Asynchronous distributed algorithms for static and dynamic directed rooted graphs

The paper provides a review of distributed graph algorithms research conducted by authors. We consider an asynchronous distributed system model represented by a strongly connected directed rooted graph with bounded edge capacity (in a sense that only a bounded number of messages can be sent through an edge in a given time interval). A graph can be static or dynamic, i.e. changing. For a static graph we propose a spanning (in- and out-) tree construction algorithm of time complexity $O(n/k+d)$, requiring $O(nd\log\Delta^+)$ message size and the same size of memory of each computing agent located in graph vertex, where $n$ is the number of vertices of the graph, $k$ is the capacity of an edge, $d$ is the maximum length of simple path in the graph, $\Delta^+$ is the maximum outdegree of the vertices. The spanning trees constructed can be used in distributed computation of a function of the multiset of values assigned to graph vertices in a time not greater than $3d$. In a dynamic graph we suppose that $k=1$ and an edge can appear, disappear, or change its end. We propose a dynamic graph monitoring algorithm than delivers information on any change to the root of the graph in $O(n)$ or $O(d)$ after the changes are stopped. We also propose graph exploration and marking algorithm with time complexity $O(n)$. The marking provided by it is used in distributed computation of a function of the multiset of values assigned to dynamic graph vertices, which can be performed in time $O(n^2)$ with messages of size $O(\log n)$ or in time $O(n)$ with messages of size $O(n\log n)$.