1. Chapter 5 Network Layer

2. Network layer
Physical layer: move bit sequence between two adjacent nodes
Data link: reliable transmission between two adjacent nodes
Network： guides packets from the source to destination, along a path that may comprise a number of routers and links.
Application
network
Data link
physical
transport

3. Network layer transport segment from sending to receiving host
application
transport
network
data link
physical
transport segment from sending to receiving host
on sending side encapsulate segments into datagrams
on receving side, deliver segments to transport layer
network layer protocols in every host, router
router examines header fields in all IP datagrams passing through it
network
data link
physical
application
transport
network
data link
physical

4. Network layer
A router is a box with input and output links. It has access to the routing table
When a packet arrives at a router, the router examines header fields of the packets and forwards the packets according to its routing table
Routing: determine the best route/path from source to destination. After that, routing tables can be constructed
routing algorithm
routing table
header value
output link
0100
0101
0111
1001 3 2 1
application
transport
network
data link
physical
application
transport
network
data link
physical 1
0111
0111 2 3
value in arriving
packet’s header
Source
Destination

5. Routing Algorithm
The routing problem can be represented using a graph. u y x w v z 2 1 3 5
Graph: G = (N,E)
N = set of routers = { u, v, w, x, y, z }
A set of link lengths/costs between these
nodes: {di,j, i, j ϵ S}, di,j= ∞ if there is
no link from i to j.
Cost/distance of path (x1, x2, x3,…, xp) = dx1,x2 + dx2,x3 + … + dxp-1,xp 0
Question: What’s the least-cost/shortest distance path between u and z ?
Routing algorithm: algorithm that finds least-cost/shortest path

6. Routing Algorithm "Good" routing algorithm: 1. simplicity
2. robustness to node/link failures
3. stability (fast adaptation to topology and traffic changes)
4. optimality (low delay and high throughput)
5. fairness.
Most of routing algorithms are distributed algorithms:
routing decisions are made at each node in a cooperative
manner. Information is exchanged with neighbors.
Bellman-Ford algorithm is a distributed algorithm.

7. Bellman-Ford Algorithm
Initial condition:
At step h=0, node i has the estimate d0(i)= , for all i S with i D and that d0(D)=0. The nodes advertise these estimates to their neighbors.
At step h 1:
Node j updates its estimate of its distance to D to some value dh(j) and advertises that value to its neighbors.
At step h+1:
Node i examines the message that it just received from its neighbors at the previous step h, e.g., j might be a neighbor of i, that is d(i,j)< and j informs i at step h that the shortest estimated distance from j to D is equal to dh(j). Node i gets similar messages from all its neighbors. Node i then know that if it chooses to send a packet to D by first sending it to its neighbor j, then the distance that the packet will travel to reach D is estimated to be d(i,j)+dh(j). Since node i can choose the neighbor to which it sends the message destined to D, node i estimates the shortest distance to D as dh+1(i) = min{d(i,j)+dh(j), j S}.

9. Bellman-Ford Algorithm
The shortest path length from node i to the destination node is the sum of the length of the arc to the node following i and the shortest path length from that node to the destination node.
Algorithm:
( =1)

10. Bellman-Ford Algorithm
Example:
Given a digraph G(N,A) with the following specifications.
N={1,2,3,4,5,6}
A={(2,1)=1, (2,3)=2, (3,1)=5, (3,2)=1, (4,2)=6, (4,5)=2, (5,3)=2, (5,4)=3, (6,4)=1, (6,5)=2}
Draw and label the arcs of the digraph.
Discuss how the Bellman-Ford algorithm can be used to determine the shortest path from any node to a destination node on the digraph.
Find the shortest paths from each of the other nodes to destination node 1 on this digraph using the Bellman-Ford algorithm.

