Compact Routing overview

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REPORT

1) Introduction2) Evolution of schemes over years3) Terminology/Algorithms Used4) Description of Schemes5) Other Topics Explored6) Conclusion7) Future Work

INTRODUCTION

Compact routing scheme aims to achieve the best possible tradeoff between

Routing table size at each node and the stretch i.e. ratio of distance between any

two nodes in given scheme to shortest distance between those two nodes in a

distributed network.

There are two types of routing being followed till date to attain shortest path

routing, but both of them have limitation in terms of scalability or header size foraccommodating larger networks.

The first way is to store next hop information of all the nodes in the routing table

at every node, in which case the routing table size would be of the order of O(n),

another approach is to have a source routing in which the header/label stores the

path to take for routing between the two nodes. In this case the label size

becomes of the order O(n) hence imposing limitation on network size.

Considering the large scale networks such as internet in the world today, the idea

of compact routing emerged. The idea is to minimize storage requirement, at the

cost of taking a slightly larger path, where the worst case stretch too is bounded.


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The first approach was the hierarchical routing, in which a tree was created from

given graph of nodes and then parent child association was employed to route.

The bounds attained were..........

In the paper Hierarchical Routing for larger networks- performance evaluationand optimization by Leonard Kleinrock and Farouk Kamoun. Aim is to reduce

routing table length by hierarchical partitioning of the network, hence

represented as a tree. The network is divided into some m level clusters. Kth level

cluster consists of all ( k-1)th level clusters recursively. The number of k-1 level

clusters constituting the kth level clusters is referred to as Degree of kth level

cluster.

Next philosophy was interval routing[7], where a depth first search enumeration

was done for the nodes of graph and the interval of nodes under the nodes was

stored, giving the range in which case the routing was to be in the said node,

otherwise the data was to be forwarded to the parent node. Till the message

reaches root which has information of all the nodes in its table. The bounds

attained by this scheme were .................

Then came landmark based routing by Cowen [1], which was based on the idea of

postal addressing system. In this scheme a set of highly connected nodes are

chosen as the landmark nodes, where any local address is identified in terms of a

well known landmark. A neighborhood of some fixed size is formed for all the

nodes. Every node has the information of the location of the neighboring nodes

and the edge to take for reaching all the landmarks nodes stored in its routing

table. Though at the landmark only the information of rest all landmarks is

stored, as any incoming packet has the edge information from landmark to the

destination node in the label itself. Every node is identified by a label L

(node_name, Landmark_name, edge_from_landmark_to_node). This works for

non weighted distributed networks and achieves the bound of RT size O(n2/3

log4/3


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n) and stretch bound of 3. This scheme uses extended dominating set and nodes

in maximum neighborhoods as candidates for landmark. It was the best bound

achieved till the time it was published.

Thorup Zwick [2] improves on the approach of Cowen [1] by using probabilistic

model for Landmark selection. It improves the bound of the landmark selection

and routing table to O(n1/2

logn) while stretch bound remains 3. This paper also

describes ways to reduce label size while routing in a tree. The minimum label size

bound achieved is (1+O(1))log2n bit. With the help of the labels of source and

destination nodes, routing can be done in a tree in constant time. For improving

LM selection , it uses a sample function recursively. In every iteration it passes a

node which exceeds the no of nodes than a size of 4*n/s and the sample returns s

no. of nodes from that set with each node assigned equal probability to be

selected as landmark .

Algorithm center(G,s)

A; WV;

While W do

{

AA sample(W,s);

C(w){vV|(w,v)< (A,v)} fore very w V

W{wV | C(w)>4n/s};

}

Return A;


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The first labeling scheme described is based on assigning weight to the nodes in

the tree as the number of descendents of the concerned node. Then segregating

the nodes as heavy or light weight nodes based on fraction of nodes under it. The

path from root to the node concerned in terms of light nodes traversed along the

path is stored. The routing information at any node consists of (v, fv, hv, Hv, Pv)

where v is the node, fv is the largest descendant of v, hv is the first heavy child of

v, Hv is the list of heavy nodes and Pv is the port numbers of corresponding heavy

nodes. Hv(0) stores the parent of the node and Pv(0) is the port of the parent of

the node.

