Is this a flow?

A railway has n stations and one-way lines (u, v, c): at most c trains an hour can run from u to v. A plan gives flow[i] trains an hour on line i. The plan is a flow from the source s to the sink t when it keeps two rules (lesson 1): every line carries between 0 and its capacity, and at every station other than s and t, the trains that arrive equal the trains that leave. Return the value of the flow, the trains leaving s minus the trains arriving at s, or None if a rule is broken.

lines = [(0, 1, 10), (0, 2, 10), (1, 2, 2),
         (1, 3, 4), (1, 4, 8), (2, 4, 9),
         (4, 3, 6), (3, 5, 10), (4, 5, 10)]
flow_value(6, lines,
           [10, 9, 0, 4, 6, 9, 5, 9, 10], 0, 5)   # 19
flow_value(6, lines,
           [10, 9, 0, 4, 6, 9, 5, 9, 11], 0, 5)   # None

In the second call the last line carries 11 on a capacity of 10. Lines may repeat, and a line may join a station to itself. Return None also when flow does not have one number per line.

Limits

• Up to 100,000 stations and 100,000 lines.
• Going through every line once per station is too slow. The big tests have 3 seconds each.