Cuts and Network Flow

The backbone analysis of any network is broadly accomplished by using Graph Theory and its Algorithms. The performance constraints are Reliability, Delay/Throughput and the goal is to minimize cost. In the backbone designing of a network the concerned points and considerations are :

1. What should be the backbone topology ?
2. Assignment of Line Capacities.
3. Flow Assignment of the lines and hence the whole network.

Some Common Definitions :

• Network : A network is a circuit which is a sequence of adjacent nodes that comes back to the starting node. A circuit containing all the nodes of a graph is known as Hamiltonian Circuit.
• Spanning Tree : A spanning tree of a graph is a sub graph containing all the nodes of the graph and some collection of arcs chosen so that there is exactly one path between each pair of nodes.
• Cut : A concept from graph theory that is useful for modelling the carrying capacity of a network is the cut. An X-Y cut is a set of arcs whose removal disconnects node X from node Y.

Cuts in weighted Graph

Four A-H cuts are shown in the figure above.

Cut 1 : AB, AE (capacity = 11)
Cut 2 : AB, ED, JF, JK (capacity = 23)
Cut 3 : BC, FG, KL (capacity = 10)
Cut 4 : CH, CL (capacity = 12)

• Minimal Cut : It is one which replacement of any of its member reconnects the Graph. In other words, in a minimal cut, all the arcs are essential. The set of arcs AB, AE and FG form A-H cut, but the cut is not minimal, because restoring arc FG does not reconnect node A to node H.
• Minimum Cut : In a weighted graph each cut has capacity. A cut with minimum capacity is minimum cut. In the diagram shown above the Cut-3 with capacity=10, is the minimum cut.
• Max Flow Min Cut Theorem : The maximum flow between any two arbitrary nodes in any graph cannot exceed the capacity of the minimum cut separating those two nodes.
• Polling : Each station on the network is polled in some predetermined order. Between polls, stations accumulate messages in their queues but do not transmit until they are polled. The process by which terminals on a line are successively invited to send data. This may occur by having a master station use a polling list to invite terminals to send data, or by each terminal sending a poll message to the next terminal in sequence so that it might send data, or via the use of a token as in a token ring to control the sending of data.
• Polling Sequences : The sequence in which terminals are invited to transmit data. This may be based on a polling list in which terminal ids are stored in the sequence that they are to be polled.
• Polling Techniques : 

A typical Polling Network

1. Roll call Polling : A master station uses one or more polling lists to determine the next terminal in sequence to be polled. Each station has to be polled in turn by the central computer (controller). After the station has transmitted its backlog of messages, it notifies the central controller with a suffix to its last packet. After receiving this suffix packet, the controller sends a poll to the next station in the polling sequence.
2. Hub Polling : The terminal currently in polled mode polls the next terminal in sequence. In this case, the go-ahead (suffix) packet contains the next station address. A monitoring channel must be provided to indicate to the appropriate station that it should start transmitting. Essentially the go-ahead is transmitted directly from one station to another.
3. Token Passing : A token is passed to the next device on the network (ring or bus) which may use it to transmit data or may let it pass to the next device.