Chapter%204%20Distributed%20Bellman-Ford%20Routing

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Chapter 4Distributed Bellman-Ford Routing

• Professor Rick Han
• University of Colorado at Boulder
• rhan_at_cs.colorado.edu

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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,

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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.

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Bellman-Ford Equation

• Distance vector RIP based on distributed
  implementation of Bellman-Ford algorithm
• Bellman-Ford equation
• Label routers iA, 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 ij or i and j not immediate
  neighbors

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Bellman-Ford Equation (2)

• Bellman-Ford equation
• D(i,j) min d(i,k) D(k,j) for all iltgtj
• k
• neighbors
• Ex. D(B,F) min d(B,k) D(k,F)
• kA,C,E

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Bellman-Ford Algorithm

• Bellman-Ford equation
• D(i,j) min d(i,k) D(k,j) for all iltgtj
• 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 dont 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

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Bellman-Ford Algorithm (2)

• Key observation
• By induction, the best (h1 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)h1 min d(i,k) D(k,j)h
• Bellman-Ford Algorithm
• D(i,j)h1 min d(i,k) D(k,j)h for all
  iltgtj, h0,1,
• k neighbors
• Iterate h0,1,2, 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)h1 for all iltgtj
• At h0, D(i,j)0 0 for ij, infinity
  otherwise
• D(i,i)h link cost on which dist. vector is
  sent - 1

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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)1d(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

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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
  iltgtj

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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, mind(B,k) D(k,F)1
• D(D,F)2 is also known

• Since diameter 3, were done and have found all
  shortest distance paths D(i,j)

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Distributed Bellman-Ford Algorithm

• Bellman-Ford Algorithm
• D(i,j)h1 min d(i,k) D(k,j)h for all
  iltgtj, h0,1,
• 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

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Distributed Bellman-Ford Algorithm (2)

• Bellman-Ford Algorithm
• D(i,j)h1 min d(i,k) D(k,j)h for all
  iltgtj, h0,1,
• 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

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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

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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

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Distance Table

• Bellman-Ford Algorithm
• D(i,j)h1 min d(i,k) D(k,j)h for all
  iltgtj, h0,1,
• 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 Neighbors k Neighbors k
Destinations j k1 k2 k3
j1 D(k1,j1)h
j2
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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
  neighbors 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 Neighbors k Neighbors k
Destinations j k1 k2 k3
j1 d(i,k1) D(k1,j1)h
j2
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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 Neighbors k Neighbors k
Destinations j k1 k2 k3
j1 d(i,k1) D(k1,j1)h
j2
Destinations j Outgoing link/port Cost
j1 k2
j2 k3
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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

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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