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Chapter 4 Distributed Bellman-Ford RoutingPowerPoint Presentation

Chapter 4 Distributed Bellman-Ford Routing

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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

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

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

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

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]}

- To calculate D(i,j), node i only needs d(i,k)’s and D(k,j)’s from neighbors

Prof. Rick Han, University of Colorado at Boulder

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]}

- 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 :
- 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

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

Bellman-Ford Algorithm Example (2) Calculate D(i,j)[2] for all other combinations of i<>j

- 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]}

- Example: for node C to F

- No one-link path from B to F, so D(B,F)[1] is infinity, same for D(D,F)[1]

Prof. Rick Han, University of Colorado at Boulder

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

- Example: for node C to F

- Since diameter = 3, we’re done and have found all shortest distance paths D(i,j)

Prof. Rick Han, University of Colorado at Boulder

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

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

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

- Physically distribute the calculation of D(i,j)[h] to router i only, and

Prof. Rick Han, University of Colorado at Boulder

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

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

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

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

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

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

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