# Redundant Trees

Document Sample

```					Redundant Trees for Preplanned Protection
&
Multipath Routing

By: Umang Patel
CSC-772 Presentation
Redundant Trees (Blue & Red)
source
• Suitable for preplanned protection
• Dual benefits for multicast networks: Routing & Protection
• Topological requirement are relaxed.

Figure -1    Figure -2   Figure -3

• Requires any 2-edge-connected topology for protection
against single-link failure & any 2-vertex-connected
topology for protection against single-vertex failure.
Problem Statement
• To design two directed trees (Blue & Red) in
such a fashion that the elimination of any
vertex (edge) in the graph (other than the
source) leaves each destination vertex
connected to the source by at least one of the
directed trees for any source, and destination
vertices in any vertex (edge) redundant graph.
Algorithm-1
(For 2-vertex connected topology)
1. Step-1

Let TB and TR contains node{s}.
Assign N(sM)=MAX, N(sM’)=0
Algorithm-1
2. Step-2

- Find a cycle [s, v1,v2 ,..., vk ,s]with k>=2.

- Put s->v1->v2->…->vk on the blue chain

- Put s->vk->vk-1->…->v1 on the red chain

- Add blue Chain to TB,red chain to TR.

- Assign. Numbers
N(SM) > N(v1)>…>N(vk)>N(SM’)

N(SM)>N(1)>N(6)>N(8)>N(5)>N(4)>N(3)
>N(SM’)
Algorithm-1
3. Step-3
- If TB contains all nodes in G, stop.
4. Step-4
- Find path [x, v1,v2 ,..., vk ,y]
s.t. x,y distinct nodes on blue chain.
If x=s, N(x)=N(sM), if y=s N(y)=N(sM’).
N(x)>N(y) , k>=1, vi not on blue chain.
- Put x->v1->v2->…->vk on the blue chain
- Put y->vk->vk-1->…->v1 on the red chain
- Add blue Chain to TB,red chain to TR.
- Assign. Numbers
N(x) > N(v1)>…>N(vk)>N(y)>N(x’)
N(x’) is max. of assigned values lower than N(x)
Algorithm-1
Interation-5
Interation-4

,
Algorithm-2
(For 2-edge connected topology)

1. Step-4 modified condition

Find path [x, v1,v2 ,..., vk ,y]
s.t. x,y distinct nodes on blue chain.
If x=s, N(x)=N(sM), if y=s N(y)=N(sM’).
N(x)>=N(y), k>=1, vi not on blue chain.
Algorithm-3
(Heuristic for low average delay on Blue tree)
1. Step-2 modified condition Running time: O(n2(m+nlogn))
-            For n nodes, 3n and nlogn link topologies.
Find cycle such that delay of -             For each topology, 100 2-connected graphs
this Cycle minus the last edge -            Bandwidth, cost, delay of links are random
integers uniformly distributed in range
is minimum among all such                   [1,10]
cycles.                                 -   Results are average over 100 runs

2. Step-4 modified condition

Find Path [x, v1,v2 ,..., vk ,y] that
has minimum delay from (s,x)
plus (x,vk) among all such
paths.
Algorithm-4
(Heuristic for reducing total cost)
1. Scaled cost                        Running time: O(n2(m+n))
For cycle [x, v1,v2 ,..., vk ,x]      -   For n nodes, 3n and nlogn link topologies.
scaled cost is                        -   For each topology, 100 2-connected graphs
((c(x,v1)+c(v1,v2)+…+c(vk,x))/k       -   Bandwidth, cost, delay of links are random
integers uniformly distributed in range
For path [x, v1,v2 ,..., vk ,y]           [1,10]
-   Results are average over 100 runs
scaled cost is
((c(x,v1)+c(v1,v2)+…+c(vk,y))/k

2. Steps 2 and 4 of the
Algorithm 1 are modified to find
low scaled cost cycle and path in
each iteration.
Algorithm-5
(Optimal algo. for maximizing bottleneck bandwidth)
1. Use bisection method on        -   For n nodes, 3n and nlogn link
topologies.
bandwidth values to find
-   For each topology, 100 2-connected
largest B such that G(B) is        graphs
2-connected.                   -   Bandwidth, cost, delay of links are
random integers uniformly distributed
in range [1,10]
2. Apply Algo. 1 or Algo. 2 for
-   Results are average over 100 runs
constructing Redundant
Trees.
