Chapter 4 Distributed Bellman-Ford Routing

1. Chapter 4Distributed Bellman-Ford Routing Professor Rick Han University of Colorado at Boulder rhan@cs.colorado.edu

2. 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 email 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) • Bellman-Ford equation: • 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 • 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) • 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 • 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 distancevector • Distributed Bellman-Ford Algorithm = Distance Vector Algorithm Prof. Rick Han, University of Colorado at Boulder

15. 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 Prof. Rick Han, University of Colorado at Boulder

16. 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 Prof. Rick Han, University of Colorado at Boulder

17. Routing Table At Router i Routing Table • Easy to derive a Routing Table from a distance table: choose the minimum distance in the row 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