edmonds-karp algorithm

an algorithm for computing the maximum flow in a flow network in $\text{[math]}$ time
algorithm unwritten

length of a flow augmenting path:
we define the length of a flow augmenting path as the number of edges in it

the time complexity of this algorithm is $\text{[math]}$

residual distance:
given a residual network $\text{[math]}$ we define a residual distance $\text{[math]}$ as the number of edges in the minimal path between $\text{[math]}$ and $\text{[math]}$

if we run the $\text{[math]}$ algorithm on a residual network $\text{[math]}$ then for each vertex $\text{[math]}$ , the distance from $\text{[math]}$ to $\text{[math]}$ , $\text{[math]}$ , becomes greater after each flow improvement.

the number of flow augmenting paths in the edmonds-karp algorithm is at most $\text{[math]}$