residuaalikuvaajien

Residuaalikuvaajien, a Finnish term, translates to residual graphs in English graph theory. A residual graph is a concept used in algorithms for network flow problems, such as the Ford-Fulkerson method and its variations. When considering a flow network, which consists of a directed graph with capacities on its edges, a residual graph is constructed to represent the remaining capacity on each edge and the possibility of reversing flow.

In a residual graph, for every edge (u, v) in the original graph with capacity c(u, v) and current flow f(u, v), there are typically two corresponding edges. The first is a forward edge (u, v) with a residual capacity of c(u, v) - f(u, v). This represents how much more flow can be pushed from u to v. The second is a backward edge (v, u) with a residual capacity of f(u, v). This backward edge signifies that flow can be "canceled" or redirected from v to u, effectively reducing the flow on the original edge (u, v).

The residual graph is dynamic; it changes as flow is augmented through the network. Algorithms like Ford-Fulkerson repeatedly find augmenting paths in the residual graph. An augmenting path is a path from the source to the sink in the residual graph that has positive residual capacity along all its edges. By pushing flow along such a path, the total flow in the network is increased, and the residual graph is updated accordingly. This iterative process continues until no more augmenting paths can be found, at which point the maximum flow for the network has been achieved. The concept of the residual graph is fundamental for understanding and implementing max-flow algorithms.