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Concept
Reachability
Résumé
In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex can reach a vertex (and is reachable from ) if there exists a sequence of adjacent vertices (i.e. a walk) which starts with and ends with .
In an undirected graph, reachability between all pairs of vertices can be determined by identifying the connected components of the graph. Any pair of vertices in such a graph can reach each other if and only if they belong to the same connected component; therefore, in such a graph, reachability is symmetric ( reaches iff reaches ). The connected components of an undirected graph can be identified in linear time. The remainder of this article focuses on the more difficult problem of determining pairwise reachability in a directed graph (which, incidentally, need not be symmetric).
For a directed graph , with vertex set and edge set , the reachability relation of is the transitive closure of , which is to say the set of all ordered pairs of vertices in for which there exists a sequence of vertices such that the edge is in for all .
If is acyclic, then its reachability relation is a partial order; any partial order may be defined in this way, for instance as the reachability relation of its transitive reduction. A noteworthy consequence of this is that since partial orders are anti-symmetric, if can reach , then we know that cannot reach . Intuitively, if we could travel from to and back to , then would contain a cycle, contradicting that it is acyclic.
If is directed but not acyclic (i.e. it contains at least one cycle), then its reachability relation will correspond to a preorder instead of a partial order.
Algorithms for determining reachability fall into two classes: those that require preprocessing and those that do not.
If you have only one (or a few) queries to make, it may be more efficient to forgo the use of more complex data structures and compute the reachability of the desired pair directly. This can be accomplished in linear time using algorithms such as breadth first search or iterative deepening depth-first search.
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