Problem set 4 (Tuesday 20.11.2001)

1. Assuming that alla routers and hosts are working properly and that all software in both is free of all errors, is there any chance, however small, that a packet will be delivered to the wrong destination?

2. Consider one sender and 32 receivers. Suppose the sender is connected to the receivers through a binary tree of routers. The sender is the root and the receivers are the leaves in the tree and in each node between has a router. The sender sents a packet first to the next router, that sends it to the router below etc until the packet reaches the the receiver.
   1. How many sendings alltogether are needed if a separate packet is sent to each receiver?
   2. How many sendings are needed, if the packet is sent as a multicast packet to each receiver?

3. Flooding is one way to multicast packets. Where else is flooding used? The problem with flooding is that very many packets are sent and the packets can remain circulating in the net forever. What different ways are there to reduce the amount of packet sendings and eventually dam the flooding?

4. Suppose node C is chosen as the center in a center-based multicast routing algorithm. Assuming that each attached router in the multicast group (= A, B, E and F) uses its least-cost path to node C, show the resulting center-based multicast routing tree. Is the resulting tree a minimum-cost Steiner tree?

            1             1
          B ------------ D ------------ E
          | .                          .|
          |  .                       .  |
          |   .                    .    |
          |    . 2             2 .      |
        4 |     .              .        | 1
          |      .           .          |
          |       .        .            |
          |        .     .              |
          |         .  .                |
          A -------- C ---------------  F            
               3             2
   
   

5. Supposing that in the net of the previous problem the cost of the link BD becomes tenfold, that is changes to 10. Find the Steiner tree that connects all the nodes A, B, E and F belonging to the group. It is not necessary to use the Steiner tree construction algorithm. It is enough to inspect the net in order to find the Steiner tree.

6. Consider the network below. Supposing that the link costs are equal on all links, what kind of "reverse path forwarding" -tree would you make for node F? How is this tree actually formed? When node F sends a broadcast packet, how many packets are really sent in the subnetwork?

   
                B --------------------------------------------- C
                |                                               |
         E -----|---------------------------- A                 |
         |      |                             |                 |
         |      |                             |                 |  
         H -----|------------ I ------------- F --------------- D
         |      |             | .                               |
         |      L             |  .                              |
         |     .              |   .                             |
         |    .               |    .                            | 
         |   .                |     .                           |
         |  .                 |      .                          |
         | .                  |       .                         |
         |.                   |        .                        |
         K -------- M ------- N -- O -- J --------------------- G