CS 457Link-State and Distance

Vector Routing Fall 2012

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0111

value in arrivingpacket’s header

routing algorithm

local forwarding tableheader value output link

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Routing vs. Forwarding

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Graph: G = (N,E)

N = set of routers = { u, v, w, x, y, z }

E = set of links ={ (u,v), (u,x), (v,x), (v,w), (x,w), (x,y), (w,y), (w,z), (y,z) }

Network Graph Abstraction

Remark: Graph abstraction is useful in other network contexts

Example: P2P, where N is set of peers and E is set of TCP connections

Network Graphs: Link Costs

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5 • c(x,x’) = cost of link (x,x’)

- e.g., c(w,z) = 5

• cost could always be 1, or inversely related to bandwidth,or inversely related to Congestion, or …

Cost of path (x1, x2, x3,…, xp) = c(x1,x2) + c(x2,x3) + … + c(xp-1,xp)

Routing question: What’s the least-cost path between u and z ?

A routing algorithm: an algorithm that finds least-cost path

or shortest path

Routing Algorithm Classification

Global or decentralized?Global:• all routers have complete

topology, link cost info• “link state” algorithms

Decentralized: • router knows physically-

connected neighbors, link costs to neighbors

• iterative process of computation, exchange of info with neighbors

• “distance vector” algorithms

Static or dynamic?Static: • Manual configuration• When routes change very

slowly over time after human intervention

Dynamic: • When routes may change

quickly– periodic update– in response to link cost

changes– In response to link

failures

Link-State Routing

Dijkstra’s algorithm• net topology, link costs

known to all nodes– accomplished via “link

state broadcast” – all nodes have same info

• computes least cost paths from one node (‘source”) to all other nodes– gives forwarding table for

that node• iterative: after k iterations,

know least cost path to k destinations

Notation:• c(x,y): link cost from node x

to y; = ∞ if not direct neighbors

• D(v): current value of cost of path from source to dest. v

• p(v): predecessor node along path from source to v

• N': set of nodes whose least cost path is definitively known

Dijsktra’s Shortest Path Algorithm

1 Initialization: 2 N' = {u} 3 for all nodes v 4 if v adjacent to u 5 then D(v) = c(u,v) 6 else D(v) = ∞ 7 8 Loop 9 find w not in N' such that D(w) is a minimum 10 add w to N' 11 update D(v) for all v adjacent to w and not in N' : 12 D(v) = min( D(v), D(w) + c(w,v) ) 13 /* new cost to v is either old cost to v or known 14 shortest path cost to w plus cost from w to v */ 15 until all nodes in N'

Notation:• c(x,y): link cost from node x to y; =

∞ if not direct neighbors

• D(v): current value of cost of path from source to dest. v

• p(v): predecessor node along path from source to v

• N': set of nodes whose least cost path is definitively known

Dijkstra’s Algorithm: Example

Step012345

N'u

D(v),p(v)2,u

D(w),p(w)5,u

D(x),p(x)1,u

D(y),p(y)∞

D(z),p(z)∞

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u,x 2,u 4,x 1,u 2,x ∞u,x,y 2,u 3,y 1,u 2,x 4,y

u,x,y,v 2,u 3,y 1,u 2,x 4,yu,x,y,v,w 2,u 3,y 1,u 2,x 4,y

u,x,y,v,w,z 2,u 3,y 1,u 2,x 4,y

V W X Y Z

Dijkstra’s Algorithm Limitations

Algorithm complexity: n nodes• each iteration: need to check all nodes, w, not in N• n(n+1)/2 comparisons: O(n2)• more efficient implementations possible: O(mlogn)

Oscillations possible when link costs change:• e.g., link cost = amount of carried traffic

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e01+e1

initially… recompute

routing… recompute … recompute

Distance-Vector Routing

Distance Vector Algorithm (1)

The Bellman-Ford EquationDefine:

dx(y) := cost of least-cost path from x to y

Then:

dx(y) = min {c(x,v) + dv(y) }

where min is taken over all V, where V is a neighbor of x

Bellman-Ford Example (2)

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5Note dv(z) = 5, dx(z) = 3, dw(z) = 3

du(z) = min { c(u,v) + dv(z), c(u,x) + dx(z), c(u,w) + dw(z) } = min {2 + 5, 1 + 3, 5 + 3} = 4

Node that achieves minimum is nexthop in shortest path ➜ forwarding table

B-F equation says:

