#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
When the BFS is used to choose the augmenting path as a shortest path from s to t in the residual network, where each edge has unit distance, the algorithm is then called Edmonds-Karp algorithm
It can be shown that with this algorithm for all vertices, the shortest path distance in the residual network increases monotonically with each flow augmentation.
This is explained in terms of contradiction. The distance between  the start and a vertex v that has not already been included is assumed to decrease and then we derive the contradiction. Let delta be the shortest path just before the decrease and delta-dash be the shortest path immediately after. We know that the decrease is one hop because BFS traversal is for neighbors. and  The distance of the vertex u is not decreased.
Moreover, we know that the edge (u,v) was not part of the shortest distance earlier because if it were the shortest distance from source to v would have been less than or equal to the shortest distance from s to u plus one hop. This in turn would have been smaller than source s to u after the current decrease  of the shortest distance from s to u  together with the one hop to v which we just now assumed. This again in turn would have been equal to the shortest distance from source to v after the decrease because the distance from s to u  plus one hop is the distance from s to v. This means that the distance from source to v before and after the decrease is the same which is a contradiction.
The same reasoning by contradiction works out in the reverse direction. Together by both reasoning in both directions, the assumption that such a vertex v exists is proved false. Hence the shortest path distance in the residual network increases monotonically.

Question: Why does streaming work with aggregates in processing graphs and tables ?
Answer : This is because the aggregate can be projected as and when the results arrive and they were the ones being done in batch.
User defined functions and user defined operators can also be used similarly.

The Microsoft StreamInsight Queries follow a five step procedure:

1)     define events in terms of payload as the data values of the event and the shape as the lifetime of the event along the time axis

2)     define the input streams of the event as a function of the event payload and shape. For example, this could be a simple enumerable over some time interval

3)     Based on the events definitions and the input stream, determine the output stream and express it as a query. In a way this describes a flow chart for the query

4)     Bind the query to a consumer. This could be to a console. For example

a.     Var query = from win in inputStream.TumblingWindow( TimeSpan.FromMinutes(3)) select win.Count();

5)     Run the query and evaluate it based on time

#codingexercise
Determine the ceiling of an input value in  binary search tree

Ceiling is the value of a node that is the smallest of all those that are higher than the given one

Int ceil(node root, int val)
{
If (root == null) return -1;
If (root.data == val) return val;
If (root.data < input) return ceil(root.right, val);
Int ceilleft = ceil(root.left, val);
If (ceilleft == -1)
   Return root.data;
Else
   Return ceilleft;
}

Similarly, the floor of a value can be found in BST.