1. LOCI: Local Clustering Service for Large Scale Wireless Sensor Networks
Vineet Mittal
Should more be added here
Committee Members:
Dr. Anish Arora (advisor)
Dr. Hakan Ferhatosmanoglu

2. Challenges in Sensor Networks
Self-configuration
ad hoc node deployment
Self-healing
node failures, message loss and corruption
Scalability
large scale deployment and limited resources

3. Hierarchical Clustering
Advantages
Network specific
facilitates distribution of control over the network
abstracts network state information: insulates changes and failures/joins in one part of the network from other parts
Application specific
energy efficient

5. Challenges in Clustering
Scalability
Local self-healing
Similar size clusters
Optimum number of overlaps between clusters
Minimum number of non-clustered nodes
Optimum number of clusters
Minimum communication overhead

6. Problem Statement
Design a distributed, local, scalable, and self-stabilizing clustering program, LOCI, that given a cluster radius interval [R, mR], where R is the physical radius and m ≥ 2, constructs network partitions such that
a unique node is designated as a leader of the cluster
all nodes in the R-neighborhood of each leader belong to that cluster
maximum distance of a node from its leader is mR
each node belongs to the cluster of the closest clusterhead

7. Outline Model LOCI Program LOCI Correctness Proof
Theoretical Performance Analysis
Simulations based Performance Analysis
Conclusions

8. Model Wireless sensor network 2-D coordinate plane
Bi-directional links
Each node has a unique ID and unit transmission range
Distance estimation capability relative to other nodes
Density: nodes are distributed as per a homogeneous spatial Poisson process of intensity λ, such that at an average within a unit area circular region there are λ nodes
Fault model: nodes and links can fail-stop, new nodes and links can join the system, and state of a node can be arbitrarily corrupted.

9. Outline Model LOCI Program LOCI Correctness Proof
Theoretical Performance Analysis
Simulations based Performance Analysis
Conclusions

10. LOCI Program Timeout and elect itself as a clusterhead
Legitimate Cluster
Wait for a random amount of time
Grow the cluster iteratively
Valid cluster
Network partition constructed R mR R mR mR R R mR

11. LOCI Program j k Dynamic Priority : <Radius, - #Overlaps>
Problem: Neighboring clusters overlap
Legitimate cluster constructed R j A B k
Radius: maximum distance d, such that all nodes ≤ d from the leader belong to the
same cluster as that of the leader
Overlaps: # of overlaps of a cluster with neighboring clusters of equal radius

12. LOCI Abstract Program timeout(can_lead(j)) → start_cluster(j) []
can_join(j, S) → join(j, S)
(3) j and k are clusterheads Λ
overlapping_clusters(j, k) → resolve_overlap(j, k)

13. Local Healing using LOCI
node join
may be subsumed by neighboring clusters if it is within mR distance
of a neighboring clusterhead
d = 1 ( ) ( ) ( ) ( )
cascading ) ( ) ( ) ( ) (
R = [1, 1] A B
new node ( ) ( ) ( ) ( )
R = [1, 2]
new node subsumed

14. Local Healing using LOCI
node join
may create a new cluster, affecting only neighboring clusters by subsuming nodes contained in them that are farther than R distance from their respective clusterheads
d = 2 ( ) ( ) ( ) ( )
R = [1, 2] ( ) ( ) ( ) ( ) ( )
new cluster

15. Local Healing using LOCI
node fail-stops
new leader may be found in the original cluster, without affecting neighboring clusters, or
remaining nodes in the original cluster may affect neighboring clusters by joining them, or
may create new cluster(s), affecting only neighboring clusters by subsuming nodes contained in them

16. Hierarchical clustering
Legitimate cluster at level 0
Legitimate cluster at level 1
Clustering at Level 0
Represent clusters by a single
node, the clusterhead R 1 R
Neighboring clusters at level 0 →
Corresponding clusterheads
are neighbors at level 1
Clustering at Level 1
i+1 i [
(2mR)
- 1 ) ] R =
(2mR) R , ( mR i
2mR – 1

