1. Distributed Asynchronous Bellman-Ford Algorithm

2. Topics Shortest-Path Algorithms:
Bellman-Ford
Dijkstra
Floyd-Warshall
Distributed Asynchronous Bellman-Ford Algorithm

3. Shortest-Path Routing
Shortest path routing: packets are delivered from source to destination along the shortest path from S to D
Shortest path from every node i to destination 1 can be found using the Bellman-Ford algorithm
Shortest paths are recomputed periodically, as link lengths might change over time due to:
varying traffic conditions
link failures and repairs
Shortest-path algorithm can be implemented as:
Centralized: all computations are performed by a single network entity and the shortest-path updates are communicated to all nodes
Decentralized or Distributed: each node performs part of the computation task in order to determine its own distance from node 1 and its corresponding outgoing link

4. Computation of Shortest Paths
Assumptions:
All cycles have positive length
Strongly connected network: if (i, j) is a link, then (j, i) is also a link
Problem: find the shortest distance Di from each node i to destination 1
These shortest distances are the unique solution of Bellman’s equation where N(i) is the set of neighbors of i – nodes j for which link (i, j) exists
Bellman-Ford algorithm:
We established convergence to the correct shortest distances for initial conditions:

5. Distributed Computation of Shortest Paths
Bellman-Ford iteration can be executed at node i in parallel with every other node
Distributed computation of shortest paths. Node i
Knows the length dij of each outgoing link (i, j)
Receives the estimates Djh from each neighbor jN(i)
Updates its shortest distance estimate according to the Bellman-Ford Iteration
Communicates the new estimate Dih+1 to all nodes k, for which iN(k)
Each node computes its own shortest distance based only on local information:
lengths of its outgoing links
shortest distance estimates for each of its neighbors

6. Distributed vs. Centralized Network Algorithms
Reduced computational complexity
Local vs. global information
Scalability as the “size” of the network increases: addition of a node does not increase the computational load for the other nodes
Survivability: if a node fails and no longer performs the algorithm, the computation is curried by the remaining nodes and the network remains operational
Convergence: does the distributed algorithm converge to the correct solution? How fast?
Efficiency: distributed versions of “optimal” network algorithms – flow control, routing – lead in suboptimal solutions

7. Distributed Algorithms: Synchronous vs. Asynchronous
Distributed Bellman-Ford algorithm:
Synchronous Implementation: the nodes update their distance estimates in an orderly, predetermined manner. For instance:
Each iteration h is executed simultaneously by all nodes
Node i updates Dih using shortest distance estimates Djh-1 from the previous iteration
Iteration h+1 starts only after every node i has executed iteration h and has updated its shortest distance estimate Dih
Asynchronous Implementation: There is no predetermined manner according to which the nodes update their distance estimates. A node does not necessarily have the latest updates for the estimates for all its neighbors. Some nodes might not update their estimates immediately after they receive updates from their neighbors.

8. Distributed Algorithms: Synchronous vs. Asynchronous
Synchronous Bellman-Ford algorithm: with initial conditions:
Terminates after at most N iterations
with, each node i knowing both its shortest distance Di and its outgoing link on the shortest path to node 1
“Synchronization” mechanisms:
Getting all nodes to agree to start the algorithm with the appropriate initial conditions
Abort and restart the algorithm – e.g., when the length of a link changes
Complexity indicates that the centralized algorithm is preferable to the synchronous distributed version
Asynchronous Implementation:
No need to synchronize the nodes
Does not require specific initial conditions

9. Asynchronous Bellman-Ford Algorithm
Operates indefinitely, by executing from time to time at each i≠1 iteration:
Dj: last estimate received from neighbor
dij: latest known length of outgoing link (i, j)
Node i transmits from time to time its latest estimate Di to all its neighbors
Neither the iterations, nor the transmission of updates need be synchronized
No assumptions are made on the initial values of Dj, jN(i) available at node i
Under a set of rather general assumptions the asynchronous Bellman-Ford algorithm finds the correct shortest distance of every node within finite time
If a number of changes in the link lengths occur up to time t0 and no other changes occur subsequently, the algorithm finds the shortest distances within finite time from t0. It is not necessary to re-initialize the algorithm after each link length change

10. Asynchronous Bellman-Ford Algorithm

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13. Asynchronous Bellman-Ford Algorithmi D

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