Achieving Minimum Coverage Breach under Bandwidth Constraints in Wireless Sensor Networks

Maggie X Cheng Lu Ruan and Weili WuDept of Comput Sci Missouri Univ

IEEE INFOCOM 2005

Outline

IntroductionMinimum Breach Problem in Sensor Networks

Problem Definition Complexity Classification of the Minimum Breach

Problem

Approximation Algorithm Integer Programming Formulation of the Minimum

Breach Problem Heuristic 1 RELAXATION Heuristic 2 MINBREACH

Simulation StudyConclusion and Extensions

IntroductionStochastically deployed sensor network1048708Oscillated between active modes and inactive modes1048708Divided into mutually exclusive subsets without consideration on subset sizes

Disjoint set covers

C1 = s1 s2

C2 = s3 s4

[Ref 1] Energy-Efficient Target Coverage in Wireless Sensor Networks infocom 2004

[Ref 2] Maximal_Lifetime_Scheduling__in_sensor_surveillance_networks

Outline

IntroductionMinimum Breach Problem in Sensor Networks

Problem Definition Complexity Classification of the Minimum Breach

Problem

Approximation Algorithm Integer Programming Formulation of the Minimum

Breach Problem Heuristic 1 RELAXATION Heuristic 2 MINBREACH

Simulation StudyConclusion and Extensions

IntroductionStochastically deployed sensor network1048708Oscillated between active modes and inactive modes1048708Divided into mutually exclusive subsets without consideration on subset sizes

Disjoint set covers

C1 = s1 s2

C2 = s3 s4

[Ref 1] Energy-Efficient Target Coverage in Wireless Sensor Networks infocom 2004

[Ref 2] Maximal_Lifetime_Scheduling__in_sensor_surveillance_networks

IntroductionStochastically deployed sensor network1048708Oscillated between active modes and inactive modes1048708Divided into mutually exclusive subsets without consideration on subset sizes

Disjoint set covers

C1 = s1 s2

C2 = s3 s4

[Ref 1] Energy-Efficient Target Coverage in Wireless Sensor Networks infocom 2004

[Ref 2] Maximal_Lifetime_Scheduling__in_sensor_surveillance_networks

Why are Bandwidth Constraints

Each Active Sensor will send the sensory data directly to the base stationldquoBandwidthrdquo is the total number of time slotsThe total number of sensors simultaneously sending to the base station must be restricted by the bandwidth

helliphellip

Bandwidth Constraints

Slot 1 Slot 2 Slot W

Problem Definition(12)

Object has equal chance of being detected from all direction If sensors lie within the area boundary the object is considered covered

Problem Definition(22)Breach If a target is not covered by any active sensor it is called ldquobreachrdquo

PROBLEM MINIMUM BREACH1048708Given a collection S of sensors a collection A of targets and the sensor-target coverage map1048708QUESTION Can we divide S into disjoint subsets such that the overall breach is at most B and each subset has at most W sensors in it

Complexity Classification of the Minimum Breach Problem mdashMINIMUM BREACH

Prove that MINIMUM BREACH problem is NP-complete

1 Divide the sensors into two disjoint subsets to minimize the overall breachmdashMINIMUM 2SET BREACH problem

2 MINIMUM SET SPLITTING (NP-Complete) Given a collection C of subsets of a finite set S Is there a partition of S into two subsets S1 and S2 such that the car

dinality of the subsets in C that are not entirely contained in either S1 or S2 (splitted) is at least |C|-B

Integer Programming Formulation of the Minimum Breach Problem

Integer Programming Formulation of the Minimum Breach Problem

Heuristic I RELAXATION

First Step the Integer Programming problem is relaxed to a Linear Programming problem and an optimal solution for LP is computed

Second Step a greedy algorithm is employed to find an integer solution based on the optimal solution obtained at the first set1048708Third Step the solution from problem is used to construct the subsets

Heuristic II MINBREACH

x1

x2

x3

xn

x = =

yki

xki

Heuristic II MINBREACHI1 denote the rows in the upper part 01-1I2 denote the rows in the lower part 01Jx represent the columns that correspond to the xki in the original (IP)Jy represent the columns that correspond to the ykj in the original (IP)

Simulation StudyAs the number of targets increase from 10 to 100 the number of sensors also increase from 10 to 100 and bandwidth increase from 2 to 20It verifies that higher f leads to higher breach rate

Conclusion and Extensions

To improve sensor coverage deploying more sensors must be accompanied by increasing bandwidth otherwise the coverage may be decreased as a result

To minimize the maximal breach is also an NP-complete problem that requires efficient approximation algorithm

[back]

~(n3)

where

n= N(N+M)W

Problem Definition(12)

Object has equal chance of being detected from all direction If sensors lie within the area boundary the object is considered covered

Problem Definition(22)Breach If a target is not covered by any active sensor it is called ldquobreachrdquo

