Space-Efficient Biconnected Components and Recognition of Outerplanar Graphs

Abstract

We present space-efficient algorithms for computing cut vertices in a given graph with n vertices and m edges in linear time using O(n + min{m, n log log n}) bits. With the same time and using O(n+m) bits, we can compute the biconnected components of a graph. We use this result to show an algorithm for the recognition of (maximal) outerplanar graphs in O(n log log n) time using O(n) bits.

Space-Efficient DFS and Applications to Connectivity Problems: Simpler, Leaner, Faster

“…Space bounds of the form O(n log(m/n)) for the same problems were mentioned by Chakraborty et al [5]. Essentially re-inventing an algorithm of Gabow [10] and combining it with machinery from [9] and with new ideas, Kammer et al [15] also demonstrated how to compute the cut vertices in O(n + m) time with O(n log log n) bits. Finally, decomposing the input graph into subtrees and processing the subtrees one by one, Chakraborty et al [5] were able to solve the BCC problem and compute the cut vertices in O(m log n log log n) time with O(n) bits.…”

“…In this subsection we will see that closely related algorithms can be used to compute the cut vertices, the bridges and the biconnected and 2-edge-connected components of an undirected graph. Our algorithms are similar to those of [3,5,15], but whereas the earlier authors indicated the space bounds only as O(n + m) or O(n log(m/n)) bits, we will strive to obtain small constant factors and indicate these explicitly.…”

“…Altogether, we have proved the following result. Kammer et al [15] consider the problem of preprocessing an undirected graph G so as later to be able to output the vertices and/or edges of a single BCC, identified via one of its edges, in time at most proportional to the number of items output. Having available the parent pointers of a DFS forest F and the array Q corresponding to F , we can solve the problem in the following way, which is the translation of the procedure of Kammer et al to our setting: Given a request to output the BCC H that contains an edge {x, y}, first follow parent pointers in parallel from x and y until one of the searches hits the other endpoint or a root in F .…”

“…The space bounds cited so far may be characterized as density-independent in that they depend only on n and not on m. If one is willing to settle for densitydependent space bounds that depend on m or perhaps on the multiset of vertex degrees, it becomes feasible to store with each gray vertex u an indication of the vertex immediately above it on the stack, which is necessarily a neighbor of u and therefore expressible in O(log(d + 1)) bits, where d is the degree of u. Since log(d+1) = O(d+1), this yields a DFS algorithm that works in O(n+m) time with O(n+ m) bits, as observed in [3,15]. One can also use Jensen's inequality to bound the space requirements of the pointers to neighboring vertices by O(n log(2+m/n)) bits.…”

“…Various bounds for these problems were claimed without proof by Banerjee, Chakraborty and Raman [3]: O(m log n log log n) time with O(n) bits for the SCC problem and O(n + m) time with m + 3n + o(n + m) bits as well as O(m log log n) time with O(n) bits for topological sorting. For the BCC problem and the computation of cut vertices, Kammer, Kratsch and Laudahn [15] described an algorithm that works in O(n + m) time using O(n + m) bits and can be seen as an implementation of an algorithm of Schmidt [18]. Essentially the same algorithm was sketched by Banerjee et al [3], who also applied it to the 2ECC problem and the computation of bridges.…”