Trees & Routing (1) BFS

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Trees & Routing (1) BFS

mercredi 11 février 2015

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Why (spanning) trees ?

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Spanning Tree

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Given a graph G=(V,E), a Spanning Tree of G is an acyclic subgraph, spanning all the nodes in V.

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Spanning Tree

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Given a graph G=(V,E), a Spanning Tree of G is an acyclic subgraph, spanning all the nodes in V.

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Breadth-first Spanning Tree (BFS)

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Given a graph G=(V,E) and a node r in V, a BFS tree of G rooted at r is a spanning tree of G such that for any node v the path leading from the root to v in the tree is of the minimum lenght possible.

r

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Breadth-first Spanning Tree (BFS)

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Given a graph G=(V,E) and a node r in V, a BFS tree of G rooted at r is a spanning tree of G such that for any node v the path leading from the root to v in the tree is of the minimum lenght possible.

r

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BFS Construction

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BFS - strategies

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«link state» (Dijkstra)

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grow on the border

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«distance vector» (Bellman-Ford)

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compute shortest paths

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Dijkstra BFS

r

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Dijkstra BFS

r root, r, broadcasts in

the curent tree, T,

«pulse»

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Dijkstra-like BFS

r

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Dijkstra-like BFS

r upon reception of

«pulse», leafs of the curent tree send

«layer» to neighbors (except to parents)

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Dijkstra-like BFS

r

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Dijkstra-like BFS

r upon reception of

the first «layer» a node choses the

sender as parent and sends back an «ack»

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Dijkstra-like BFS

r upon reception of

an «ack» a node register the sender

as its child

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Dijkstra-like BFS

r upon reception of

an «ack» a node register the sender

as its child

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Dijkstra-like BFS

r

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Dijkstra-like BFS

r root, r, broadcasts in

the curent tree, T,

«pulse»

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Dijkstra-like BFS

r

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Dijkstra-like BFS

r upon reception of

«pulse», leafs of the curent tree send

«layer» to neighbors (except parent)

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Dijkstra-like BFS

r upon reception of

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Dijkstra-like BFS

r upon reception of

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Dijkstra-like BFS

r upon reception of

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Dijkstra-like BFS

r upon reception of

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Dijkstra-like BFS ?

r

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Dijkstra-like BFS ?

r

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Dijkstra-like BFS - Complexity

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Time Complexity: O(Diam²(G))

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Message Complexity: O(nDiam(G)+|E|)

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Bellman-Ford-like BFS

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each node maintains :

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its shortes distance to the root d (initially d=infinity)

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its parent to the root on the shortest distance

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Bellman-Ford-like BFS

r d=0 parent=r

d=∞

parent=❉ d=∞

parent=❉

d=∞

parent=❉

d=∞

parent=❉ d=∞

parent=❉

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Bellman-Ford-like BFS

r root, r, broadcasts

d=0 to all its neighbors

d=0 parent=r

d=∞

parent=❉ d=∞

parent=❉

d=∞

parent=❉

d=∞

parent=❉ d=∞

parent=❉

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Bellman-Ford-like BFS

r d=0 parent=r

d=1 parent=r d=1

parent=r

d=1 parent=r

d=∞

parent=❉

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Bellman-Ford-like BFS

r d=0 parent=r

d=1 parent=r d=1

parent=r

d=1 parent=r

d=∞

parent=❉

each node

broadcasts its d to all its neighbors

except parent

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Bellman-Ford-like BFS

r d=0 parent=r

d=1 parent=r d=1

parent=r

d=1 parent=r

d=2 parent=n2

n1 n2

n3

n3 n4

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Bellman-Ford-like BFS

r d=0 parent=r

d=1 parent=r d=1

parent=r

d=1 parent=r

d=2 parent=n2

n1 n2

n3

n3 n4

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Bellman-Ford-like BFS - Complexity

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Time Complexity: O(Diam(G))

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Message Complexity: O(|E|)

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BFS - Complexity

Time Messages Dijkstra-like D² E+nD Bellman-Ford-

like D nE

Awebuch-Peleg Dlog³n E+nlog³n

Lower Bound D E

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Spanning Tree and Routing

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Routing

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«link state» (Dijkstra like)

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Open Shortest Path First (OSPF)

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«distance vector» (Bellman-Ford like)

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Border Gateway Protocol (BGP)

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Interior Gateway Routing Protocol (IGRP, CISCO property)