Efficient Algorithms for Maximum Lifetime Data Gathering and Aggregation in Wireless Sensor Networks Selected from Elsevier: Computer Networks Konstantinos

Efficient Algorithms for Maximum Lifetime Data Gathering and Aggregation in

Wireless Sensor Networks

Selected from Elsevier: Computer Networks

Konstantinos Kalpakis,Koustuv Dasgupta,

Parag Namjoshi

Presentation by Shu-Ping Lin

Outline

Introduction The Data Gathering Problem Maximum Lifetime Data Gather with

Aggregation Greedy CMLDA Incremental CMLDA Experimental Results Conclusions

Outline

Introduction The Data Gathering Problem Maximum Lifetime Data Gather with A

ggregation Greedy CMLDA Incremental CMLDA Experimental Results Conclusions

Introduction

Rapid development of sensor results from advances in Micro-sensor technology Low-power analog/digital electronics Bigger memory size

Obstacles arise from Limited energy Computing capabilities Communication resources available

Introduction (cont’d)

In this paper authors consider a system of sensors that are homogeneous and highly energy-constrained.

Replenishing energy via replacing battery on hundreds of nodes is infeasible.

The basic operation is systematic gathering of sensed data to be eventually transmitted to a base station.


The key idea of data aggregation is to combine data from different sensors to eliminate redundant transmissions.

Address-centric versus data-centric.


This paper derives novel algorithms for data gathering and aggregation in sensor networks.

A near-optimal polynomial-time algorithm is proposed, but it is computationally expensive for large sensor networks.

Then two heuristics based on GREEDY and INCREASMENTAL concept are derived.

Outline


ggregation Greedy MLDA Incremental MLDA Experimental Results Conclusions

The Data Gathering Problem

System Model A network of n sensor nodes and a base station

node t labeled n+1 distributed over a region. Each sensor produces one data packet whose si

ze is k bits per unit time as it monitors its vicinity. Each time unit is referred as a round. Each sensor has the ability to transmit data to a

ny other sensor in the network. Each sensor i has a initial battery with finite, no

n-replenishable energy .

i

The Data Gathering Problem (cont’d)

Energy Model Energy model is based on the first order radio

model. A sensor consumes to run t

he transmitter or receiver circuitry and for the transmitter amplifier. Energy consumed by a sensor i in receiving a

k-bit packet is given by ……(1) Energy consumed in transmitting a packet to s

ensor j is given by ..(2) where di,j is the distance between nodes i and j.

bitnJelec / 50

2pJ/bit/m 100 amp

kRx eleci

kdkTx jiampelecji 2,,

The Data Gathering Problem (cont’d)

Problem Statement Lifetime T of the system to be the number of

rounds until the first sensor is drained out. A data gathering schedule specifies how the

data packets from all the sensor are collected and transmitted to the base station.

The objective of data gathering problem is to find a schedule that maximizes the system lifetime T.

Outline

Introduction The Data Gathering Problem Maximum Lifetime Data Gathering w

ith Aggregation Greedy MLDA Incremental MLDA Experimental Results Conclusions

Maximum Lifetime Data Gathering with Aggregation

Data aggregation performs in-network fusion of data packets in an attempt to minimize the number and size of transmissions and thus save energy.

Aggregation can be performed when the data from different sensor are highly correlated.

Simplistic assumption An intermediate sensor can aggregate multiple

incoming packets into a single outgoing packet.

Maximum Lifetime Data Gathering with Aggregation (cont’d)

The Maximum Lifetime Data Aggregation (MLDA) problem Finding a gathering schedule with

maximum lifetime, where sensors are permitted to aggregate incoming data packets.


Let fi,j be the total number of packets that node i transmits to node j in a schedule S with lifetime T rounds.

The energy constraint for each sensor i is

The schedule S induces a flow network G which is a directed graph having edges (i,j) with capacity fi,j whenever fi,j ≥ 0.

i

n

jiij

n

jjiji RxfTxf

1,

1

1,,


Theorem 1 Let S be a schedule with lifetime T, and let G be

the flow network induced by S. Then, for each sensor s, the maximum flow from s to the base station t in G is ≥ T.

Proof Each data packet transmitted from a sensor mus

t reach the base station. In terms of network flows, this implies that senso

r s must have a maximum s-t flow of size ≥ T to the base station in the flow network G.


t

Round 1

t

Round 2

t

Flow Network G


A necessary condition for a schedule to have lifetime T is that each node in the induced flow network can push flow T to the base station t.

