459: 450: 446: 439: 427: 425: 423: 422:Hasse diagram 419: 406: 402: 393: 380: 376: 371: 363: 362:Hasse diagram 358: 354: 350: 334: 311: 307: 302: 297: 294: 290: 286: 280: 276: 270: 266: 260: 256: 250: 246: 240: 236: 206: 202: 186: 185:partial order 182: 168: 156: 144: 139: 137: 123: 114: 111: 107: 103: 99: 96:is formed by 95: 87: 85: 83: 79: 74: 72: 68: 65:(also called 64: 60: 56: 52: 48: 44: 40: 36: 32: 23: 19: 3423: 3371: 3365: 3340: 3336: 3330: 3313: 3309: 3303: 3283: 3279:Lothaire, M. 3273: 3228: 3222: 3173:(1): 10503, 3170: 3166: 3160: 3125: 3121: 3115: 3090: 3084: 3078: 3045: 3039: 3026: 3006: 2992: 2964: 2957: 2938: 2928: 2893: 2880: 2843: 2839: 2833: 2816: 2812: 2802: 2782: 2775: 2755: 2748: 2729: 2711: 2691: 2684: 2664: 2650: 2630: 2623: 2603: 2596: 2569: 2563: 2543: 2536: 2516: 2509: 2489: 2482: 2462: 2413: 2407: 2395: 2386: 2377: 2368: 2343: 2337: 2331: 2319: 2307: 2295: 2285: 2276: 2234: 2221: 2204: 2200: 2184: 2164: 2152: 2147:, pp. 50–51. 2117: 2078: 2074:Pearl, Judea 2067: 2030: 2021: 2004:math/0310423 1994: 1988: 1969:Royle, G. F. 1965:McKay, B. D. 1959: 1948: 1934: 1911: 1900: 1867: 1860: 1840: 1830: 1810: 1803: 1783: 1776: 1748: 1741: 1721: 1711: 1691: 1684: 1674: 1665: 1646: 1616: 1582: 1558: 1497: 1448: 1414: 1401: 1390: 1373:Family trees 1371: 1338: 1334:epidemiology 1325: 1310: 1298:causal loops 1290: 1273: 1258: 1250:data streams 1244: 1229: 1225: 1212:longest path 1200: 1176: 1160:object files 1149: 1141:Applications 1112: 1098: 1094: 1088: 1073: 1069:dense graphs 1062: 1058: 1047: 1040: 1036: 1021: 1017: 1006: 991:each of its 985:condensation 970: 957:condensation 935: 927: 908: 899: 881: 860: 852:arborescence 850: 848: 843: 837: 835: 826: 822: 816: 814: 757:denotes the 749: 745: 729:real numbers 711: 560: 538: 523: 511: 503: 473: 469: 404: 400: 378: 374: 367: 348: 332: 309: 305: 298: 292: 288: 284: 278: 274: 268: 264: 258: 254: 248: 244: 238: 234: 204: 200: 178: 132:and ends at 115: 91: 81: 77: 75: 66: 46: 42: 35:graph theory 28: 18: 2998:Pach, János 2458:Zhang, Ping 1905:. See also 1508:Price model 1500:Price model 1460:Price model 1454:Papers" by 1415:history DAG 1386:patrilineal 1382:matrilineal 1322:moral graph 1292:follow any 1236:logic gates 1179:spreadsheet 1164:source code 1156:spreadsheet 989:contracting 961:contracting 915:orientation 911:total order 887:linear time 735:: a matrix 725:eigenvalues 88:Definitions 31:mathematics 3443:Categories 3180:1903.03667 2846:(10): 10, 2813:Biometrika 1897:Harary, F. 1841:Algorithms 1600:References 1407:algorithms 1404:randomized 1207:milestones 1152:scheduling 1146:Scheduling 1067:bound for 763:self-loops 220:can reach 3425:MathWorld 3233:CiteSeerX 3135:1310.8224 2402:, p. 469. 2314:, p. 496. 2302:, p. 495. 2267:247570676 2115:(2001) , 1950:MathWorld 1530:⁡ 1476:prior art 1261:compilers 1184:makefiles 1137:to find. 891:postorder 857:orienting 818:multitree 790:multitree 733:bijective 689:− 670:− 626:− 612:− 589:∑ 483:reachable 390:) in the 122:reachable 116:A vertex 3398:14796451 3357:21028055 3281:(2005), 3265:17045308 3215:32601403 3152:10228152 3070:14325149 2920:15449409 2872:21504603 2662:(2007), 2460:(2013), 2440:17570052 2326:, p. 38. 2233:(1979). 