The simple online VM assignment problem can be formulated in terms of connected components. The connected components of a graph are a partitioning of the graph into subgraphs. A connected component of a directed graph is a maximal set of vertices such that for every pair of vertices and in , there is a path connecting them. Hence two vertices are in the same connected component if and only if there exists a path between the vertices. If a graph contains only one connected component then it is said to be a connected graph.
The directed graph is a connected graph while the directed graph may or may not be connected. It will be connected if and only if there at least one such that . Note that since is connected, the graph of where there such that such that , will also be connected.
Given an existing , , and their associated functions, we calculate where
The VM assignment problem may now be described as: