1. CSCI 465 Data Communications and Networks Lecture 16
Martin van Bommel
CSCI 465 Data Communications & Networks

2. Least Cost Routing Algorithms
Basis for routing decisions
Network of nodes connected via bi-directional links
Link has cost associated with it in each direction
If just minimizing hop count – each link has cost 1
Usually link cost inversely proportional to capacity
Link costs may include queue lengths
For each pair of nodes, find path with least cost
May be different for each direction
Dijkstra’s or Bellman-Ford algorithms
CSCI 465 Data Communications & Networks

3. CSCI 465 Data Communications & Networks
Dijkstra’s Algorithm
Finds shortest paths from given source nodes to all other nodes
Develop paths in order of increasing length
Algorithm runs in stages
Add node with next shortest path at each stage
Terminates when all nodes added
CSCI 465 Data Communications & Networks

4. CSCI 465 Data Communications & Networks
Dijkstra’s Method
Step 1: Initialization
T = {s} set of nodes incorporated so far (s = start node)
L(n) = w(s, n) for n  s initial path costs are link costs
Step 2: Get Next Node
Find node x  T with minimum L(x)
Add x to T and add the new edge leading to x
Step 3: Update Least-Cost Paths
L(n) = min [ L(n), L(x) + w(x, n) ] for all n  T
CSCI 465 Data Communications & Networks

5. Dijkstra’s Algorithm Example

6. Dijkstra’s Iterations
L(2)
Path
L(3)
L(4)
L(5)
L(6) 1
{1} 2
1–2 5
1-3
1–4  -
{1,4} 4
1-4-3
1-4–5 3
{1, 2, 4}
{1, 2, 4, 5}
1-4-5–3
1-4-5–6
{1, 2, 3, 4, 5} 6
{1, 2, 3, 4, 5, 6}
1-2
1-4

7. Bellman-Ford Algorithm
Find shortest paths from given node subject to constraint that paths contain at most one link
Find shortest paths with constraint of paths of at most two links
and so on …
Uses Lh(n) – cost of least-cost path to node n using no more than h links
CSCI 465 Data Communications & Networks

8. CSCI 465 Data Communications & Networks
Bellman-Ford Method
Step 1: Initialization
L0(n) =  for all n  s
Lh(s) = 0 for all h
Step 2: Update
for each successive h ≥ for each n  s, compute Lh(n) = minj [ Lh( j ) + w( j, n) ]
Connect n with the predecessor node j that achieves minimum (eliminate previous links from earlier iterations)
Path from s to n terminates with j to n link
CSCI 465 Data Communications & Networks

9. Bellman-Ford Example

10. Bellman-Ford Iterationsh
Lh(2)
Path
Lh(3)
Lh(4)
Lh(5)
Lh(6)  - 1 2
1-2 5
1-3
1-4 4
1-4-3
1-4-5 10
1-3-6 3

11. Routing Algorithm Comparison
Dijkstra: Each node
needs to know compete topology
needs to know link costs of all links in network
must exchange information with all other nodes
Bellman-Ford: Each node
needs link cost to neighboring nodes
needs total cost from s to each neighbor
can exchange information with direct neighbors
can update costs and paths based on information from neighbors and knowledge of link costs
CSCI 465 Data Communications & Networks

12. Routing Algorithm Evaluation
Dependent upon
Processing time of algorithms themselves
Amount of information required from other nodes
Very implementation specific
Both converge under static topology and costs
converge to same solution
If link costs change, attempt to catch up
If dependent on traffic, may have instability
CSCI 465 Data Communications & Networks