2192:(1973), 2059:18710118 1983:(2004), 1981:Wilf, H. 1719:(1992), 1673:(1975), 1402:In many 1364:causing 1326:marrying 940:, where 839:polytree 827:mangrove 806:polytree 98:vertices 59:vertices 53:with no 3433:DAGitty 3206:7324613 3185:Bibcode 3107:8536863 3050:Bibcode 3041:Science 2970:Bibcode 2863:3102622 2360:0403596 2259:0519066 2009:Bibcode 1899:(ed.), 1754:Bibcode 1574:strings 1135:NP-hard 1129:or the 1052:is the 981:NP-hard 944:is the 553:in the 550:A003024 100:and by 49:) is a 3396: 3386: 3355: 3291: 3263: 3253: 3235: 3213: 3203: 3150: 3105: 3068: 3014: 2980: 2945: 2918: 2908: 2870: 2860: 2790: 2763: 2736: 2699: 2672: 2638: 2611: 2584: 2551: 2544:Timing 2524: 2497: 2470: 2438: 2428: 2358: 2265: 2257: 2247: 2172: 2125: 2057: 2047: 1997:: 33, 1922: 1875: 1848: 1818: 1791: 1764: 1729: 1699: 1653: 1623: 1510:, the 1045:where 739:is an 272:, and 37:, and 3394:S2CID 3353:S2CID 3261:S2CID 3175:arXiv 3148:S2CID 3130:arXiv 3103:S2CID 3036:(PDF) 2916:S2CID 2890:(PDF) 2436:S2CID 2197:(PDF) 2083:(PDF) 2055:S2CID 1999:arXiv 1568:is a 1449:In a 997:empty 987:, by 975:or a 963:each 938:(−1)| 825:or a 485:from 160:A DAG 102:edges 94:graph 63:edges 3384:ISBN 3341:C-27 3289:ISBN 3251:ISBN 3211:PMID 3066:PMID 3012:ISBN 2978:ISBN 2943:ISBN 2906:ISBN 2868:PMID 2788:ISBN 2761:ISBN 2734:ISBN 2697:ISBN 2670:ISBN 2636:ISBN 2609:ISBN 2582:ISBN 2549:ISBN 2522:ISBN 2495:ISBN 2468:ISBN 2426:ISBN 2263:OCLC 2245:ISBN 2170:ISBN 2156:For 2123:ISBN 2045:ISBN 1920:ISBN 1873:ISBN 1846:ISBN 1816:ISBN 1789:ISBN 1762:ISBN 1727:ISBN 1697:ISBN 1651:ISBN 1621:ISBN 1578:trie 1498:The 1294:path 1240:bits 1117:and 1089:The 861:root 555:OEIS 524:The 449:edge 420:. A 368:The 351:, ≤) 335:, ≤) 299:The 242:and 224:(or 194:and 179:The 110:path 67:arcs 61:and 41:, a 3376:doi 3345:doi 3318:doi 3243:doi 3201:PMC 3193:doi 3140:doi 3095:doi 3058:doi 3046:149 2898:doi 2858:PMC 2848:doi 2821:doi 2574:doi 2418:doi 2348:doi 2209:doi 2035:doi 1409:in 1397:Git 1259:In 1030:or 849:An 481:is 412:to 212:to 80:or 47:DAG 29:In 3445:: 3422:, 3418:, 3392:, 3382:, 3351:, 3339:, 3314:38 3312:, 3259:, 3249:, 3241:, 3209:, 3199:, 3191:, 3183:, 3171:10 3169:, 3146:, 3138:, 3124:, 3101:, 3091:27 3089:, 3064:, 3056:, 3044:, 3038:, 3004:, 3000:; 2976:, 2937:, 2914:, 2904:, 2892:, 2866:, 2856:, 2842:, 2817:82 2815:, 2811:, 2724:; 2720:; 2658:; 2580:, 2448:^ 2434:, 2424:, 2356:MR 2354:, 2344:22 2342:, 2261:. 2255:MR 2253:. 2243:. 2229:; 2203:, 2199:, 2137:^ 2111:; 2107:; 2103:; 2092:^ 2053:, 2043:, 2007:, 1993:, 1987:, 1979:; 1975:; 1967:; 1947:, 1943:, 1914:, 1887:^ 1760:, 1635:^ 1607:^ 1551:. 1527:ln 1495:. 1470:. 1441:. 1105:. 1071:. 1063:mn 1022:mn 836:A 833:. 815:A 804:A 788:A 557:). 504:A 477:, 472:→ 443:A 403:→ 386:, 377:→ 360:A 317:, 308:→ 296:. 291:≤ 287:≤ 277:→ 267:→ 262:, 257:→ 247:→ 237:→ 203:≤ 92:A 84:. 3401:. 3378:: 3360:. 3347:: 3325:. 3320:: 3298:. 3268:. 3245:: 3195:: 3187:: 3177:: 3155:. 3142:: 3132:: 3126:3 3110:. 3097:: 3073:. 3060:: 3052:: 3021:. 2987:. 2972:: 2952:. 2923:. 2900:: 2875:. 2850:: 2844:6 2828:. 2823:: 2797:. 2770:. 2743:. 2706:. 2679:. 2645:. 2618:. 2591:. 2576:: 2558:. 2531:. 2504:. 2477:. 2443:. 2420:: 2363:. 2350:: 2290:. 2269:. 2216:. 2211:: 2205:5 2179:. 2087:. 2062:. 2037:: 2011:: 2001:: 1995:7 1929:. 1882:. 1855:. 1825:. 1798:. 1771:. 1756:: 1736:. 1706:. 1679:. 1660:. 1630:. 1539:) 1536:n 1533:( 1439:q 1435:q 1431:q 1368:. 1099:C 1095:C 1065:) 1061:( 1059:O 1048:ω 1043:) 1041:n 1039:( 1037:O 1024:) 1020:( 1018:O 1013:m 1009:n 942:χ 936:χ 934:| 930:! 928:n 923:n 767:I 755:I 750:I 746:A 737:A 697:. 692:k 686:n 682:a 676:) 673:k 667:n 664:( 661:k 657:2 650:) 645:k 642:n 637:( 629:1 623:k 619:) 615:1 609:( 604:n 599:1 596:= 593:k 585:= 580:n 576:a 539:n 534:n 497:. 495:v 491:u 487:u 479:v 474:v 470:u 414:v 410:u 405:v 401:u 396:≤ 388:v 384:u 379:v 375:u 349:S 347:( 343:≤ 339:S 333:S 331:( 327:≤ 323:≤ 319:v 315:u 310:v 306:u 293:w 289:v 285:u 279:w 275:u 269:w 265:v 259:v 255:u 249:w 245:v 239:v 235:u 230:u 226:v 222:v 218:u 214:v 210:u 205:v 201:u 196:v 192:u 188:≤ 134:v 130:u 126:u 118:v 45:(

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index