The graph relabeling problem and its variants

Fast Sequential and Parallel Vertex Relabelings of Km,m

“…§4 gives results in both parallel and sequential settings when the labeling is parity. Similarly, in §5, when precise labeling is concerned, results are shown in the parallel and sequential settings, improving the previous result in [3]. In §6 we give a conclusion and discuss open problems.…”

“…When the precise labeling is considered, the parallel time complexity increases by a factor of log m while the processor complexities remain m and m log m . We also show that, when graphs are restricted to K m,m , the time complexities to solve the problem on a sequential RAM machine are O(m) and O(m log m) for the cases of parity and precise labelings, respectively, thereby improving the result in [3] for this specific case.…”

“…In this problem we are given a graph and two vertex labelings and we are asked to transform the graph from one labeling to the other using a sequence of label flips, where each flip interchanges the labels of two adjacent vertices. Some instances of this problem were initially explored by Kantabutra [12] and later by Agnarsson et al [2,3]. In particular, Agnarsson et al showed solving the Vertex Relabeling Problem on arbitrary graphs can be done sequentially in θ(n 2 ), where n is the number of vertices in G [3].…”

“…Some instances of this problem were initially explored by Kantabutra [12] and later by Agnarsson et al [2,3]. In particular, Agnarsson et al showed solving the Vertex Relabeling Problem on arbitrary graphs can be done sequentially in θ(n 2 ), where n is the number of vertices in G [3].…”

On the parallel complexity of hierarchical clustering and CC‐complete problems

“…We next discuss and define the Lexicographically First Maximal Matching Problem (LFMMP) which we need in the proofs of Theorems 4.1, 4.2, and 4.3. The concept of matching is well studied in graph theory and computer science [23]. Given an undirected graph G = (V , E), which consists of a set of vertices V and a set of edges E, a matching is a set of edges M = {e 1 , .…”