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

1
Efficient Algorithms for Maximum Lifetime Data
Gathering and Aggregation in Wireless Sensor
NetworksSelected from Elsevier Computer
Networks

• Konstantinos Kalpakis,
• Koustuv Dasgupta,
• Parag Namjoshi
• Presentation by Shu-Ping Lin

2
Outline

• Introduction
• The Data Gathering Problem
• Maximum Lifetime Data Gather with Aggregation
• Greedy CMLDA
• Incremental CMLDA
• Experimental Results
• Conclusions

3
Outline

• Introduction
• The Data Gathering Problem
• Maximum Lifetime Data Gather with Aggregation
• Greedy CMLDA
• Incremental CMLDA
• Experimental Results
• Conclusions

4
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

5
Introduction (contd)

• 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.

6
Introduction (contd)

• The key idea of data aggregation is to combine
  data from different sensors to eliminate
  redundant transmissions.
• Address-centric versus data-centric.

7
Introduction (contd)

• 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.

8
Outline

• Introduction
• The Data Gathering Problem
• Maximum Lifetime Data Gather with Aggregation
• Greedy MLDA
• Incremental MLDA
• Experimental Results
• Conclusions

9
The Data Gathering Problem

• System Model
• A network of n sensor nodes and a base station
  node t labeled n1 distributed over a region.
• Each sensor produces one data packet whose size
  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
  any other sensor in the network.
• Each sensor i has a initial battery with finite,
  non-replenishable energy .

10
The Data Gathering Problem (contd)

• Energy Model
• Energy model is based on the first order radio
  model.
• A sensor consumes to
  run the 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
  sensor j is given by
  ..(2) where di,j is the distance between
  nodes i and j.

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The Data Gathering Problem (contd)

• 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.

12
Outline

• Introduction
• The Data Gathering Problem
• Maximum Lifetime Data Gathering with Aggregation
• Greedy MLDA
• Incremental MLDA
• Experimental Results
• Conclusions

13
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.

14
Maximum Lifetime Data Gathering with Aggregation
(contd)

• The Maximum Lifetime Data Aggregation (MLDA)
  problem
• Finding a gathering schedule with maximum
  lifetime, where sensors are permitted to
  aggregate incoming data packets.

15
Maximum Lifetime Data Gathering with Aggregation
(contd)

• 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.

16
Maximum Lifetime Data Gathering with Aggregation
(contd)

• 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 must
  reach the base station.
• In terms of network flows, this implies that
  sensor s must have a maximum s-t flow of size T
  to the base station in the flow network G.

17
Maximum Lifetime Data Gathering with Aggregation
(contd)
t
t
t
Round 1
Round 2
Flow Network G
18
Maximum Lifetime Data Gathering with Aggregation
(contd)

• 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.

19
Maximum Lifetime Data Gathering with Aggregation
(contd)

• Finding a near-optimal admissible flow network
• flow variable indicating the flow that
  k sends to the base station t over edge (i,j).

20
Maximum Lifetime Data Gathering with Aggregation
(contd)

• Objective
• maximize T
  (4)
• Constraints
• 
  (5)
• 
  
• 
  (6)
• 
  
• 
  (7)
• 
  (8)
• 
  
• 
  (9)
• where k1,2,..,n and all variables are
  non-negative integers.

Flow Conservation Constraints
21
Maximum Lifetime Data Gathering with Aggregation
(contd)

• 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
22
Maximum Lifetime Data Gathering with Aggregation
(contd)

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

23
Maximum Lifetime Data Gathering with Aggregation
(contd)

• 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.

24
Maximum Lifetime Data Gathering with Aggregation
(contd)
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Maximum Lifetime Data Gathering with Aggregation
(contd)

• 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.

26
Maximum Lifetime Data Gathering with Aggregation
(contd)

• 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.

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Maximum Lifetime Data Gathering with Aggregation
(contd)
40
20
1
4
1
4
60
40
40
20
40
2
3
2
3
40
Infeasible!!
28
Maximum Lifetime Data Gathering with Aggregation
(contd)
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Maximum Lifetime Data Gathering with Aggregation
(contd)

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

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Maximum Lifetime Data Gathering with Aggregation
(contd)

• Worst-case running time of MLDA
• Lemma 1 an e-optimal solution to a linear
  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

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Maximum Lifetime Data Gathering with Aggregation
(contd)

• 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
  
• 
  

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Maximum Lifetime Data Gathering with Aggregation
(contd)

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

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Maximum Lifetime Data Gathering with Aggregation
(contd)

• 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

34
Outline

• Introduction
• The Data Gathering Problem
• Maximum Lifetime Data Gather with Aggregation
• Greedy CMLDA
• Incremental CMLDA
• Experimental Results
• Conclusions

35
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
  station t.
• Greedy heuristic is to compute a maximum lifetime
  schedule for the super-sensor
• with base station, and then use this schedule
  to construct aggregation trees for the sensors.

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Greedy CMLDA (contd)
37
Greedy CMLDA (contd)
38
Greedy CMLDA (contd)
39
Greedy CMLDA (contd)
O(n2)
O(n2)
O(m15log m)
O(n)
O(n3)
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Greedy CMLDA (contd)

• 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).

41
Greedy CMLDA (contd)

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

42
Outline

• Introduction
• The Data Gathering Problem
• Maximum Lifetime Data Gather with Aggregation
• Greedy CMLDA
• Incremental CMLDA
• Experimental Results
• Conclusions

43
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 .

44
Incremental CMLDA (contd)

• Phase I
• The same with GREEDY heuristic
• The only difference is that we do not compute
  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.

45
Incremental CMLDA (contd)

• 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.

46
Incremental CMLDA (contd)

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

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Incremental CMLDA (contd)
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
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Incremental CMLDA (contd)
Total flow sent from super-sensor i equals to T
49
Incremental CMLDA (contd)

• 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.

50
Incremental CMLDA (contd)

• 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.

51
Incremental CMLDA (contd)
52
Incremental CMLDA (contd)

• Note that these capacities obtained from Phase
  III are fractional non-negative numbers.
• We scale the provisioned capacities by a factor
  of , where ?max is the maximum energy
  consumed by any sensor.
• Then we floor all the capacities to obtain flow
  network with integer capacities.

53
Incremental CMLDA (contd)

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

54
Incremental CMLDA (contd)
55
Incremental CMLDA (contd)

• 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)

56
Outline

• Introduction
• The Data Gathering Problem
• Maximum Lifetime Data Gather with Aggregation
• Greedy CMLDA
• Incremental CMLDA
• Experimental Results
• Conclusions

57
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.

58
Experimental Results (contd)

• 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.

59
Experimental Results (contd)
60
Experimental Results (contd)

• 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.

61
Experimental Results (contd)

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

62
Experimental Results (contd)
63
Experimental Results (contd)
64
Experimental Results (contd)

• 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.

65
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.