Now we must consider the problem of finding a flow network G with maximum T, that allows each sensor to push flow T to the base station, while respecting the energy constraints at all sensor.


Finding a near-optimal admissible flow network

: flow variable indicating the flow that k sends to the base station t over edge (i,j).

)(,kji


Objective:

maximize T (4)

Constraints:

(5)

(6)

(7)

(8)

(9)

where k=1,2,..,n and all variables are non-negative integers.

i

n

jiij

n

jjiji RxfTxf

1,

1

1,,

ki andn 1,2,...,i ,1

1

)(,

1

)(,

n

j

kji

n

n

kij

1

1

)(,

1

)(,

n

j

kjk

n

j

kkjT

1n1,2,...,j andn 1,2,...,i 0 ,)(

, jikji f

Tn

i

kni

1

)(1,

Flow Conservation Constraints


The linear relaxation of this integer program can be computed in polynomial time.

Then we can obtain a very good approximation for the optimal admissible flow network by first fixing the edge capacities to the floor of their values obtained from the linear relaxation so that the energy constrains are all satisfied.

Get edge capacities fi,j


Finally we solve the linear program (4) subject to constraints (6)-(9) without requiring anymore that the flows are integers.


How to get a schedule from an admissible flow network?

A schedule is a collection of directed trees that span all the sensors and the base station, with one such tree for each round.

These trees are called aggregation trees that may be used for one or more rounds.



Definition 1 Given an admissible flow network G with

lifetime T and a directed tree A rooted at the base station t with lifetime f.

(A, f)-reduction G’ of G is the flow network that results from G after reducing by f, the capacities of all of its edges that are also in A.

Definition 2 An (A, f)-reduction G’ of G is feasible if the

maximum flow from node v to the base station t in G’ is ≥ T - f for each node v in G’.


If A is an aggregation tree with lifetime f and the (A, f)-reduction of G is feasible, then the (A, f)-reduced flow network G’ of G is an admissible flow network with lifetime T – f.

Therefore, we can devise a simple iterative procedure to construct a schedule for an admissible flow network G with lifetime T.


1

2

4

3

40

40 40

1

2

4

3

20

20

40

40

60

Infeasible!!



Computing a maximum lifetime data gathering schedule described above is referred to MLDA algorithm.


Worst-case running time of MLDA Lemma 1: an ε-optimal solution to a lin

ear program with n variables can be found in time.

Lemma 2: given a flow network G = (V, E) with integral edge capacities bounded by U, a maximum s-t flow can be computed in

))log(1

log( 55.3 nUnnO

))log)/log(),(min( 22/13/2 UEVEEVO


Lemma 3: the lifetime of the sensor network is ≤ . Proof: Let dmin be the minimum distance of a

sensor from the base station t. Based on Eqs. (1) and (2), the minimum total energy expended by all the sensors in on round is at least

=

)1()/(max Okelec

)()1( 2 kdkkn ampelecelec kdkn ampelec 2

min

kdkn

n

imummi

sensorstheallofenergytotal

ampelec

2min

max

round onein consumedenergy n

)1(max Okelec


Theorem 3: The worst-case running time of the MLDA algorithm is O(n15log n), where n is the number of sensors.

Proof The linear program (4) has O(n3) variable

s and using lemma 3 we know that the time to compute an є-approximate solution to the linear program (4) is

n) logO(nn) log)(1

log)(( 15535.33 nnOtLP


The GETSCHEDULE procedure makes O(T) calls to the GETTREE routine.

GETTREE routine involves O(V2E) maxflow computations whose running time is

O(n8/3 log n). Thus, the running time of GETSCEEDULE is

≤ Total worst-case running time of MLDA is

n) logO(nn log1 20/33/8222 nnntEVT MZXFLOW

n) logO(nn) logO(nn) logO(nn) log(2 1520/3153/20 nOtLP

Outline



Greedy CMLDA

Let sensors be partitioned into m clusters

each consisting of at most c sensors. We refer to each cluster as a super-sensor. Let super-sensor consist only of the base st

ation t. Greedy heuristic is to compute a maximum lifeti

me schedule for the super-sensor

with base station, and then use this schedule to construct aggregation trees for the sensors.

m ,...,, 21

1m

m ,...,, 21

Greedy CMLDA (cont’d)




O(n2)

O(n2)

O(m15log m)

O(n)

O(n3)


The worst-cast running time of GREEDY CMLDA heuristic is O(m15 log m + n3).

