Distributed Calculation of Shortest Paths

1
Distributed Calculation of Shortest Paths

• Motivation
• Since network conditions change, there is need to
  periodically re-calculate the routing tables.
• Reasons for preferring distributed calculation
• congestion at central node
• survivability ( what if central node fails )
• Goal Distributed calculation of shortest paths.
  In fact a node needs to know only the next hop
  to each destination, not necessarily the entire
  path, nor the length of the path.
• Basis a node knows the information about its
  neighborhood and has to learn the rest.

2
Distributed Bellman-Ford

• Assumptions
• every node knows the weights of its adjacent
  outgoing links
• network remains connected (failures treated
  separately)
• the protocol calculates paths from all nodes to
  some node dest (separate calculations for each
  destination).
• Bellman-Ford algorithm equations
  Initialization
• After N-1 steps, the equations converge to
• where N(i) is the set of neighbors of i

3
Synchronous Distributed Bellman-Ford

• At step h , every node i calculates
  from received from neighbors and
  sends to all neighbors.
• Then h is incremented.
• Drawbacks
• needs synchronization between nodes - hard to do.
• needs mechanism for early end of the protocol, if
  some link weight changes during its operation
• restart needs reset ( inefficient use of previous
  information)

4
Asynchronous Distributed Bellman-Ford (distance
vector Routing Protocol )

• Protocol
• every node i performs
  from time to time
• from time to time, every node i sends calculated
  to all its neighbors.
• Assumptions
• nodes do not postpone forever update of Di
• nodes do not postpone forever sending update to
  neighbors
• on each link there is a DLC that ensures that
  messages sent on each link arrive and preserve
  order
• Variables at node i
• Di estimate of distance from i to dest
• Di(j) estimate of distance from neighbor j to
  dest as received in the last message from j
• Algorithm for node i
• whenever there is a change in some wij or in Di
  (j), perform and
  update neighbors by sending them Di select the
  neighbor that provides the minimum as your
  next-hop preferred neighbor towards dest .

5
Asynchronous Distributed Bellman-Ford (continued)

• Theorem (not easy to prove)
• For any given initial condition, if changes in
  wij cease, the protocol converges in finite
  time to a state where all Di reflect the
  shortest distances to dest.
• Note the protocol should be performed separately
  for each dest . Update messages can contain
  estimated distances for several or all
  destinations.

6
Example of update step
Table at node J
7
stability

• unbounded number of steps
• the number should be redefined as a
  bounded large number
• to overcome the problem
• run in parallel a protocol with weights1 and
  stop when variables reach N

8
Link State Routing Protocol

• Protocol
• every node i broadcasts from time to time, like
  in PI sequence number, its identity, the
  identity of its neighbors, the weights of its
  adjacent links
• every node i uses the received information to
  maintain an updated picture of the network
  topology
• every node i performs a Dijkstra algorithm to
  find shortest path to each node .