10.- Graph Algorithms

Centro de Investigación y Estudios Avanzados del Instituto Politécnico Nacional

• Introduction• Routing Algorithms

• Computation of Shortest Path• Distance Vector Routing• Link-State Routing• Interval Routing

Sistemas Distribuidos

Jorge Antonio Perez Espinoza

Graph

Graph G:G = (V, E)V = Set of nodes 0….N-1.E = Set of edges representing links: (A,B),(A,C)

• Each edge w(i, j) has a weight.

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Graphs on Distributed System

• The topology of a distributed system is represented by a graph where the nodes represent processes, and the links represent communication channels.

• Static vs. Dynamic Topology.

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ROUTING ALGORITHMS • Routing is a fundamental problem in networks.

• Discover and maintain an acyclic path from the source of a message to its destination.

• The routing table is updated when the topology changes.

• A path can have many attributes: hops, delay.

COMPUTATION OF SHORTEST PATHGiven a graph G:G = (V, E)V = Set of nodes 0….N-1.E = Set of edges representing links.• Each edge w(i, j) has a weight.

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Chandy and Misra Algorithm• A refinement of the Bellman-Ford Algorithm.

• Designed to work with a single initiator node 0.

• D(i): Distance to node 0

• Each node knows the weights of all edges incident on it.

Initially :D(0) = 0∀i : i > 0 : D(i) = ∞.

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For each node:• 1. D, the current shortest distance of node 0 to itself. Initially

D(0) = 0, D(i: i > 0) = ∞

• 2. A node called parent: Initially parent (j) = j

• 3. A variable deficit, representing the number of unacknowledged messages. Initially deficit=0.

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program shortest path {for process i >0};

define D, S : distance;{S denotes the shortest distance received through a message}parent : processdeficit : integer;N: set of neighbors of process i;

initially D= ∞, parent = i, deficit = 0

{for process 0}send (w(0,i), 0) to each neighbor i;deficit := |N(0)|;do ack -> deficit := deficit – 1 od;{deficit = 0 signals termination}

{for process i > 0}do message = (S ,k) ^ S < D if parent ≠ k or i -> send ack to parent fi;

parent := k; D := S; send (D + w(i,j), i) to each neighbor j ≠ parent;

deficit := deficit + |N(i)| -1message (S,k) ^ S ≥ D -> send ack to senderack -> deficit := deficit – 1deficit = 0 ^ parent ≠ i -> send ack to the parentod

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Example

Lemma :• When the algorithm terminates, let k = parent (i). If D(k) is the

distance of the shortest path from k to 0, then D(i) = D(k) + w(k, i) is the distance of the shortest path from i to 0 and the shortest path includes k.

• Proof: Suppose this is false :The shortest path from 0 to i is via j , where j ≠ k.

Message D(i)>D(j)+w(j,i).

Message D(i)>D(k)+w(k,i).

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DISTANCE VECTOR ROUTING

Shortest path routing, but handles topology changes.Routing Table

Destination NextHop DistanceD[i, j] = 0 when i = j,= 1 when j is a neighbor of i, and

= ∞ when i = j and j is not a neighbor of i

∀k ≠ i :D[i,k]=minj(w[i,j]+D[j,k])

Each node j periodically broadcasts its distance vector to its immediate neighbors

DISTANCE VECTOR ROUTING

RT- U

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Following this, the distance vectorsare corrected, and routing table is eventually recomputed.

Count to Infinity Problem

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1.- Node A. -> B(A,A,1)2.- Node B. -> C(A,B,2)3.- Node C. -> B(A,C,3)4.- Node B. -> C(A,B,4)5.- Node C. -> B(A,C,5)....

Split Horizont Method

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LINK-STATE ROUTING• This is an alternative method of shortest path routing.• Converges faster.• Each node i periodically broadcasts the weights of all edges (i,j)

incident on it (this is the link state) to all its neighbors

Link State Protocol

INTERVAL ROUTING• Consider a connected network of N nodes.• The conventional routing table used to route a message from

one node to another has N − 1 entries, one for each destination node.

EntryDestination # Port

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Can we do something to reduce the growth ofthe routing tables?

EntryDestination # Port

??????

Interval routing• Santoro and Khatib first proposed interval routing for tree

topologies only.• Each node has two ports:• port 0 is connected to the node with a higher id.• port 1 is connected with the node of lower id

Interval Routing• For a set of N nodes 0 ….N − 1, define the interval [p, q) between

a pair of p and q as follows:• if p < q then [p,q) = p, p + 1, p + 2 , . . ., q − 2, q − 1• if p ≥ q then [p, q) = p, p + 1, p + 2, . . .,N − 1, 0, 1, . . ., q − 2, q − 1

As an example, if N = 8, then [5, 5) = 5, 6, 7, 0, 1, 2, 3, 4.

Routing for a tree Network

1. Label the root as node 0.2. Do a preorder traversal of the tree, and label the successive nodes in ascending order starting from 1.3. For each node, label the port towards a child by the node number of the child. Then labelthe port towards the parent by L(i) + T(i) + 1 mod N, where

• L(i) is the label of the node i• T(i) is the number of nodes in the subtree under node i (excluding i)

Subtree under i: L(i),L(i)+T(i) + 1 mod N)Comp: [L(i)+T(i)+1 mod N,L(i))

Adaptation to changes in the topology .• Every time a new node is added to a network, or an existing

node is removed from the network, in general all node labels and port labels have to be recomputed.

Prefix Routing

if X = Y → Deliver the message locally X = Y → Find the port with the longest prefix of X as its label; Forward the message towards that portfi