By appropriately choosing the number of super-sensors m, we can achieve a significant reduction in the actual time.

For example, for m = n3/16, the worst-cast running time is O(n3).


The rationale is to greedily construct the tree by choosing minimum energy consumption edge at every iteration.

Outline



Incremental CMLDA

In solving MLDA problem, we are essentially interested in provisioning the (edge) capacities of an admissible flow network G.

This proposed heuristic builds such a flow network by incrementally provisioning capacities on its edges.

INCREMENTAL MLDA heuristic consists of four phases .

Incremental CMLDA (cont’d)

Phase I The same with GREEDY heuristic The only difference is that we do not com

pute schedule w.r.t. super-sensor, only linear relaxation of the integer program of MLDA is run.

After running the linear relaxation we get the capacity between every pair of super-sensor and , such that the system of super-sensors has a lifetime T.

jif ,

ji


In phase II we determine the capacity provisions between s and the remaining sensors within the same super-sensor, as well as between s and each of the super-sensor , such that The sum of provisioned capacities from all the

sensors in to each super-sensor equals obtained from Phase I. each sensor s in can push T packets to the

remaining super-sensors.

m ,...,, 21

ji

jif ,

i


Our objective is to minimize the maximum energy consumed by any sensor within the super-sensor , thereby extending the lifetime of the sensors.

i

Incremental CMLDA (cont’d)Energy required for transmission within the same super-sensor i

Energy required for transmission between other super-sensor j

Flow sent from super-sensor i to j must equal to

jif ,


Total flow sent from super-sensor i equals to T


From phase II, we obtain the capacity provisions between any sensor s and all other sensors in the same super-sensor.

In phase III, we need to determine the capacities that need to be provisioned between individual sensors in different super-sensors.

Incremental CMLDA (cont’d) Consider two distinct super-sensor and We provision capacities between pairs of

sensors from and , while ensuring that Total capacity provisioned from each sensor

to all the sensors in equals the provisioning obtained from Phase II.

Total capacity provisioned from each sensor

to all the sensors in equals the provisioning obtained from Phase II.

rl

r

l

li

rif ,

lrj

ljf ,

r



Note that these capacities obtained from Phase III are fractional non-negative numbers.

We scale the provisioned capacities by a factor of , where єmax is the maximum energy consumed by any sensor.

Then we floor all the capacities to obtain flow network with integer capacities.

max


Using this flow network we finally compute the integral system lifetime by MLDA algorithm, and a data gathering schedule S using the GETSCHEDULE algorithm.



Worse-case analysis Phase I: O(m15 log m) Phase II: O(c5m10 log (cm)) Phase III: O(c10 log c)

Worst-case running time of Incremental CMLDA is

O(m15 log m) + O(c5m10 log (cm)) + O(c10 log c)

Outline



Experimental Results

Consider a network of sensors randomly distributed in a 50mX50m field.

The number of sensor in the network is varied to be 40, 50, 60, 80 and 100.

Performance ration RM is defined as the ratio of the system lifetime achieved using MLDA to the lifetime given by the LRS protocol.

Experimental Results (cont’d)

The depth of a sensor v is defined to be its average number of hops from the base station in the schedule.

Construct initial cluster Pick a sensor i farthest from the base

station and form a cluster that includes i and its c-1 nearest neighbors.



The lifetime of a schedule obtained using the INCREMENTAL CMLDA heuristic is always within 3% of optimal solution.

The lifetime of a schedule give by the MLDA algorithm near-optimal.

The algorithms proposed in this paper outperform the LRS protocol in terms of system lifetime.


The average depth of a data gathering schedule attained by these heuristics is slightly higher than that of the LRS.




The GREEDY and INCREMENTAL CMLDA heuristics significantly outperform the LRS protocol.

The average depth of a data gathering schedule attained by these heuristics is only slightly higher than that of the LRS.

Conclusions

This paper proposed a polynomial-time near-optimal algorithm (MLDA) for solving the maximum lifetime data gathering problem for sensor networks.

Three heuristics are proposed and formulated as linear programming problem.

Future research Aggregation rate should be included. Depth (delay) constraint is considered.

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Efficient Algorithms for Maximum Lifetime Data Gathering and Aggregation in Wireless Sensor Networks Selected from Elsevier: Computer Networks Konstantinos