Distance Vector Algorithm (3)Definitions• Dx(y) = estimate of least cost from x to y• Distance vector: Dx = [Dx(y): y є N ]

Local information• Node x knows cost to each neighbor v: c(x,v)• Node x maintains Dx = [Dx(y): y є N ]• Node x also maintains distance vectors

received from its neighbors– Dv = [Dv(y): y є N ] for each neighbor v

Distance Vector Algorithm (4)Basic idea: • Each node periodically sends its own

distance vector estimate to its neighbors• When a node x receives new DV estimate

from a neighbor, it updates its own DV using B-F equation:

Dx(y) ← minv{c(x,v) + Dv(y)} for each node y ∊ N

• Eventually, the estimate Dx(y) converges to the actual least cost dx(y)

Distance Vector Algorithm (5)

Iterative, asynchronous: each local iteration caused by:

• local link cost change • DV update message from

neighbor

Distributed:• each node notifies

neighbors, but only when its DV changes– neighbors then notify their

neighbors if necessary

wait for (change in local link cost or msg from neighbor)

recompute estimates

if DV to any dest has

changed, notify neighbors

Each node:

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∞ ∞ ∞∞ ∞ ∞

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cost to

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from

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from

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∞ ∞ ∞

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time

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node x table

node y table

node z table

Dx(y) = min{c(x,y) + Dy(y), c(x,z) + Dz(y)} = min{2+0, 7+1} = 2 Dx(z) = min{c(x,y) +

Dy(z), c(x,z) + Dz(z)} = min{2+1, 7+0} = 3

32

x y z

xyz

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∞ ∞ ∞∞ ∞ ∞

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from

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time

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node x table

node y table

node z table

Dx(y) = min {c(x,y)+Dy(y), c(x,z)+Dz(y)} = min {2+0, 7+1} = 2

Dx(z) = min {c(x,y)+Dy(z), c(x,z)+Dz(z)} = min {2+1, 7+0} = 3

Example 2Initial Distances to your neighbors

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Info atnode

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Distance to node

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Distance to node

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Distance to node

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Distance to node

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Distance to node

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Distance to node

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Distance to node

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Final Distances

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Distance to node

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Final Distances After Link Failure

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View From a Node

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Next hop

E’s routing table

The Bouncing Effect

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dest costA

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C Sends Routes to B

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B Updates Distance to A

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C Sends Routes to B

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dest cost

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dest cost

How Are These Loops Caused?

• Observation 1:– B’s metric increases

• Observation 2:– C picks B as next hop to A– But, the implicit path from C to A includes

itself!

Solution 1: Holddowns

• If metric increases, delay propagating information– in our example, B delays advertising route– C eventually thinks B’s route is gone, picks

its own route– B then selects C as next hop

• Adversely affects convergence

Other “Solutions”

• Split horizon– B does not advertise route to C

• Poisoned reverse– B advertises route to C with infinite

distance

Works for two node loops– does not work for loops with more nodes

Example Where Split Horizon Fails

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• When link breaks, C marks D as unreachable and reports that to A and B.

• Suppose A learns it first. A now thinks best path to D is through B. A reports D unreachable to B and a route of cost=3 to C.

• C thinks D is reachable through A at cost 4 and reports that to B.

• B reports a cost 5 to A who reports new cost to C.

• etc...

Avoiding the Bouncing Effect

Select loop-free paths• One way of doing this:

– each route advertisement carries entire path– if a router sees itself in path, it rejects the route

BGP does it this way

Space proportional to diameter

Cheng, Riley et al

Distance Vector in Practice

• RIP and RIP2– uses split-horizon/poison reverse

• BGP/IDRP– propagates entire path– path also used for effecting policies

LS vs. DV Algorithms

The two algorithms take complimentary approaches:

• Link State: Tell everyone what you know about your neighbors

• Distance vector: Tell your neighbors what you know about everyone

Comparison of LS and DV Algorithms

Message complexity• LS: with n nodes, E links,

O(nE) msgs sent • DV: exchange between

neighbors only

Speed of Convergence• LS: O(n2) algorithm requires

O(nE) msgs– may have oscillations

• DV: convergence time varies– may have routing loops– count-to-infinity problem

Robustness: what happens if router malfunctions?

LS: – node can advertise

incorrect link cost– each node computes only

its own table

DV:– DV node can advertise

incorrect path cost– each node’s table used by

others • error propagates thru

network