Internet Routing Algorithm

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Department of Computer Engineering

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Internet Routing Algorithms


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Routing algorithms

Shortest Path

Path that has the least cost

Widest Path

Path that has the most widthFor different purpose, different approaches available

Shortest path algorithms are applicable to IP networks

Widest path algorithms are useful for telephone networks

Introduction


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Communication network is made up of nodes and links

In IP network , a node is called a router

A link in IP node is called an IP trunk

Traffic flows from start (source) to end node (destination)

Terminologies Used


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Route setup configured manually is called static route

Route decided on the fly is called dynamic route

Shortest path may be distance

Distance can be measured in terms of time

In networks normally shortest path is either no. of hops or time

Terminologies Used


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Sample Six Node Network

Cost of each link given

By eyeballing we can see that the shortest path from node-1 to

node-6 is 1-4-3-6 with a cost of 3 units

How to find this from an algorithmic standpoint?


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Bellman Ford Algorithm

Centralized Approach

Distance Vector Approach

Dijkstras Algorithm

Centralized Approach

Distance Vector Approach

Candidate Path Caching Algorithm

K-path Algorithm

Shortest Path Algorithms


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Source node needs to know the cost of the Shortest Path to all the nodesimmediately prior to the destination node

We use two generic node i and j , in a network of N nodes k is an intermediate node Track two sets of information

Bellman-Ford equation

dij = finite value if directlyconnected

dij= if not

Bellman Ford Algorithm : Centralized Approach


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Bellman Ford Algorithm :Centralized Approach

Centralized Approach: uses number of hops in algorithm


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Q. Find the shortest path between 1 to 6 for the given network.

What to remember1. Intermediate nodes k are directly connected to destination node.

In our case destination node is j and intermediate nodes k are3,4 and 5. Therefore values between k and destination nodes arefinite and knowndkj = d36 and d46 and d56 d36 = 1 d46=15 d56 = 1

2. What we have to calculate is the paths from source node to allintermediate nodes ie Dik = D13 and D14 and D15

3. According to algorithms , we have to calculate

D16(3) = D13

(2) + d36D16

(2) = D14(1) + d46

D16(3) = D15

(2) + d56

Let us calculate D terms

Bellman Ford Algorithm : Manual Calculations


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Initially D16(1) =

Let us calculate D13

D13(2) = D12

(1) + d23 = 1+2 = 3 path 1-2-3or D13

(2) = D14(1) + d43 = 1+1 = 2 path 1-4-3 (min)

or D13(3) = D14

(2) + d43 = 2+1 = 3 path 1-2-4-3

or D13(4)

= D15(3)

+ d53 = 4+1 = 5 path 1-2-4-5-3


D14(1) = 1 min Dij in terms of hops and cost.

or D14(2) = D12

(1) + d24 = 1+1 = 2


D15(2) = D14

(1) + d45 = 1+2 = 3min Dij in terms of hops and cost.



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Therefore shortest path is 1-4-3-6 = 3

D16 =


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Bellman Ford Algorithm : Distance Vector Approach (Time)

Presents a complementary viewof centralize Bellman-Ford

In centralized approach intermediate node k was connected to

destination nodeIn distance vector approach intermediate node k is directlyconnected to source node.

Source node needs to know the cost of the all the nodesimmediately after the source ( complementary view)

Source node is directly connected to all intermediate nodes ki.


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Algorithm uses a concept of time domain dependency.

At time t, node ican only know what it has received from kregarding

distance to nodej, i.e., Dikj(t)

Uses hop based cost in terms of discrete time windows ( traffic/load)

t is now related with hop or hop is replaced by t ( h -> t)

f C


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The computation of the shortest path by each node is done based

on link cost ( traffic / load) information received at given time t.This cost may vary time to time and there may be different paths at

different time.

In real scenario nodes (intermediate) are always distributing their

cost to all directly connected nodes in order to find out SP.


D t t f C t E i i


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D t t f C t E i i


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Dijkstras Algorithm : Centralized Approach

Computing path from a source node view

It works on the notion of neighbouring node set

Need to have the link costs of all links

Computes shortest path to all the destinations

Does not depends upon all neighbouring intermediates k

It always calculates optimal intermediate node and then moves next

D t t f C t E i i


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Divides the list of N nodes into two list.

