1. Chapter 4 Distributed Bellman-Ford Routing
Professor Rick Han
University of Colorado at Boulder

2. Prof. Rick Han, University of Colorado at Boulder
Announcements
Reminder: Programming assignment #1 is due Feb. 19
Homework #2 available on Web site, due Feb. 26
Hand back HW #1 next week
OH cancelled yesterday, send me to meet
Next, more on IP routing, …
Prof. Rick Han, University of Colorado at Boulder

3. Recap of Previous Lecture
ARP
IP Forwarding Tables
Destination and Output Port
IP Routing
Distributed algorithm to create Forwarding Tables
Calculate shortest path to each node
Distance Vector (RIP)
Presentation should have been better by me, textbook, etc.
Prof. Rick Han, University of Colorado at Boulder

4. Bellman-Ford Equation
Distance vector & RIP based on distributed implementation of Bellman-Ford algorithm
Bellman-Ford equation:
Label routers i=A, B, C, …
Let D(i,j) = distance for best route from i to remote j
Let d(i,j) = distance from router i to neighbor j
Set to infinity if i=j or i and j not immediate neighbors
Prof. Rick Han, University of Colorado at Boulder

5. Bellman-Ford Equation (2)
D(i,j) = min {d(i,k) + D(k,j)} for all i<>j k
neighbors
Ex. D(B,F) = min {d(B,k) + D(k,F)}
k=A,C,E
Prof. Rick Han, University of Colorado at Boulder

6. Bellman-Ford Algorithm
Bellman-Ford equation:
D(i,j) = min {d(i,k) + D(k,j)} for all i<>j
k neighbors
Bellman-Ford Algorithm solves B-F Equation:
To calculate D(i,j), node i only needs d(i,k)’s and D(k,j)’s from neighbors
Problem: don’t know D(k,j)’s
Solution:
For each node i, first find shortest distance path from i to j using one link, D(i,j)[1]
Shortest distance path using two or fewer links, D(i,j)[2], must depend on the shortest distance path using one link, namely D(i,j)[2] = min {d(i,j) + D(i,j)[1]}
Prof. Rick Han, University of Colorado at Boulder

7. Bellman-Ford Algorithm (2)
Key observation:
By induction, the best (h+1 or fewer)-hop path between nodes i and j must be arise from an i-to-neighbor link connected with a (h or fewer)-hop path from neighbor to j :
D(i,j)[h+1] = min {d(i,k) + D(k,j)[h]}
Bellman-Ford Algorithm:
D(i,j)[h+1] = min {d(i,k) + D(k,j)[h]} for all i<>j, h=0,1, …
k neighbors
Iterate h=0,1,2, … until reach diameter DM of graph
D(i,j)[DM] is the originally desired B-F solution D(i,j) !
At each h, calculate D(i,j)[h+1] for all i<>j
At h=0, D(i,j)[0] = {0 for i=j, infinity otherwise}
D(i,i)[h] = link cost on which dist. vector is sent - 1
Prof. Rick Han, University of Colorado at Boulder

8. Bellman-Ford Algorithm Example
Suppose C wants to find shortest path to each destination
First, calculate shortest one-link paths from each node: easy, D(i,j)[1]=d(i,j)
D(C,B)[1], D(C,D)[1], and
D(B,A)[1], D(B,E)[1], D(B,C)[1], and
D(D,E)[1], D(D,C)[1], and
D(A,B)[1], D(A,E)[1], D(A,F)[1], and
D(E,A)[1], D(E,B)[1], D(E,D)[1], D(E,F)[1], and
D(F,A)[1], D(F,E)[1]
Prof. Rick Han, University of Colorado at Boulder

9. Bellman-Ford Algorithm Example (2)
Second, calculate shortest 2-or-fewer hop paths from each node:
Example: for node C to F
D(C,F)[2] = min (d(C,k) + D(k,F)[1]) for all j
k neighbors
= min {d(C,B) + D(B,F)[1], d(C,D) + D(D,F)[1]}
No one-link path from B to F, so D(B,F)[1] is infinity, same for D(D,F)[1]
Calculate D(i,j)[2] for all other combinations of i<>j
Prof. Rick Han, University of Colorado at Boulder

10. Bellman-Ford Algorithm Example (3)
Third, calculate shortest 3-or-fewer hop paths from each node:
Example: for node C to F
D(C,F)[3] = min {d(C,B) + D(B,F)[2], d(C,D) + D(D,F)[2]}
No more unknowns:
D(B,F)[2] is known by now and was calculated in the last iteration, = min{d(B,k) + D(k,F)[1]}
D(D,F)[2] is also known
Since diameter = 3, we’re done and have found all shortest distance paths D(i,j)
Prof. Rick Han, University of Colorado at Boulder

