Random processes for generating task-dependency graphs
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We investigate random processes for generating task-dependency graphs of order n with m edges and a specified number of initial vertices and terminal vertices. In order to do so, we consider two random processes for generating task-dependency graphs that can be combined to accomplish this task. In the (x, y) edge-removal process, we start with a maximally connected task-dependency graph and remove edges uniformly at random as long as they do not cause the number of initial vertices to exceed x or the number of terminal vertices to exceed y. In the (x, y) edge-addition process, we start with an empty task-dependency graph and add edges uniformly at random as long as they do not cause the number of initial vertices to be less than x or the number of terminal vertices to be less than y. In the (x, y) edge-addition process, we halt if there are exactly x initial vertices and y terminal vertices. For both processes, we determine the values of x and y for which the resulting task-dependency graph is guaranteed to have exactly x initial vertices and y terminal vertices, and we also find the extremal values for the number of edges in the resulting task-dependency graphs as a function of x, y, and the number of vertices. Furthermore, we asymptotically bound the expected number of edges in the resulting task-dependency graphs. Finally, we define a random process using only edge-addition and edge-removal, and we show that with high probability this random process generates an (x, y) task-dependency graph of order n with m edges.
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