## 1.18 Exercises

Exercises are given fractional (floating point) numbers, to allow for interpolation of new exercises. Exercise 2.5 is distinct, for example, from exercises 2.0 and 3.0. Exercises marked with a ♢ have solutions or hints at 24.1   Solutions for An Overview of Networks.

1.0. Give forwarding tables for each of the switches S1-S4 in the following network with destinations A, B, C, D. For the next_hop column, give the neighbor on the appropriate link rather than the interface number.

A         B         C
│         │         │
S1────────S2────────S3
│
│
S4───D


2.0. Give forwarding tables for each of the switches S1-S4 in the following network with destinations A, B, C, D. Again, use the neighbor form of next_hop rather than the interface form. Try to keep the route to each destination as short as possible. What decision has to be made in this exercise that did not arise in the preceding exercise?

A───S1──────S2───B
│       │
│       │
D───S4──────S3───C


2.5. In the network of the previous exercise, suppose that destinations directly connected to an immediate neighbor are always reached via that neighbor; eg S1’s forwarding table will always include ⟨B,S2⟩ and ⟨D,S4⟩. This leaves only routes to the diagonally opposite nodes undetermined (eg S1 to C). Show that, no matter which next_hop entries are chosen for the diagonally opposite destinations, no routing loops can ever be formed. (Hint: the number of links to any diagonally opposite switch is always 2.)

2.7.♢ Give forwarding tables for each of the switches A-E in the following network. Destinations are A-E themselves. Keep all route lengths the minimum possible (one hop for an immediate neighbor, two hops for everything else). If a destination is an immediate neighbor, you may list its next_hop as direct or local for simplicity. Indicate destinations for which there is more than one choice for next_hop. 3.0. Consider the following arrangement of switches and destinations. Give forwarding tables (in neighbor form) for S1-S4 that include default forwarding entries; the default entries should point toward S5. The default entries will thus automatically forward to the “possible other destinations” shown below right.

Eliminate all table entries that are implied by the default entry (that is, if the default entry is to S3, eliminate all other entries for which the next hop is S3).

A───S1
│         D
│         │
C───S3────────S4─────────S5────... possible other destinations
│         │
│         E
B───S2


4.0. Four switches are arranged as below. The destinations are S1 through S4 themselves.

S1──────S2
│       │
│       │
S4──────S3

(a). Give the forwarding tables for S1 through S4 assuming packets to adjacent nodes are sent along the connecting link, and packets to diagonally opposite nodes are sent clockwise.
(b). Give the forwarding tables for S1 through S4 assuming the S1–S4 link is not used at all, not even for S1⟷S4 traffic.
5.0. Suppose we have switches S1 through S4; the forwarding-table destinations are the switches themselves. The tables for S2 and S3 are as below, where the next_hop value is specified in neighbor form:
S2: ⟨S1,S1⟩ ⟨S3,S3⟩ ⟨S4,S3⟩
S3: ⟨S1,S2⟩ ⟨S2,S2⟩ ⟨S4,S4⟩

From the above we can conclude that S2 must be directly connected to both S1 and S3 as its table lists them as next_hops; similarly, S3 must be directly connected to S2 and S4.

(a). The given tables are consistent with the network diagrammed in exercise 4.0. Are the tables also consistent with a network in which S1 and S4 are not directly connected? If so, give such a network; if not, explain why S1 and S4 must be connected.
(b). Now suppose S3’s table is changed to the following. Find a network layout consistent with these tables in which S1 and S4 are not directly connected. Do not add additional switches.
S3: ⟨S1,S4⟩ ⟨S2,S2⟩ ⟨S4,S4⟩

While the table for S4 is not given, you may assume that forwarding does work correctly. However, you should not assume that paths are the shortest possible. Hint: It follows from the S3 table above that the path from S3 to S1 starts S3 ⟶ S4; how will this path continue? The next switch along the path cannot be S1, because of the hypothesis that S1 and S4 are not directly connected.

6.0. (a) Suppose a network is as follows, with the only path from A to C passing through B:

... ──A────B────C── ...


Explain why a single routing loop cannot include both A and C. Hint: if the loop involves destination D, how does B forward to D?

(b). Suppose a routing loop follows the path A──S1──S2── … ──Sn──A, where none of the Si are equal to A. Show that all the Si must be distinct. (A corollary of this is that any routing loop created by datagram-forwarding either involves forwarding back and forth between a pair of adjacent switches, or else involves an actual graph cycle in the network topology; linear loops of length greater than 1 are impossible.)

7.0 Consider the following arrangement of switches:

S1─────S4─────S10──A──E
│      │
│      │
S2─────S5─────S11──B
│      │
│      │
S3─────S6─────S12──C──D──F


Suppose S1-S6 have the forwarding tables below. For each of the following destinations, suppose a packet is sent to the destination from S1.

(a). A
(b). B
(c). C
(d).♢ D
(e). E
(f). F

Give the switches the packet will pass through, including the initial switch S1, up until the final switch S10-S12.

S1: (A,S4), (B,S2), (C,S4), (D,S2), (E,S2), (F,S4)

S2: (A,S5), (B,S5), (D,S5), (E,S3), (F,S3)

S3: (B,S6), (C,S2), (E,S6), (F,S6)

S4: (A,S10), (C,S5), (E,S10), (F,S5)

S5: (A,S6), (B,S11), (C,S6), (D,S6), (E,S4), (F,S2)

S6: (A,S3), (B,S12), (C,S12), (D,S12), (E,S5), (F,S12)

7.5 Suppose a set of nodes A-F and switches S1-S6 are connected as shown.

A────S1───5───S4────D
│         │
1         1
│         │
B────S2───2───S5────E
│         │
8         1
│         │
C────S3───4───S6────F


The links between switches are labeled with weights, which are used by some routing applications. The weights represent the cost of using that link. You are to find the path through S1-S6 with lowest total cost (that is, with smallest sum of weights), for each of the following transmissions. For example, the lowest-cost path from A to E is A–S1–S2–S5–E, for a total cost of 1+2=3; the alternative path A–S1–S4–S5–E has total cost 5+1=6.

(a).♢ A→F
(b). A→D
(c). A→C

(d). Give the complete forwarding table for S2, where all routes are selected for lowest total cost.

8.0 In exercise 7.0, the routes taken by packets A-D are reasonably direct, but the routes for E and F are rather circuitous.

(a). Assign weights to the seven links S1–S2, S2–S3, S1–S4, S2–S5, S3–S6, S4–S5 and S5–S6, as in exercise 7.5, so that destination E’s route in exercise 7.0 becomes the optimum (lowest total link weight) path.
(b). Assign weights to the seven links that make destination F’s route in exercise 7.0 optimal. (This will be a different set of weights from part (a).)

Hint: you can do this by assigning a weight of 1 to all links except to one or two “bad” links; the “bad” links get a weight of 10. In each of (a) and (b) above, the route taken will be the route that avoids all the “bad” links. You must treat (a) entirely differently from (b); there is no assignment of weights that can account for both routes.

9.0 Suppose we have the following three Class C IP networks, joined by routers R1–R4. There is no connection to the outside Internet. Give the forwarding table for each router. For networks directly connected to a router (eg 200.0.1/24 and R1), include the network in the table but list the next hop as direct or local.

R1── 200.0.1/24
│
│
│
R4──────────R2── 200.0.2/24
│
│
│
R3── 200.0.3/24