Chapter 4 distributed bellman ford routing
This presentation is the property of its rightful owner.
Sponsored Links
1 / 19

Chapter 4 Distributed Bellman-Ford Routing PowerPoint PPT Presentation


  • 281 Views
  • Uploaded on
  • Presentation posted in: General

Chapter 4 Distributed Bellman-Ford Routing. Professor Rick Han University of Colorado at Boulder [email protected] Announcements. Reminder: Programming assignment #1 is due Feb. 19 Homework #2 available on Web site, due Feb. 26 Hand back HW #1 next week

Download Presentation

Chapter 4 Distributed Bellman-Ford Routing

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Chapter 4Distributed Bellman-Ford Routing

Professor Rick Han

University of Colorado at Boulder

[email protected]


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]}

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]}

  • 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)

  • 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


    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


    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

    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 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


    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


  • Login