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After the transitive closure is constructed, as depicted in the following figure, in an O(1) operation one may determine that node d is reachable from node a. The data structure is typically stored as a Boolean matrix, so if matrix[1][4] = true, then it is the case that node 1 can reach node 4 through one or more hops.
The Floyd–Warshall algorithm [5] can be used to compute the transitive closure of any directed graph, which gives rise to the reachability relation as in the definition, above. The algorithm requires (| |) time and (| |) space in the worst case. This algorithm is not solely interested in reachability as it also computes the shortest path ...
The transitive closure of a DAG is the graph with the most edges that has the same reachability relation as the DAG. It has an edge u → v for every pair of vertices ( u , v ) in the reachability relation ≤ of the DAG, and may therefore be thought of as a direct translation of the reachability relation ≤ into graph-theoretic terms.
closure 1. For the transitive closure of a directed graph, see transitive. 2. A closure of a directed graph is a set of vertices that have no outgoing edges to vertices outside the closure. For instance, a sink is a one-vertex closure. The closure problem is the problem of finding a closure of minimum or maximum weight. co-
A transitive orientation is an orientation such that the resulting directed graph is its own transitive closure. The graphs with transitive orientations are called comparability graphs; they may be defined from a partially ordered set by making two elements adjacent whenever they are comparable in the partial order. [8] A transitive orientation ...
A transitive orientation of a graph is an acyclic orientation that equals its own transitive closure. Not every graph has a transitive orientation; the graphs that do are the comparability graphs. [8] Complete graphs are special cases of comparability graphs, and transitive tournaments are special cases of transitive orientations.
“It’s not what you feed, it’s the way you feed it,” explains Burton. “Your treat delivery technique can have a powerful impact on the outcome of your training.”
A correct evaluation order is a numbering : of the objects that form the nodes of the dependency graph so that the following equation holds: () < (,) with ,. This means, if the numbering orders two elements a {\displaystyle a} and b {\displaystyle b} so that a {\displaystyle a} will be evaluated before b {\displaystyle b} , then a ...