Routing information stored at v =(v, fv,hv,Hv,Pv)= O(b) words Label of any node consists of (v, Lv)

where Lv is the port number of the light levels from root to the node v.

Routing Algorithm

Label(v)=(v , Lv) at node w

If w=v, done

else check if v (w , fw)

a) if not a descendentf/w to parent of w using Pv[0]

b) else if descendent

check if v (hw , fw) search Hw and get corresponding Pw else if light descendent search Lv[lw]

In the second approach of labeling a single heavy child is considered and rest of

the light child are assigned weights. All the port numbers are stored as one string


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and port numbers are encoded in varying lengths as per bits required to code

them .

If for vertex v, v is heavy child & v0, v1,.,vd-1 light children s.t.sv >= sv0>= sv1>=.>=svd-1 ; assign edge (v, vi ) for 0


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Peleg [3] tried to solve the question of how to identify distance between nodes

by the labels of the source and destination nodes. The scheme is based on Tree

separators and assigning the corresponding distance to each node in given branch

in the subtree. A separator splits the tree into branches having nearly n/2

members each. No separator is selected when a leaf node is obtained. Each node

is then identified by concatenation labels from all its separators. For n vertex

weighted trees with M bit edge weights, label size O(Mlogn+log2n) bit .

Label of a node p triples J(v)=(I(v0), dist(v,v0,T), i), Tree separator v0 labeledas (I(v0), 0, 0).If vertex v is internal to subtrees at level p-1 & becomesseparator at level p then

Label(v)= J1(v) . J2(v).Jp(v)

The distance between any two nodes is given by sum of distances of the label

with respect to a common separator in which the two nodes are not on same

branch.

Using the concept of Peleg and Thorup Zwick scheme of labeling the nodes in

O(1+logn), Brady& Cowen [4] came up with the scheme which is meant for power

law graphs(the degree distribution follows a power law) like internet. The paper

achieves stretch bound of 3 and RT size of O(n1/2

).. This scheme identifies a highly

connected node h and builds a core of diameter d around it, named as d-core.

while the remaining area is named as d-fringe. A minimum spanning tree is

formed for all nodes of the graph using h as the root, while separate trees are


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formed in the fringe for all connected components. Also the edges not used in the

tree formation which are removed to make graph acyclic in the fringe are taken

and minimum spanning tree is created using those edges. Now Peleg label is

associated with every such label.

For routing between source and destination all the trees are explored for

presence of these two nodes( Source and Destination). The tree with minimum

Peleg distance between the two nodes is chosen to route and the labeling scheme

by thorup zwick is used for routing.

Conclusion

Now the role of Thorup/Zwick and Brady/Cowen scheme to be deployed in

current world internet scenario is being explored. A new kind of protocol based

on philosophy of BGP protocol is envisaged which supports IPv4.

Other Topics studied

1) Wormhole Routing2) Voronoi Diagrams3) TZ Also a tree based routing scheme was developed by TZ in [5] to identify

the distance between any two nodes in a graph.


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References

1) Compact Routing with Minimum stretch. Lenore J. Cowen: J. of Algorithms,38:170-183, 2001

2) Compact Routing Schemes. Mikkel Thorup and Uri Zwick : In proceedings ofthe 13

thAnnual ACM Symposium on Parallel Algorithms and Architectures,

pages 1-10. ACM, July 2001

3) Proximity-Preserving Labeling Schemes and Their Applications. David Peleg: WG99 Proceedings of the 25

thInternational Workshop on Graph

Theoretic Concepts in Computer Science, pages 30-41.

4) Compact routing on power-law graphs with additive stretch. A. Brady and L.Cowen. In ALENEX, 2006.

5) Approximate distance oracles. Mikkel Thorup and Uri Zwick: In Proc. 33rdACM Symp. On Theory of Computing, pages 183-192, May2001.

6) Compact routing on Internet-like graphs. D. Krioukov, K. Fall, and X. Yang. InINFOCOM, 2004.

7) Labelling and implicit routing in networks. N. Santoro and R. Khatib :TheComputer Journal, 28(1):5-8, 1985.

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Compact Routing overview