Simple DFS on the Complement of a Graph and on Partially Complemented Digraphs

A complementation operation on a vertex of a digraph changes all outgoing arcs into non-arcs, and outgoing non-arcs into arcs. A partially complemented digraph $\widetilde{G}$ is a digraph obtained from a sequence of vertex complement operations on $G$. Dahlhaus et al. showed that, given an adjacency-list representation of $\widetilde{G}$, depth-first search (DFS) on $G$ can be performed in $O(n + \widetilde{m})$ time, where $n$ is the number of vertices and $\widetilde{m}$ is the number of edges in $\widetilde{G}$. To achieve this bound, their algorithm makes use of a somewhat complicated stack-like data structure to simulate the recursion stack, instead of implementing it directly as a recursive algorithm. We give a recursive $O(n+\widetilde{m})$ algorithm that uses no complicated data-structures

Faster graph algorithms via switching classes

2012 Summer.Includes bibliographical references.The runtime of an algorithm is intimately related to how an instance is represented. Recall that the runtimes of the first generation of graph algorithms were expressed as functions of n := |V|. This analysis was natural since at this time graphs were represented in n2 space via their adjacency matrix. It was soon noticed that if m := |E| = o(n2), then a variety of graph algorithms could be sped-up by computing the adjacency-list from the adjacency matrix, then running the algorithm on the more efficient adjacency-list representation. This motivated the introduction of m to the runtime of graph algorithms and it is now customary in algorithm design to assume that a graph instance is given in the form of its adjacency-list. For instance, a graph algorithm is not considered to run in linear time unless it runs in O(n + m) time. An O(n2) bound is not considered linear, even though the two bounds are the same in the worst case. Let m͂ be the size of the minimum representative of a graph G's switching class (w.r.t. to some switching operation). It is shown that better bounds for several classical graph algorithms can be obtained by modifying them so that their running time is a function of n+m͂ rather than of n+m. This is significant because m͂ is O(m) but m is not O(m͂). This is accomplished by first computing the so-called partially complemented adjacency list (pc-list) from an adjacency list, then designing an algorithm that is amenable to the more efficient pc-list representation. The pc-list data-structure is generalization of the adjacency list that has a natural correspondence to switching classes. Using this approach, better bounds are obtained for bipartite maximum matching, graph diameter, and vertex-weighted all-pairs shortest path

A General Label Search to Investigate Classical Graph Search Algorithms

International audienceMany graph search algorithms use a labeling of the vertices to compute an ordering of the vertices. We generalize this idea by devising a general vertex labeling algorithmic process called General Label Search (GLS), which uses a labeling structure which, when specified, defines specific algorithms. We characterize the vertex orderings computable by the basic types of searches in terms of properties of their associated labeling structures. We then consider performing graph searches in the complement without computing it, and provide characterizations for some searches, but show that for some searches such as the basic Depth-First Search, no algorithm of the GLS family can exactly find all the orderings of the complement. Finally, we present some implementations and complexity results of GLS on a graph and on its complement