Algorithm-6
(Algorithm for enhancing QoP)
1. QoP of pair of single-link    Running time: O(n2m)
-   For n nodes, 3n and nlogn link
recovery trees equals total       topologies.
number of cycles and          -   For each topology, 100 2-connected
paths used in construction        graphs
process.                      -   Bandwidth, cost, delay of links are
random integers uniformly distributed
2. Step 2 and step 4 of              in range [1,10]
algorithm 2 is modified to    -   Results are average over 100 runs
find minimum hop cycle and
paths in each iteration.
MPLS Domain protection using Redundant Trees

1. Assume MPLS domain represented by graph G(N,L). N is set
of nodes and L is set of links between nodes. G is two-edge
connected and therefore can protect against single link
failure.

2. Protection paths for all working path terminating in an
egress router are calculated simultaneously.
MPLS Domain protection using Redundant Trees

1. Initialization:
- Find spanning tree of graph G rooted in egress router A.
- Let P be set of nodes for which the protection paths have
been established. Initially P={e}
MPLS Domain protection using Redundant Trees
2. Repeat until all nodes are protected (P=N)
- Select one of branches of spanning tree attached to
egress node and mark all nodes except egress node.
- scan all marked nodes to find node i that has link to
unmarked node j.
- consider ring path consisting of links of spanning tree
spanning tree between j and e.
MPLS Domain protection using Redundant Trees
-   Place two protection paths along the ring: one in clock-
wise , the other in counterclockwise direction. The paths
originate in two nodes of the ring that are adjacent to
egress router and follow the ring all the way to egress
node. Merge the created protection paths with the
protection paths established in previous iterations. All
nodes on the ring are now connected to both protection
MPLS Domain protection using Redundant Trees
-   In the subsequent iteration of the algorithm consider
new graph constructed by treating all nodes in P as a
single node that will act as the egress node and removing
all links that connect two protected nodes.
MPLS Domain protection using Redundant Trees
Redundant Multicast Trees for optical networks

Multicast Session from S to {6,8,14,12}   Edge-disjoint paths from S to node 6
Redundant Multicast Trees for optical networks

Edge-disjoint paths from S to nodes 6,8   Edge-disjoint paths from S to nodes 6,8.11
Redundant Multicast Trees for optical networks
Edge-disjoint paths from S to nodes 6,8,11,12
Redundant Multicast Trees for optical networks
Protecting multicast sessions in WDM Mesh
Networks using Redundant Trees

Multicast Session (S->1,2) Protected
- Working Tree: S->1(λ1), 1->2(λ1)
- Protection Tree: S->2(λ1), 2->1(λ1)

Multicast Session (S->1,2,3), Not-protected
- Working Tree: S->1(λ2), 1->2(λ2), 1->3(λ1)
ILP for finding Light Trees
Redundant Trees For Multipath Routing
1. Every Node in the network has two preferred neighbors to
drain: red and blue.
2. Source marks packet with color and intermediate node
forwards packet according to color of packet.
3. Path from any source to drain on blue and red trees are
4. Network can be viewed as two trees rooted at drain and the
paths on these trees are directed towards drain.
Redundant Trees For Multipath Routing
1. Every Node in the network has two preferred neighbors to
drain: red and blue.
2. Source marks packet with color and intermediate node
forwards packet according to color of packet.
3. Path from any source to drain on blue and red trees are
4. Network can be viewed as two trees rooted at drain and the
paths on these trees are directed towards drain.
Distributed construction of colored trees
Analysis of Multipath Routing

1. System Model
- Source A and destination B
- n disjoint routes between them
- Each route can support m=1 connection
- Connection arrival is Poisson Process with rate λ
- When connection request arrives, no knowledge about
availability of routes
- One(or more) routes are selected randomly to attempt
reservation
- Overall period of reservation and connection duration
time is exponentially distributed with mean 1/μ
Single-Path Reservation

-   For each connection arrival, one route out of n is randomly
chosen to attempt bandwidth reservation on it.
bandwidth reservation.
-   System can be modeled as n+1 state birth-death process
with transition rates

-   Markov chain is depicted below
Greedy Multipath-Path Reservation

-   For each connection arrival, all n routes are chosen to
attempt bandwidth reservation on it.
bandwidth reservation on any route.
-   System can be modeled as n+1 state birth-death process
with transition rates λs= λ and μs= μ
-   Markov chain is depicted below
Greedy Multipath-Path Reservation

Flow-balance
equation

probability of
being in state s

Summation of
all probabilities

probability of
being in state 0
Probability of
successful
connection
Thank You !

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 5 posted: 7/11/2012 language: English pages: 35
How are you planning on using Docstoc?