11. Distributed Bellman-Ford Algorithm
D(i,j)[h+1] = min {d(i,k) + D(k,j)[h]} for all i<>j, h=0,1, …
k neighbors
One way to implement in a real network:
Flood d(i,j) first to every router in the network
Calculate B-F Algorithm in each router
Drawbacks:
Generates lots of overhead
Requires much computation on each router
Duplication of many of calculations on each router
Consider an alternative to distribute calculations
Prof. Rick Han, University of Colorado at Boulder

12. Distributed Bellman-Ford Algorithm (2)
D(i,j)[h+1] = min {d(i,k) + D(k,j)[h]} for all i<>j, h=0,1, …
k neighbors
Key observations:
We had to calculate D(i,j)[h] for each node i in the graph, at each step h in the iteration
At every iteration h, we only needed information about the h-1 or fewer hop paths to calculate D(i,j)[h]
Prof. Rick Han, University of Colorado at Boulder

13. Distributed Bellman-Ford Algorithm (3)
Therefore, in a real network,
Physically distribute the calculation of D(i,j)[h] to router i only, and
No duplication
Less calculation
Exchange the results of your D(i,j)[h] with neighboring routers at each iteration h
Less overhead
Satisfies condition that D(i,j)[h] only needs info on h-1 or less hop paths.
At iteration h, d(i,j) within radius h-1 will be propagated to all routers within radius h-1
Prof. Rick Han, University of Colorado at Boulder

14. Distributed Bellman-Ford Algorithm (4)
In practice, convergence will eventually occur even if different routers are slow to propagate or calculate their D(i,j)[h] and/or d(i,j)
Bertsekas and Gallagher proved this, in the absence of topology changes
Distributed routing algorithm where each router only performs a small but sufficient part of the overall B-F algorithm
Node i calculates and sends D(i,j)[h] to its neighbors – this is a distance vector
Distributed Bellman-Ford Algorithm = Distance Vector Algorithm
Prof. Rick Han, University of Colorado at Boulder

15. Prof. Rick Han, University of Colorado at Boulder
Distance Table
Bellman-Ford Algorithm:
D(i,j)[h+1] = min {d(i,k) + D(k,j)[h]} for all i<>j, h=0,1, …
k neighbors
Each router i must maintain a “distance table”:
Must store d(i,k), D(k,j)[h] for each neighbor k and destination j
Costs D(k,j)[h]
Neighbors k
Destinations j k1 k2 k3 j1
D(k1,j1)[h] … j2
Prof. Rick Han, University of Colorado at Boulder

16. Prof. Rick Han, University of Colorado at Boulder
Distance Table (2)
In reality, each cell in distance table stores d(i,k) + D(k,j)[h], not just D(k,j)[h]
Must store d(i,k) or receive it within a neighbor’s distance vector advertisement
If d(i,k) is a hop, then d(i,j)=1 always, so no need to store
Costs D(k,j)[h]
Neighbors k
Destinations j k1 k2 k3 j1
d(i,k1) + D(k1,j1)[h] … j2
Prof. Rick Han, University of Colorado at Boulder

17. Prof. Rick Han, University of Colorado at Boulder
Routing Table
Easy to derive a Routing Table from a distance table: choose the minimum distance in the row
Costs D(k,j)[h]
Neighbors k
Destinations j k1 k2 k3 j1
d(i,k1) + D(k1,j1)[h] … j2
Routing Table At Router i
Destinations j
Outgoing link/port
Cost j1 k2 j2 k3
Prof. Rick Han, University of Colorado at Boulder

18. Routing Information Protocol (RIP)
RIP is a specific realization of the distance vector or distributed Bellman-Ford routing algorithm
Distance vectors are carried over UDP over IP
RIP uses hop count as its shortest path metric, so d(i,j)=1
Distance vectors are sent every 30 seconds
When a routing table changes, a router can send triggered updates to neighbors before 30 sec
Can lead to network storms, so limit rate: wait 5 seconds between sending new routing update and the update that caused routing table to change
Prof. Rick Han, University of Colorado at Boulder

19. Alternative Shortest Path Calc.
Compute a shortest path tree
Observation:
shortest path to nodes further from the root must go through a branch of the shortest path tree closer to the root
Strategy: expand outwards, calculating the shortest path tree from the root
Prof. Rick Han, University of Colorado at Boulder