Note: understanding of upstream and downstream network streams

Feasible stream with no source sink

Arc traffic restriction condition B (u, v) <= f (u, v) <= C (u, v), (u, v) ε E

F (u, v) = B (u, v) + F1 (u, v ),

Σ (B (u, v) + F1 (u, v) = Σ (B (V, W) + F1 (V, W ))

Σ B (u, v)-Σ B (V, W) = Σ F1 (V, W)-Σ F1 (u, v)

 

If a feasible stream exists, 0 <= F1 (u, v) <= C (u, v)-B (u, v), and the lower bound stream is full.

 

You can add super source S and super sink T.

Assume that the traffic limit of an edge (u, v) is [B, c], equivalent to S → v traffic B, u → T traffic B, U → v traffic c-B.

We can perform the maximum stream from S to T. If the load is full, the lower stream can flow full, that is, there is a feasible stream.

 

 

Active and reachable streams

The upstream and downstream feasible stream with a source sink S → T, connects a traffic INF edge from t to S, and then performs the same as the non-source sink.

 

 

Maximum stream with source sink

 

Connect a traffic INF edge from t to S, add super source S and super sink T, and perform the maximum flow from S to T.

If the load is full, a feasible stream exists. The lower-bound stream is guaranteed to be full.

Perform the largest stream from S to t to get the answer.

 

 

Minimum stream with source sink

Connect a traffic INF edge from t to S, add super source S and super sink T, and perform the maximum flow from S to T.

If the load is full, a feasible stream exists. The lower-bound stream is guaranteed to be full.

Perform the largest stream from t to S to get the smallest stream. (Face hitting possible)