On Non-Disjoint Dominating Sets for the Lifetime of

Wireless Sensor NetworksAkshaye Dhawan

Characteristics of Sensor Networks

• Low cost• Usually deployed in large numbers• Constraints – energy. Limited power

supply in the form of a battery. • But the fact that sensors are deployed in

large numbers means that there is significant overlap of their monitoring regions

• Idea: Use a subset of these sensors

Dominating Sets

• A dominating set of a graph is a subset of all the nodes such that each node is either in the dominating set or adjacent to some node in the dominating set.

Connected Dominating Set

• A subset of nodes such that they are dominating and the subgraph induced by the nodes in the dominating set is connected (i.e. they communicate without the help of any node not in the CDS)

Usefulness of (C)DS

• At any point of time only the (C)DS nodes have to be active instead of all nodes

• Related closely to clustering – when we choose clusterheads such that every node is either a cluster head or has at least one clusterhead in its neighborhood -> maps to dominating set problem

• Q. Min Dominating Set (min cardinality)

Fault Tolerance

• Since probability of node failure is fairly high we may use the k-dominating set approach

• Each node v V has at least k dominators in its neighborhood.

Key ideas

• Objective : Maximize network lifetime

• Approach: Find a number of disjoint dominating sets

• Activate these successively

• So the problem now becomes one of finding the maximum number of disjoint dominating sets – max domatic partition

• Domatic Number D(G) of a graph G

Problem Statement and Model

• G=(V,E) each network node is represented by v V and there is an edge {u,v} E iff u and v are within communication range

• Undirected edges is the assumption

• n=|V| is known to all nodes

• Nv : Neighbor set of v

• Nv+ : Nv U {v}

Definitions

• δv : |Nv | (number of neighbors of v)

• Δ : max v V δv (Max and Min degree

• δ : min v V δv in the network)

• b v : time a node v can be in the dominating set ( < total energy)

Problem Definition

• Schedule S is a set of pairs (D1,t1) … (Dk, tk) where Di is a Dominating Set and ti is the time during which Di is active i.e. D1 is active in [0,…,t1] and generally Di is active in k

• Lifetime of a schedule L(S) = Σi=1 ti

• Maximum Clustering Lifetime problem asks for a schedule S with max length L(S) such that

Example

Conventions

• SOPT : Optimal Schedule

• LOPT =L(SOPT ) : Lifetime of Optimal Schedule

• Sv(s1: s2) = 1 if v Di and Di is active in the interval [s1…s2]

• Standard Mathematical Result used in Analysis:

The Uniform Case

• bV=b for all v V initially• • Proof idea: v can be covered by itself or by

a neighbor in the entirety of the schedule. Since each node is active for b, result follows

• Use technique shown in [5] to get an efficient approx algo with an approximation ration O(log n)

Idea behind algorithm

• Randomized: Each node v randomly picks a color in the range [1,…, δv

(2) / 3(logn)]• δv

(2) : denotes the minimum degree of a node in N v

+

• Idea is to interpret different color classes as a domatic partition of G.

• The schedule S simply follows by activating each of the color classes one after another

g

• Proof Idea: show that with high probability, each color class forms a valid dominating set.

• Ci is the set of nodes that randomly choose cv = I in line 4

General case

• Each node v can have its own initial battery bv•

• Again a randomized approach. Instead of

choosing a single color cv each node chooses bv many colors in a certain range

• Once again by restricting color range we can guarantee that each node has many diff color classes in its neighborhood

t

Open Problems

• Maximum Lifetime Connected Dominating Set – extending domatic partition to connected domatic partition appears to be non trivial

• Assumption that n is known. Any way to get rid of this?

Why non-disjoint sets?

• Each sensor has 2 units

• Disjoint lifetime: 2 units

• Non disjoint: ({p1,p2},1)

({p2,p3},1) and ({p3,p1},1)

3 units

• But it’s a more complex problem since using one set now drains lifetime of other sets

References• [1]J. Carle and D. Simplot-Ryl, "Energy-Efficient area monitoring for

sensor networks", Computer, Vol. 37, Issue 2, pp. 40-46, Feb. 2004• [2] J. Wu and H. Li, "On Calculating connected dominating set for

efficient routing in ad hoc wireless networks", in Proc. of the 3• [3] J. Wu, F. Dai, M. Gao, and I. Stojmenovic, On Calculating

PowerAware Connected Dominating Set for Efficient Routing in Ad Hoc Wireless Networks, Journal of Communications and Networks, Vol. 5, No. 2, pp. 169-178, March 2002.

• [4] Wattenhofer et al. “Maximizing Lifetime of Dominating Sets”, WMAN 2005

• [5] Feige et al. “Approximating the domatic number”. SIAM Journal of Computing. 32(1):172-195, 2003