Графтағы төбелердің арасындағы қолжетімділік

Whether one vertex can be reached from another in a graph

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. If you will be making many queries, then a more sophisticated method may be used; the exact choice of method depends on the nature of the graph being analysed. In exchange for preprocessing time and some extra storage space, we can create a data structure which can then answer reachability queries on any pair of vertices in as low as time. Three different algorithms and data structures for three different, increasingly specialized situations are outlined below.

The Floyd–Warshall algorithm 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 distance between all pairs of vertices. For graphs containing negative cycles, shortest paths may be undefined, but reachability between pairs can still be noted.

A related problem is to solve reachability queries with some number of vertex failures. For example: "Can vertex still reach vertex even though vertices have failed and can no longer be used?" A similar problem may consider edge failures rather than vertex failures, or a mix of the two. The breadth first search technique works just as well on such queries, but constructing an efficient oracle is more challenging. Another problem related to reachability queries is in quickly recalculating changes to reachability relationships when some portion of the graph is changed. For example, this is a relevant concern to garbage collection which needs to balance the reclamation of memory (so that it may be reallocated) with the performance concerns of the running application.