Distributed algorithms on rooted undirected graphs

Distributed algorithms of solving problems on undirected graphs are considered. In section 2, a model is defined featuring a root as a starting and ending point of the algorithm execution. Synchronous and asynchronous versions of the model are described. In section 3, algorithms of solving any problems are suggested based on collecting information on the whole graph in the root or in any vertex, as well as, on the graph labeling (its vertices and/or edges), if required. Emphasis is made on the time of the algorithm execution or on saving memory in vertices and total size of transferred messages, if this time is minimal. The rest of the paper considers optimizations for particular problems: creation of Maximal Independent Set (MIS), Finding Set of Bridges (FSB), creation of Minimum Spanning Tree (MST) in a edge-weighted graph. In section 4, a modification of general algorithms for these problems is suggested decreasing the estimate of memory size of vertices and messages. Section 5 includes lower-bound estimates of solution complexity for these problems. In section 6, for synchronous model, the time of algorithms execution with graph labeling is decreased to the lower bound for problems with single-valued solution depending on only simple cycles of the graph, in particular, FSB, MST and the problem of Hamiltonian cycle search. In section 7, time-optimal algorithms for FSB and MST are considered for both synchronous and asynchronous models. Conclusion summarizes the results and outlines the directions for further research.