11. Bellman-Ford Algorithm
Solution:
(a) The labeled digraph is as follows:

12. Bellman-Ford Algorithm
(b) The Bellman-Ford algorithm can be implemented on the above diagram.
Label each node i with (di, j) where di (the first element of the label) is the distance from node i to the destination node through the neighboring node j.
Initialization: label node i with (di0,.) where
Shortest distance labeling of all nodes: for each successive run h > 0, determine j which minimizes dih+1=d(i,j)+djh, and update label at each node i with (dih+1,j).
Repeat step 3 until no change of any di occurs.

13. Bellman-Ford Algorithm
(c) The execution results of Bellman-Ford algorithm on the above digraph is shown in the following table
From table, we can get all the shortest paths:
2 -> 1
3 -> 2 -> 1
4 -> 5 -> 3 -> 2 -> 1
5 -> 3 -> 2 -> 1
6 -> 5 -> 3 -> 2 -> 1

14. Link-State Routing Algorithm
Network topology, link costs known to all nodes
accomplished via “link state broadcast”
all nodes have same info
Then each router uses Dijkstra’s algorithm
To compute its least cost paths to all other nodes
yields forwarding table for that node
iterative: after k iterations, know least cost path to k destinations
Notation:
c(x,y): link cost from node x to y; = ∞ if not direct neighbors
D(v): current value of cost of path from source to destination v
p(v): predecessor node along path from source to v
N': set of nodes whose least cost path definitively known

15. Dijsktra’s Algorithm at node u
Notations:
c(x,y): link cost from node x to y
D(v): current value of cost of path from source u to destination v
p(v): predecessor node along path from source u to v
N': set of nodes whose least cost path definitively known
1 Initialization:
2 N' = {u}
3 for all nodes v
if v adjacent to u
then D(v) = c(u,v)
else D(v) = ∞ 7
8 Loop
9 find w not in N' such that D(w) is a minimum
10 add w to N'
11 update D(v) for all v adjacent to w and not in N' :
D(v) = min( D(v), D(w) + c(w,v) )
13 /* new cost to v is either old cost to v or known
shortest path cost to w plus cost from w to v */
15 until all nodes in N'

16. Step 0: Initialization z x u y w v Source = node u2 1 3 5
Source = node u
Routing algorithm is run at node u to find the least cost paths to all the other nodes.
Because no direct connection
Step 1 2 3 4 5 N' u ux
uxy
uxyv
uxyvw
uxyvwz
D(v),p(v)
2, u
2,u
D(w),p(w)
5, u
4,x
3,y
D(x),p(x)
1, u
D(y),p(y) ∞
2,x
D(z),p(z)
4,y

17. Iteration 1 z x D(v) = min( D(v), D(w) + c(w, v))u y x w v z 2 1 3 5
D(v) = min( D(v), D(w) + c(w, v))
- Smallest cost in row 0 and node x is not in N’
- Then, add x to N’
Step 1 2 3 4 5 N' u ux
uxy
uxyv
uxyvw
uxyvwz
D(v),p(v)
2, u
D(w),p(w)
5, u
D(x),p(x)
1, u
D(y),p(y) ∞
D(z),p(z)
2, u3,y
3,y
2,u4, x
2, u
4, x
2, x ∞
D(x) =1
C(x, v)
D(v) = min(2, 1+ 2) = 2
D(w) = min(5, 1+ 3) = 4
D(y) = min(∞, 1+ 1) =2

18. Iteration 2 z x D(v) = min( D(v), D(w) + c(w, v))u y x w v z 2 1 3 5
Iteration 2
D(v) = min( D(v), D(w) + c(w, v))
Compute for the neighbors of y that are not in N’
D(w) = min(4, 2 + 1)= 3
D(z) = min(∞, 2+ 2) = 4
Lower cost  change
Step 1 2 3 4 5 N' u ux
uxyv
uxyvw
uxyvwz
D(v),p(v)
2,u
D(w),p(w)
5,u
4,x
D(x),p(x)
1,u
D(y),p(y) ∞
2, x
D(z),p(z)
uxy
2,u
3,y
4, y

