Directed acyclic graph - Wikipedia

• A directed acyclic graph (DAG) is a mathematical structure of vertices and edges where it is impossible to return to a starting point by following directed paths.
• The defining characteristic of a DAG is its ability to be topologically ordered, meaning vertices can be arranged linearly to respect all edge directions.
• DAGs serve as the fundamental representation for finite partially ordered sets, where reachability between points defines a specific hierarchy.
• The transitive closure of a DAG represents the maximum number of edges possible while maintaining the same reachability, effectively mapping the entire partial order.
• Transitive reduction identifies the most efficient version of a DAG by removing redundant edges while preserving the essential reachability between all vertices.
• These structures are critical across diverse fields, including evolutionary biology, citation networks, and computational task scheduling.

That is, it consists of vertices and edges (also called arcs), with each edge directed from one vertex to another, such that following those directions will never form a closed loop.

• Transitive reductions and Hasse diagrams simplify the visualization of partial orders by minimizing edges and using vertical orientation.
• A directed graph is acyclic if and only if it possesses at least one topological ordering, where every edge points from an earlier vertex to a later one.
• The number of unique directed acyclic graphs on n labeled vertices grows rapidly, following a specific recurrence relation studied by Robinson.
• There is a bijective proof linking DAG adjacency matrices to (0,1) matrices with exclusively positive real eigenvalues.
• Specialized DAG structures include multitrees, which allow at most one path between any two vertices, and polytrees, which are oriented undirected trees.

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.

• Topological sorting, the process of ordering vertices linearly, can be achieved in linear time using Kahn's algorithm or depth-first search.
• Any undirected graph can be converted into a DAG through acyclic orientation, where the number of possible orientations relates to the graph's chromatic polynomial.
• Cyclic directed graphs can be transformed into DAGs by contracting strongly connected components into a condensation or by removing feedback sets.
• Finding the smallest feedback vertex or arc set to break all cycles in a graph is an NP-hard computational problem.
• Transitive closure and reduction algorithms determine reachability between all vertex pairs, with performance bounds tied to matrix multiplication speeds.
• The closure problem involves finding a vertex set with optimal weight where no edges exit the set.

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.

• Topological ordering allows for linear-time solutions to shortest and longest path problems in DAGs, which are significantly more efficient than algorithms for general graphs.
• Finding the longest path in an arbitrary graph is NP-hard, but it is computationally simple in a DAG, facilitating project management and optimization.
• Dependency graphs in spreadsheets and software compilation use DAGs to prevent circular dependencies and ensure a consistent update order.
• The Program Evaluation and Review Technique (PERT) utilizes DAGs where edges represent tasks and vertices represent project milestones.
• The 'critical path' of a project is identified as the longest path in a PERT chart, determining the minimum time required for project completion.
• Data processing networks use DAGs to model the flow of information through various processing elements from input to output.

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.

• DAGs are fundamental to electronic circuit design, where they represent static combinational logic blocks and hierarchical component references.
• In computer science, dataflow programming and compiler optimization utilize DAGs to manage parallel processing and eliminate redundant subexpressions.
• The acyclic dependencies principle dictates that large software systems should organize module relationships as a DAG to prevent circularity.
• Causal structures are inherently acyclic because time-ordered events cannot loop back to affect the past, reflecting our intuition of causality.
• Probabilistic models like Bayesian networks and influence diagrams use DAGs to calculate event likelihoods and evaluate the impact of specific interventions.

The lack of a cycle follows because the time associated with a vertex always increases as you follow any directed path in the graph, so you can never return to a vertex on a path.

• All directed acyclic graphs possess a topological ordering that establishes a clear causal or temporal sequence.
• Family trees function as DAGs rather than simple trees due to pedigree collapse from marriages between relatives.
• Distributed revision control systems like Git utilize DAGs to track version history and complex merges.
• Computational geometry employs history DAGs to efficiently locate points within evolving geometric structures.
• Citation networks in academia, law, and patents form DAGs because documents can only reference previously existing works.
• Transitive reduction of citation DAGs reveals unique structural mechanisms across different professional fields.

Despite the name, these graphs are not necessarily trees because of the possibility of marriages between relatives causing pedigree collapse.

• Main path analysis uses citation links within directed acyclic graphs to identify the most significant chains of academic influence.
• The Price model provides a mathematical framework for citation networks, allowing for analytic calculations of properties like the logarithmic scaling of path lengths.
• Directed acyclic graphs serve as a powerful tool for data compression by representing shared subsequences as merged paths.
• Directed acyclic word graphs (DAWGs) improve upon the trie structure by allowing paths to rejoin, significantly reducing the space needed to store sets of strings.
• Binary decision diagrams apply DAG principles to represent binary functions, compressing decision trees by merging paths that lead to identical outcomes.

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 text provides a comprehensive bibliography of foundational research on directed acyclic graphs (DAGs) and their algorithmic properties.
• Key mathematical concepts covered include transitive closures, reductions, and the counting of labeled acyclic digraphs.
• Algorithmic applications focus heavily on topological sorting, including unique orderings and depth-first search implementations.
• The references highlight the utility of DAGs in diverse fields such as causal poly-trees in AI and circular dependency analysis in software engineering.
• Advanced graph problems like single-source shortest paths and maximal closures are explored through seminal texts like Cormen's 'Introduction to Algorithms'.
• The collection bridges theoretical graph theory with practical computational complexity and NP-completeness.

Multitrees: enriching and reusing hierarchical structure.

• The text provides a comprehensive list of academic references spanning graph theory, compiler design, and decision-making under uncertainty.
• It highlights the use of directed acyclic graphs (DAGs) in diverse fields such as gene regulatory networks, epidemiology, and causal learning.
• Several citations focus on the structural analysis of information, including citation networks and the transitive reduction of scientific papers.
• The references include foundational work on data flow procedure languages and advanced backend optimization in computer science.
• The collection emphasizes the intersection of mathematics and biology, specifically regarding pedigrees, genotypes, and genealogical visualization.

A general theory of bibliometric and other cumulative advantage processes.

• The text provides a comprehensive bibliography of foundational and modern research on Directed Acyclic Graphs (DAGs).
• Key applications mentioned include the Price model for network growth and the construction of compact word graphs.
• The references highlight the historical development of Binary Decision Diagrams (BDDs) from the late 1950s through the 1980s.
• Algorithmic foundations are anchored in standard texts like 'Introduction to Algorithms' by Cormen et al.
• The section concludes with practical resources, including online tools for creating and visualizing DAG structures.

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

• Topological sorting can be done in linear time with Kahn’s algorithm or depth-first search.
• Finding the smallest feedback vertex or arc set to break all cycles is NP-hard.

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.

• Topological ordering enables linear-time shortest and longest path algorithms in DAGs.
• The critical path in a PERT chart is the longest path, giving the minimum project completion time.

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.

• Main path analysis uses citation links in DAGs to identify the most significant chains of academic influence.
• Directed acyclic word graphs (DAWGs) improve on tries by allowing paths to rejoin, greatly reducing storage.

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.