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IOTA / TANGLE // Heart Beacon Cycle Time - Space Meter synergy

The Heart Beacon Cycle Time - Space meter describes an improvement to an acyclic algorithm based on firefly synchronization that uses the median / mean of the firefly heartbeat synchronization algorithm to strive to sync to the closest OPTEMPO Heart Beacon micro - macro cycle. See slide #22 this page LINK

The Tangle is based on the concept of a Directed acyclic graph DAG. A directed acyclic graph (DAG!) is a directed graph that contains no cycles. A rooted tree is a special kind of DAG and a DAG is a special kind of directed graph. The DAG 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 can model many different kinds of information. A spreadsheet can be modeled as a DAG, with a vertex for each cell and an edge whenever the formula in one cell uses the value from another; a topological ordering of this DAG can be used to update all cell values when the spreadsheet is changed. Similarly, topological orderings of DAGs can be used to order the compilation operations in a makefile. 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. 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. More abstractly, the reachability relation in a DAG forms a partial order, and any finite partial order may be represented by a DAG using reachability.

mportant 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[1] or acyclic digraphs.[2]     



Topological sorting is the algorithmic problem of finding a topological ordering of a given DAG. It can be solved in linear time.[16] 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.[17] Alternatively, a topological ordering may be constructed by reversing a postorder numbering of a depth-first search graph traversal.[16]

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[18] or alternatively, for some topological sorting algorithms, by verifying that the algorithm successfully orders all the vertices without meeting an error condition.[17]

Construction from cyclic graphs[edit]

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.[19]

The yellow directed acyclic graph is the condensation of the blue directed graph. It is formed by contracting each strongly connected component of the blue graph into a single yellow vertex.

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.[20] 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.[21] 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[edit]

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.[22] 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.[23]

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.[