Connected Component (graph Theory) - Algorithms

Algorithms

It is straightforward to compute the connected components of a graph in linear time (in terms of the numbers of the vertices and edges of the graph) using either breadth-first search or depth-first search. In either case, a search that begins at some particular vertex v will find the entire connected component containing v (and no more) before returning. To find all the connected components of a graph, loop through its vertices, starting a new breadth first or depth first search whenever the loop reaches a vertex that has not already been included in a previously found connected component. Hopcroft and Tarjan (1973) describe essentially this algorithm, and state that at that point it was "well known".

There are also efficient algorithms to dynamically track the connected components of a graph as vertices and edges are added, as a straightforward application of disjoint-set data structures. These algorithms require amortized O(α(n)) time per operation, where adding vertices and edges and determining the connected component in which a vertex falls are both operations, and α(n) is a very slow-growing inverse of the very quickly growing Ackermann function. A related problem is tracking connected components as all edges are deleted from a graph, one by one; an algorithm exists to solve this with constant time per query, and O(|V||E|) time to maintain the data structure; this is an amortized cost of O(|V|) per edge deletion. For forests, the cost can be reduced to O(q + |V| log |V|), or O(log |V|) amortized cost per edge deletion.

Researchers have also studied algorithms for finding connected components in more limited models of computation, such as programs in which the working memory is limited to a logarithmic number of bits (defined by the complexity class L). Lewis & Papadimitriou (1982) asked whether it is possible to test in logspace whether two vertices belong to the same connected component of an undirected graph, and defined a complexity class SL of problems logspace-equivalent to connectivity. Finally Reingold (2008) succeeded in finding an algorithm for solving this connectivity problem in logarithmic space, showing that L = SL.