A-level Mathematics/OCR/D2/Flows in a Network


 * Weights on arcs are called capacities.
 * Start of network called the source (S); end of network called the sink (T).
 * Arcs which are carrying their full capacity are said to be saturated.
 * A cut is a continuous line that separates S from T and does not pass through any nodes.
 * With the exception of S and T, the sum of flows into a node must equal sum of flows out of node.
 * Maximum flow - minimum cut theorem:
 * The flow through a network cannot exceed the value of any cut.
 * The maximum flow equals the value of the minimum cut.
 * Meaning that if you can find, by inspection, a flow and a cut which have the same value, then the maximum flow - minimum cut theorem tells you that you have obtained the maximum flow.
 * Labelling procedure finds the max flow by systematically augmenting any initial flow:
 * 1) Being with any initial flow or zero flow.
 * 2) Replace each arc with two arcs: one showing the amount by which the flow can be increased (its excess capacity) and the other showing the amount by which the flow can be decreased (its potential backflow).
 * 3) If S is still connected to T then find a new flow from S to T and alter the excess capacities and potential backflows as necessary.
 * 4) Repeat Step 3 until S is disconnected from T.
 * 5) If S is disconnected from T then the max flow is the sum of all the flows of Steps 1 and 3.
 * If a network has more than one source, create a new supersource; if it has more than one sink, create a new supersink.
 * If a node has a restricted capacity then replace it by two unrestricted nodes connected by an arc of the relevant capacity.
 * For networks where arcs have both upper and lower capacities, the value of a cut is:
 * the sum of the upper capacities that cross the cut line in the direction from S to T minus the sum of the lower capacities that cross the cut line in the direction from T to S.


 * For such a network, for the labelling procedure backflows are calculated using the minimum capacities. E.g. for a flow of 5 along an arc with an upper capacity of 8 and a lower capacity of 1, the excess capacity is 3 and the potential backflow is 4.