19. Iteration 3 z x D(v) = min( D(v), D(w) + c(w, v))u y x w v z 2 1 3 5
D(v) = min( D(v), D(w) + c(w, v))
Iteration 3
D(w) = min(3, 2 + 3) =3
Step 1 2 3 4 5 N' u ux
uxy
uxyvw
uxyvwz
D(v),p(v)
2,u
D(w),p(w)
5,u
4,x
3,y
D(x),p(x)
1,u
D(y),p(y) ∞
2,x
D(z),p(z)
4,y
uxyv
3,y
4,y

20. Iteration 4 z x D(v) = min( D(v), D(w) + c(w, v))u y x w v z 2 1 3 5
D(v) = min( D(v), D(w) + c(w, v))
Iteration 4
D(z) = min(4, 3 + 5) = 4
Step 1 2 3 4 5 N' u ux
uxy
uxyv
D(v),p(v)
2,u
D(w),p(w)
5,u
4,x
3,y
D(x),p(x)
1,u
D(y),p(y) ∞
2,x
D(z),p(z)
4,y
uxyvw
4,y

21. Iteration 5 z x D(v) = min( D(v), D(w) + c(w, v))u y x w v z 2 1 3 5
D(v) = min( D(v), D(w) + c(w, v))
Iteration 5
D(z) = min(4, 4 + 0) = 4
Step 1 2 3 4 5 N' u ux
uxy
uxyv
uxyvw
D(v),p(v)
2,u
D(w),p(w)
5,u
4,x
3,y
D(x),p(x)
1,u
D(y),p(y) ∞
2,x
D(z),p(z)
4,y
uxyvwz

22. Resulting shortest-path tree from u:y x w v z
To v: uv
To x: ux
To y: uxy
To w: uxyw
To z: uxyz
Resulting forwarding table in u: v x y w z
destination
Next-hop node
cost 2 1 2 3 4

23. Dijkstra’s algorithm: example
D(v)
p(v)
D(w)
p(w)
D(x)
p(x)
D(y)
p(y)
D(z)
p(z) v
Step N' u ∞
7,u
3,u
5,u u 1 uw ∞
11,w
6,w
5,u w 2
uwx
14,x
11,w
6,w 3
uwxv
14,x
10,v 4
uwxvy
12,y 5
uwxvyz w 3 4 v x u 5 7 y 8 z 2 9
Notes:
construct shortest path tree by tracing predecessor nodes

24. Optimal Routing Advantage:
The algorithm is very flexible to the choice of initial estimates dj0. It can be executed at each node i in parallel with every other nodes.
Disadvantage:
The routing uses only one path per original-destination pair, thereby potentially limiting the throughput; its capability to adapt the changing traffic conditions is limited by its susceptibility to oscillations, which is due to the abrupt traffic shifts resulting when some of the shortest paths change due to changes in link lengths.
Example:

25. Optimal Routing
Source A is attached to two routers. Initially, A sends all its traffic to D via node R1. The delay along link (R1, D) is equal to 10 and delay along link (R2, D) is equal to 1+e. These routers advertise these delays to node A which then finds out that the preferred path to D starts with R2 and not with R1. Source A then sends all its traffic to R2 and not with R1.
As a result, the situation alternates between the two paths. A better routing strategy would consist in source A sending half of its traffic to R1 and the other half to R2.
Optimal routing can eliminate both of these disadvantages. It directs traffic exclusively along paths which are shortest with respect to some link lengths that depend on the traffic flows carried by the links.

26. Optimal Routing
E.g. The given input r is to be divided into the two path flows 1 & 2, so as to minimize a cost function (based on M|M|1 approximation)

27. Optimal Routing

28. Optimal Routing

29. Optimal Routing

30. Optimal Routing
The figure shows the optimal solution for r in the range [0, C1+C2) for possible inputs.