It first starts with permanent list S , which represents nodes already

considered

It prepares tentative list Sfor nodes which are not considered

As algorithm progresses, list S expands with new nodes while list S

shrinks when nodes are added in S

List S is expanded by considering a neighbouring node with least

cost

Algorithm stops when list Sbecomes empty

D t t f C t E i i


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Dijkstras Algorithm : Centralized Approach : Illustration

Initially S= {1} and S = {2, 3, 4, 5, 6}

Node 1 is directly connected to 2 and 4. Both path have same cost.There is a tie . So select anyone. Let us select node 2. Now we get

S= {1,2} and S = {3, 4, 5, 6}



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Node 2 is directly connected to 3 and 4. Now calculate the cost forboth from node 1 and select minimum one. Now we get

There is no improvement in the path of 1 to 4, so select (1,4). Now

S= {1,2,4} and S = {3, 5, 6}

Keeps the minimum




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D13 = min { D13 , D14 + d43 } = min { 3,1+1} = 2

D15 = min { D15 , D14 + d45 } = min {,1+2} = 3

D16 = min { D16 , D14 + d46 } = min {,1+15} = 16

S= {1,2,4,3} and S = { 5, 6}

Keeps the minimum

Similarly, it includes nodes 5 and finally node 6.

We get shortest path : 1-4-3-6 = 3




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Dijkstras Algorithm : Iterative Steps



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Very similar to centralized algorithm:

The computation of the shortest path by each node is done

separately based on link cost information received

Link cost of a link received by one node could be different

from another node time to time

Link cost seen by node iat time tregarding link k-m:

dikm(t)

Dijkstras Algorithm : Distributed Approach



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Dijkstras Algorithm : Distributed Approach



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Shortest Path with Candidate Path caching

In certain situations, possible or candidate paths to a

destination are known ahead of time In this case, only the updated link cost are needed

Then, the computation is fairly simple

Such situations arise if a router cashes a path, or more

commonly, in off-line traffic engineering computation

Consider the four paths from 1 to 6:



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Shortest Path Computation with Candidate Path caching



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Widest path routing has been around since the early days of

dynamic call routing in the telephone network

The notion here is different from shortest path routing in that

the path with the maximum width is to be chosen

Since a path is made of links, the link with the smallest

width dictates the width of the path Consider width (bandwidth) of link l-m as seen by node iat

time t: bikm(t)

Main implication: path cost is non-additive (concave)

Widest Path Computation with Candidate Path caching



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From 1 to 5:

Widest path is 1-4-5

Widest Path : Illustration



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Widest Path Computation with Candidate Path caching



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Widest Path : Dijkstras Based



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S = {1}, S = {2, 3, 4, 5}

B12 = 30, B14 = 20, B13 = B15 = 0

MaxjSB1j= 30 is attained atj= 2

Now, S = {1, 2}, S = {3, 4, 5}

Next compare: B13 = max {B13, min{B12,b23}} = 10 // use 1-2-3

B14 = max {B14, miin{B12,b24}} = 20 // stay on 1-2

B15 = max{B15,min{B12,b25}} =max {0, min{30,0}} = 0

MaxjSB1j= max{B13,B14,B15} = 20, attained for j=4.

Now, S = {1, 2, 4}, S = {3, 5}

And so on [ Bij : add node according to j in the list ]

Widest Path : Dijkstras Based : Illustration (I)



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Next compare: node 3 and node 5

B13 = max {B13, min{B14,b43}} = 15 // use 1-4-3

B15 = max {B15, miin{B14,b45}} = 20 // stay on 1-4-5

Therefore widest path is 1-4-5 = 20

Widest Path : Dijkstras Based : Illustration



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Widest Path : Dijkstras Based : Illustration (II)



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Bellman-Ford-based Widest path routing

Similar to additive version:



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When we need to generate the shortest path, the next shortest

path, up to the K-th shortest path

Naive idea:

Run the shortest path algorithm

Delete each link one at a time, and re-run shortest path

algorithm on the reduced graph

Pick the best from this set of paths

and so on

K Shortest Paths Algorithm



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K Shortest Paths Algorithm



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E-book ofNetwork Routing

Algorithms, Protocols, and Architectures

By

Deepankar Medhi

Karthikeyan Ramasamy

And

TCP/IP by Forozoun

Resources



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Thank You

Documents

Internet Routing Algorithm