17. Outline Model LOCI Program LOCI Correctness Proof
Theoretical Performance Analysis
Simulations based Performance Analysis
Conclusions

18. LOCI correctness proof
Variables
d.j = distance of node j from the clusterhead
r.j = circular radius of the cluster
c.j = ID of the cluster to which j belongs
o.j = set of tuples that contains ID and radius of the overlapping
clusters
State predicates
Valid Cluster (VC.j) = circular radius of the cluster is r.j and all
the nodes in the cluster have correct values
Legitimate Cluster (LC.j) = circular radius of the cluster is R and
all the nodes in the cluster have correct values

19. LOCI correctness proof
Theorem 1: I is an invariant of LOCI
I (Invariant) Ξ if a node belongs to a cluster then that cluster is a valid cluster and the values of all the nodes in that cluster are correct
Theorem 2: F is a fixpoint of LOCI
F (Fixpoint) Ξ all nodes belong to a legitimate cluster and there is a legitimate path from every node to its clusterhead
Theorem 3: Starting from an invariant state I, the system
eventually reaches a state in F
Theorem 4: Starting at an arbitrary state, every computation of the system reaches a state in I

20. Outline Model LOCI Program LOCI Correctness Proof
Theoretical Performance Analysis
Simulations based Performance Analysis
Conclusions

21. Performance Analysis
Theorem 5: The convergence time of LOCI from invariant
state to fixpoint state is O(R4) rounds
Communication complexity
Cluster formation O(R3)
Individual node O(R3)

22. Percentage of uncovered nodes
For m≥2, LOCI constructs network partitions
For m<2, in the worst case the percentage of uncovered nodes using LOCI is m
1.0
1.2
1.5
1.7
2.0
p (%)
69.8
56.5
32.1
12.7
0.0

23. Number of clusters constructed
Overhead in number of clusters constructed is 3
√3R R 2R
(√2/3)R
Minimum number of clusters = n
Maximum number of clusters (LOCI) = 3n

24. Outline Model LOCI Program LOCI Correctness Proof
Theoretical Performance Analysis
Simulations based Performance Analysis
Conclusions

25. (unit transmission range)
Simulation Results
Assumptions
Nodes are in a regular grid
Nodes are aware of their neighbor set
(unit transmission range)

26. R = 2, m = 2, 10 x 10 Grid

27. R = 2, m = 2, 10 x 10 Grid

28. Simulation Results Overhead in the number of clusters (1-D network)64
1.31 (1.7)
1.29
1.41
1.50
256
1.36
1.32
1.34
1.31
625
1.37
1.35
1.38
1.30
Overhead in the number of clusters (2-D network)
N =
R=1
R=2
R=4
R=8 64
2.21
2.15
2.00
1.00
256
1.99
625
1.89
2.12
2.20
2.25
N: Number of nodes
R: Radius of the cluster

29. Conclusions
A distributed, local, scalable, and self-stabilizing clustering scheme, LOCI, is presented that partitions the network into bounded radius clusters
Convergence time and self-healing in LOCI are scalable both in time and space
LOCI bounds the overhead in the number of clusters constructed by a constant
Clusters constructed by LOCI form a Voronoi tessellation

30. Future Work Convergence time of O(R2 log(R))
Integrating LOCI in the “Line in the Sand” tracking service to achieve scalability and fault tolerance

32. Number of clusters constructed
R (Radius) ≈ D (Diameter of the network)
Overhead in number of clusters constructed is 6 R R 2R R R R

33. Cluster assignment R 2R+1 Assign clusterheads in surrounding region
Priority < n, id >
n: [0, 6] R
2R+1

34. LOCI correctness proof
Variables
d.j = distance of node j from the clusterhead
r.j = circular radius of the cluster
c.j = ID of the cluster to which j belongs
o.j = set of tuples that contains ID and radius of the overlapping
clusters
State predicates
VC.j = j is the clusterhead of a valid cluster
LC.j = j is the clusterhead of a legitimate cluster
H.j = variables stored at a node j have correct values
LP.j = legitimate path from a node j to its clusterhead