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### Routing with d-CDS

Based on paper entitled “Distributed Routing Algorithms for Wireless Ad Hoc Networks using d-hop Connected d-hop Dominating Sets”

A presentation by

Michael Rieck, Drake University

Subhankar Dhar, San Jose State University

Sukesh Pai, Microsoft Corporation

HPC ASIA 2002, December 19, Bangalore

© M.Rieck, S.Dhar, S.Pai

Presentation Overview

- Ad Hoc Networks – Overview
- Definitions
- Some New Algorithms
- Performance Evaluations
- Summary and Conclusions

Overview: What is an Ad Hoc Network ?

- A set of identical nodes
- Self-organizing and adaptive
- Nodes move freely and independently and communicate with other nodes via wireless links
- Nodes within the transmission range are connected to each other
- There has been a growing number of applications of these networks, particularly in those places where no network infrastructure is available

Definitions

- G will denote a connected graph, representing an ad hoc network
- V denotes the set of all vertices in the graph G
- Gd is the d-closure of G, that is
- vertices are the same as those of G
- but has an edge between two vertices u and v if and only if 0 < d(u,v) d
- d(u,v) is the minimal number of edges along a path connecting u and v

Definitions

- A subset D of V is dominating if every vertex not contained in D is adjacent to some vertex in D
- Backbone-basedrouting typically involves a connected dominating set
- A subset D of V is a d-hop dominating set of G if it is a dominating set for Gd ,i.e.if every vertex of G is within a distance d of some vertex in D
- D is d-hop connectedin G if it is connected in Gd

Definitions

- In cluster-based routing, the nodes of the network are organized into local clusters, and from each cluster certain nodes are selected to be clusterheads and are responsible for maintaining the routing information
- Clusters may or may not be allowed to overlap
- The d-hop neighborhoods of a d-hop dominating set can be used for clusters

Definitions

- For a fixedd-hop dominating set D define the following:
- A path in G is D-routing if the vertices in D along the path, together with its end-points form a connected set in the d-closure Gd
- The set D has the shortest path property if each pair of vertices can be connected by a shortest path which is also D-routing

- A dominating set with the shortest path property can be used to easily implement shortest path routing through the network

The d-CDS Algorithm

- Thed-hop Connected Dominating Set algorithm (d-CDS)that we have developed has both the desirable features of cluster-based and backbone-based routing.
- Moreover, the set produced by d-CDS is a d-hop connected d-hop dominating set that has the shortest path property.

The d-CDS Algorithm

- For each pair of vertices x and y satisfying d(x,y) = d+1, consider all of the shortest paths from x to y.
- Consider the set of vertices that lie strictly between x and y along such a path. Let E(x,y) be the vertex in this set with the highest ID. Call this vertex E(x,y).
- Construct the set Dd(G) by including all such E(x,y), and only these vertices.

The d-CDS Algorithm

Theorem: Assume that the connected graph G has radius at least d+1. Then the set Dd(G) is a d-hop connected d-hop dominating set. Moreover, any two vertices of G can be connected by a Dd(G)-routing path that is also a shortest path in G.

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Example

The Algorithm of Jie Wu & Hailan Li

Basic: For each vertex z inG, does z have neighbors x and y such that x and y are not adjacent? The vertex z is then admitted to a set which we will call WuLi0 (G) if and only if the answer to this question is “yes”.

Rule 1: For each vertex z in WuLi0 (G), does z have a neighbor z' in WuLi0 (G), whose ID is higher than that of z, and which is such that all of the neighbors of z are also neighbors of z' ? If so, z is deemed to be superfluous. The set WuLi1(G) consists of all the vertices from WuLi0 (G) for which the answer to the question is “no”.

The Algorithm of Jie Wu & Hailan Li

Rule 2: For each vertex z in WuLi1 (G), does z have two neighbors from WuLi1 (G), which are themselves adjacent, and which have IDs larger than that of z, and which are such that their combined neighbors include all of the neighbors of z? The set WuLi2(G) consists of all the vertices from WuLi1(G) for which the answer is “no”.

- Note that the sets WuLi0 (G), WuLi1(G) and WuLi2(G) are all connected dominating sets.

Wu-Li Rules 1 & 2 applied to 3-closure, #flood rounds = 2d = 6

Wu-Li with Rule 1 applied to 3-closure, #flood rounds = 2d = 6

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Example

The 6 d-CDS Algorithm

- For comparison, we apply Wu-Li to Gd rather than G. This produces a d-hop connected d-hop dominating set.
- In contrast with the Wu-Li algorithm, ourd-CDS algorithmworks directly with the graph G, rather than Gd, in order to obtain a d-hop dominating set.
- Moreover, this algorithm has a more efficient implementation.

The 6 Generalized d-CDS Algorithm

- For each pair of vertices x and y, a distance f apart, consider all paths from x to y whose length does not exceed g.
- Consider the set Sd,e,g(x,y) of all vertices that lie on at least one of these paths (including the endpoints), and which are within a distance d of x and a distance e of y.
- Define Ed,e,g(x,y) to be the vertex with the largest ID among these vertices.
- Define Dd,e,f,g(G) to be the set of such Ed,e,g(x,y) for all pairs x and y, as above.

Gen. 3-CDS = D 6 3,3,4,5(G), #flood rounds = d+2= 5

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Example

Wu-Li Rules 1 & 2 applied to 3-closure 6

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Example – Wu-Li, D-routing path

3-CDS 6

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Example – d-CDS, D-routing path

3- Gen. CDS = D 6 3,3,4,5(G)

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Example – Gen. d-CDS, D-routing path

Performance Evaluation 6

- Metrics used
- Message costs
- Dominating set size
- Cumulative routing path length

- We compared our algorithms(d-CDS, Gen. d-CDS, altered Wu-Li)against the algorithm of Wu and Li as well as the Max-Min algorithm of Amis et al.

Results 6

Dominating Set size for various algorithms

Results 6

Message costs for various algorithms

Results 6

Cumulative path lengths (%)

The y-axis represents the the ratio of the difference in the cumulative path lengths. If LWL is the cumulative path length for Wu-Li and LGEN is the cumulative path length for Generalized d-CDS, then the y-axis shows (LWL - LGEN) / LGEN.

Results 6

- Overall, our generalized d-CDS algorithm performed better for
- message costs
- cumulative routing path lengths

- the shortest path property trades off with the size of the dominating set

Summary and Conclusions 6

- We proposed a novel approach of finding a d-hop dominating set in an ad hoc wireless network.
- This set is also d-hop connected and has a certain shortest path property in some special cases.
- This is the basis of our routing scheme which is also very efficient from a cost perspective.
- We are exploring cost efficient alternatives to Rule 2 in the Wu-Li algorithm. While we recognize that Rule 2 plays a very useful role in controlling the size of the set, it also sacrifices the shortest path property, and is costly to compute.
- We are also exploring a hierarchical version of the d-CDS.

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