#algorithm discussion continued
Today we continue reading the flow networks. We were looking at max flow min cut theorem and the properties associated with it. We saw how it applied to Ford Fulkerson method. Each iteration of the method augmented the flow by considering a path in the residual network until there is no more. For each path considered, the residual capacity of that path is found and augmented to the flow network.
How the path is chosen is not mentioned by the method. This is usually done with Breadth First Search
We saw that with BFS, the shortest path distance for all vertices in the residual network increases monotonically with each flow augmentation. We saw that in the forward direction, when we assumed the distance to decrease, we contradicted ourselves because the next vertex  v was one hop away from the current vertex u and since this vertex was not already considered its distance was the sum of the shortest distance to current vertex and one hop. This would mean that the distance to the new vertex before and after the decrease would be the same which is a contradiction.Therefore this distance increases monotonically.
Maximum flow problems have a direct application in some combinatorial problems. The multiple-source multiple-sink maximum flow. One such example is to find the maximum matching in a bipartite graph. This can be done in O(VE) time in a G = (V,E) graph
A matching is a subset of edges M such that for all vertices v in V, at most one edge of M is incident on v. We say that a vertex v belongs to V is matched by the matching M if some edge in M is incident on V and unmatched otherwise. A maximum matching is one where the number of matchings is maximum. A graph is called bipartite when all its vertex set can be partitioned into disjoint sets L and R and all edges in E go between L and R.
To find this maximum matching, we convert the bipartite graph into a flow network in which the flows correspond to matchings. A source s and a sink v are added to either sides of the bipartite graph. We assign directed edges from L to R with unit capacity for all edges with flows and zero otherwise. Since all the flows are thus integers, the flow network is called integer valued.
If M is a matching in the bipartite graph G, then there is an integer valued flow f in flow network G' with cardinality of f to be the same as that of M. Conversely if f is an integer valued flow in G', then there is a matching M in G with cardinality M to be same as that of f. Since the flow f is maximum as found by the Ford-Fulkerson method, the cardinality of maximum matching M equals the value of maximum flow.
#codingexercise
Given an array of heights of poles. Find the no of poles which are visible if you are standing at the kth pole.
int Getvisible(List<int>h, int n, int k)
{
var left = new int[n]{};
var right = new int[n]{};
int count = 1; // the current pole

left[0] = h[0];
for (int i = 1; i < n; i++)
   left [i] = max(left[i-1], h[i]);

   right[n-1] = h[n-1];
    for (int i = n-2; i >= 0; i--)
       right[i] = max(right[i+1], h[i]);

for (int i =k-1; i >= 0; i--){
      count++;
      if(h[i] != left[k])
          break;
}
for (int i =k+1; i < n; i++){
      count++;
      if (h[i] != right[k])
          break;
}
return count;
}