# Question

Let G = (V, E) be a flow network with source s, sink t, and an integer capacity c (u, v) on each edge (u, v) ¬ E. Let C = max (u, v) Ec (u, v). a. Argue that a minimum cut of G has capacity at most C |E|.

b. For a given number K, show that an augmenting path of capacity at least K can be found in O(E) time, if such a path exists. The following modification of FORD-FULKERSON-METHOD can be used to compute a maximum flow in G.

MAX-FLOW-BY-SCALING (G, s, t)

1 C ← max (u. v)¬ Ec (u, v)

2 initialize flow f to 0

3 K ← 2⌊lgC⌋

4 while K ≥ 1

5 do while there exists an augmenting path p of capacity at least K

6 do augment flow f along p

7 K ← K/2

8 return f

c. Argue that MAX-FLOW-BY-SCALING returns a maximum flow.

d. Show that the capacity of a minimum cut of the residual graph Gf is at most 2K |E| each time line 4 is executed.

e. Argue that the inner while loop of lines 5-6 is executed O (E) times for each value of K.

f. Conclude that MAX-FLOW-BY-SCALING can be implemented so that it runs in O (E2 lg C) time.

b. For a given number K, show that an augmenting path of capacity at least K can be found in O(E) time, if such a path exists. The following modification of FORD-FULKERSON-METHOD can be used to compute a maximum flow in G.

MAX-FLOW-BY-SCALING (G, s, t)

1 C ← max (u. v)¬ Ec (u, v)

2 initialize flow f to 0

3 K ← 2⌊lgC⌋

4 while K ≥ 1

5 do while there exists an augmenting path p of capacity at least K

6 do augment flow f along p

7 K ← K/2

8 return f

c. Argue that MAX-FLOW-BY-SCALING returns a maximum flow.

d. Show that the capacity of a minimum cut of the residual graph Gf is at most 2K |E| each time line 4 is executed.

e. Argue that the inner while loop of lines 5-6 is executed O (E) times for each value of K.

f. Conclude that MAX-FLOW-BY-SCALING can be implemented so that it runs in O (E2 lg C) time.

## Answer to relevant Questions

Prove that any sorting network on n inputs has depth at least lg n.Show that the depth of SORTER [n] is exactly (lg n) (lg n + 1)/2.Explain how a vertex u of a directed graph can end up in a depth-first tree containing only u, even though u has both incoming and outgoing edges in G.An Euler tour of a connected, directed graph G = (V, E) is a cycle that traverses each edge of G exactly once, although it may visit a vertex more than once. a. Show that G has an Euler tour if and only if in-degree (v) = ...Calculate the reversible work out of the two-stage turbine shown in Problem 6.41, assuming the ambient is at 25C. Compare this to the actual work which was found to be 18.08 MW.Post your question

0