routing algorithms

Routing Algorithms
Suggested Reading:
Bertsekas and Gallager: 5.1 (5.1.1, 5.1.2), 5.2.3 (on Bellman-Ford and
Dijkstra’s)

Kurose and Rose (2001 Edition): 4.1, 4.2, 4.3, 4.4
Also routing protocol


Outline
™
™
™

Flooding
Distributed Bellman Ford Algorithm
Dijkstra’s Shortest Path First Algorithm


Problem
Given a topology, link costs, and a source-destination (SD) pair,
determine a route from S to D so that the route has the minimum
cost (i.e., is the shortest).


Example network
The shortest route A to B:
R1, R2, R5, R8

A

1

R1

1

R2

R4

2

2

R3

4

4

R6

3

R5

2

R7

3

2

R8

B


Routing Metrics
™

Metrics
¾ Link state: up or down (stability)
¾ Delay to send an average size packet (Make high speed links
attractive, but closeness counts)
¾ Bandwidth
¾ Link utilization

For simplicity, our examples assume that the cost for the path is
additive.


Example network
In this simple case, solution is clear from inspection
A

1

R1

1

R2

R4

2

2

R3

4

4

R6

3

R5

2

R7

3

2

R8

B


Flooding
Routers forward packets to all ports
except the ingress port.
R1
Advantages:
™ Simple.
™ Every destination in the network is reachable.
Disadvantages:
™ Some routers receive a packet multiple times.
™ Packets can go round in loops forever.
™ Inefficient.


Spanning Trees
Find the lowest cost route from each of
(R1, …, R7) to R8 (the same as from R8 to the rest for this case)

1

R1

1

R2

R4

2

2

R3

4

4

R6

3

R5

2

R7

3

2

R8

A Spanning Tree
1

R1

1

R2

R4

2

2

R3

4

R5

2

4

R6

3

R7

2
3

R8

The solution is a spanning tree with R8 as the source of the tree.
™ Tree: There are no loops.
™ Spanning: All nodes included.
™ Will see two algorithms that build spanning trees automatically:
™

™
™

The distributed Bellman-Ford algorithm
Dijkstra’s shortest path first algorithm


Bellman Ford Algorithm*
™

Finds the shortest paths, from a given source node, say node 8, to all other
nodes (i is the index for destinations).
General idea:
–First find the shortest single arc path,
–Then the shortest path of at most two arcs, etc.
–Let dij be the cost between node i to j; and dij=∞ if (i,j) is not directly linked

Let Di(h) be the shortest distance from 8 to i using at most h hops/arcs. (h

can be regarded as time also.)
–Di(1) = d8i for i≠8, D8(h) = 0 for all h.
–Di(h+1) = min {j} [Dj(h) +dji] for i≠8.

If all weights are positive, algorithm terminates in N-1 steps, where N is the
no. of nodes in the network.
*Notations are similar to those used in Bertsekas and Gallager.


Bellman-Ford Algorithm
R1

Example

∞

∞

1

2

R3

R1

1

R2

2

∞

∞

4

R4

R3

4

R5

R6
2

3

R8

∞

2

2

∞

R7

2

1

R2
4

3

∞

∞

4

R4

R5

∞

1

∞

2

2

4

3

R7

3

R6
2
3

R8

2


Bellman-Ford Algorithm
6

4

1

R1

2

1

R2

2

4

R5

4

R3

2

R4

6
3

4

R6

3

R7

2

2

2

3

R8

Solution
5

R1

4

1

1

R2

2
4

R3

4

5

2

R4

3

2

R5

R6

2

R7 3

2
R8


Routing Table for R8
Destination

R1

R5

R2

R5

R3

R3

R4

R5

R5

R5

R6

R6

R7

R7

Next Hop


Distributed Bellman-Ford
Let N(i) be a set of neighbors of node i:
Di(h+1) = min {j in N(i)} [Dj(h) +dji] for i≠8; D8(h+1) = 0
Only need to keep/eachange distance information from/with neighbors.

Bellman-Ford Algorithm
Questions:
1.
2.

3.

How long can the algorithm take to run?
How do we know that the algorithm always
converges?
What happens when link costs change, or when
routers/links fail?


A Problem with Bellman-Ford
“Bad news travels slowly”
R1

1

R2

1

R3

1

R4

Consider the calculation of distances from/to R4:
Time
0
1
2
3
…

R3
R1
R2
3,R2
2,R3
1, R4
3,R2
2,R3
3,R2
3,R2
4,R3
3,R2
5,R2
4,R3
5,R2
“Counting
to
…
…infinity” …

R3

d34=inf.

R4 fails


Counting to Infinity Problem
Solutions
1.

2.

3.

Set infinity = “some small integer” (e.g. 16). Stop
when count = 16.
Split Horizon: Because R2 received lowest cost path
from R3, it does not advertise cost to R3
There are many problems with (and fixes for) the
Bellman-Ford algorithm.


Dijkstra’s Shortest Path First Algorithm

™
™
™
™

Routers send out update messages whenever the state of a
link changes. Hence the name: “Link State” algorithm.

Each router calculates lowest cost path to all others, starting
from itself.
At each step of the algorithm, router adds the next shortest
(i.e. lowest-cost) path to the tree.
Finds spanning tree routed on source router.


Dijkstra’s Shortest Path First Algorithm
Example
Step 1:

Shortest path set, S = {R8}. Candidate set, C = {R3 , R5 , R7 , R6 }

Step 2:

S = {R8 , R5},

C = {R3 , R7 , R6 , R2 }.
Step 3:

S = {R8 , R5 , R6},

C = {R3 , R7 , R2 , R4 }.
Step 4:

R5

R8
R6

R5

R8

S = {R8 , R5 , R6 , R7 },

C = {R3 , R2 , R4}.

R6
R5

R7
R8


Dijkstra’s SPF Algorithm
Step 8 : S = {R8 , R5 , R6 , R7 , R2 , R1 , R4 },
C = {}.

R1

1

R2

1

R4

R6

2

R3

4

R5

2

R7

3
R8