PROBLEM MINIMUM BREACH1048708Given a collection S of sensors a collection A of targets and the sensor-target coverage map1048708QUESTION Can we divide S into disjoint subsets such that the overall breach is at most B and each subset has at most W sensors in it

Complexity Classification of the Minimum Breach Problem mdashMINIMUM BREACH

Prove that MINIMUM BREACH problem is NP-complete

1 Divide the sensors into two disjoint subsets to minimize the overall breachmdashMINIMUM 2SET BREACH problem

2 MINIMUM SET SPLITTING (NP-Complete) Given a collection C of subsets of a finite set S Is there a partition of S into two subsets S1 and S2 such that the car

dinality of the subsets in C that are not entirely contained in either S1 or S2 (splitted) is at least |C|-B

Integer Programming Formulation of the Minimum Breach Problem

Integer Programming Formulation of the Minimum Breach Problem

Heuristic I RELAXATION

First Step the Integer Programming problem is relaxed to a Linear Programming problem and an optimal solution for LP is computed

Second Step a greedy algorithm is employed to find an integer solution based on the optimal solution obtained at the first set1048708Third Step the solution from problem is used to construct the subsets

Heuristic II MINBREACH

x1

x2

x3

xn

x = =

yki

xki

Heuristic II MINBREACHI1 denote the rows in the upper part 01-1I2 denote the rows in the lower part 01Jx represent the columns that correspond to the xki in the original (IP)Jy represent the columns that correspond to the ykj in the original (IP)

Simulation StudyAs the number of targets increase from 10 to 100 the number of sensors also increase from 10 to 100 and bandwidth increase from 2 to 20It verifies that higher f leads to higher breach rate

Conclusion and Extensions

To improve sensor coverage deploying more sensors must be accompanied by increasing bandwidth otherwise the coverage may be decreased as a result

To minimize the maximal breach is also an NP-complete problem that requires efficient approximation algorithm

[back]

~(n3)

where

n= N(N+M)W

Problem Definition(22)Breach If a target is not covered by any active sensor it is called ldquobreachrdquo

PROBLEM MINIMUM BREACH1048708Given a collection S of sensors a collection A of targets and the sensor-target coverage map1048708QUESTION Can we divide S into disjoint subsets such that the overall breach is at most B and each subset has at most W sensors in it

Complexity Classification of the Minimum Breach Problem mdashMINIMUM BREACH

Prove that MINIMUM BREACH problem is NP-complete

1 Divide the sensors into two disjoint subsets to minimize the overall breachmdashMINIMUM 2SET BREACH problem

2 MINIMUM SET SPLITTING (NP-Complete) Given a collection C of subsets of a finite set S Is there a partition of S into two subsets S1 and S2 such that the car

dinality of the subsets in C that are not entirely contained in either S1 or S2 (splitted) is at least |C|-B

Integer Programming Formulation of the Minimum Breach Problem

Integer Programming Formulation of the Minimum Breach Problem

Heuristic I RELAXATION

First Step the Integer Programming problem is relaxed to a Linear Programming problem and an optimal solution for LP is computed

Second Step a greedy algorithm is employed to find an integer solution based on the optimal solution obtained at the first set1048708Third Step the solution from problem is used to construct the subsets

Heuristic II MINBREACH

x1

x2

x3

xn

x = =

yki

xki

Heuristic II MINBREACHI1 denote the rows in the upper part 01-1I2 denote the rows in the lower part 01Jx represent the columns that correspond to the xki in the original (IP)Jy represent the columns that correspond to the ykj in the original (IP)

Simulation StudyAs the number of targets increase from 10 to 100 the number of sensors also increase from 10 to 100 and bandwidth increase from 2 to 20It verifies that higher f leads to higher breach rate

Conclusion and Extensions

To improve sensor coverage deploying more sensors must be accompanied by increasing bandwidth otherwise the coverage may be decreased as a result

To minimize the maximal breach is also an NP-complete problem that requires efficient approximation algorithm

[back]

~(n3)

where

n= N(N+M)W

Complexity Classification of the Minimum Breach Problem mdashMINIMUM BREACH

Prove that MINIMUM BREACH problem is NP-complete

1 Divide the sensors into two disjoint subsets to minimize the overall breachmdashMINIMUM 2SET BREACH problem

2 MINIMUM SET SPLITTING (NP-Complete) Given a collection C of subsets of a finite set S Is there a partition of S into two subsets S1 and S2 such that the car

dinality of the subsets in C that are not entirely contained in either S1 or S2 (splitted) is at least |C|-B

Integer Programming Formulation of the Minimum Breach Problem

Integer Programming Formulation of the Minimum Breach Problem

Heuristic I RELAXATION

First Step the Integer Programming problem is relaxed to a Linear Programming problem and an optimal solution for LP is computed

Second Step a greedy algorithm is employed to find an integer solution based on the optimal solution obtained at the first set1048708Third Step the solution from problem is used to construct the subsets

Heuristic II MINBREACH

x1

x2

x3

xn

x = =

yki

xki

Heuristic II MINBREACHI1 denote the rows in the upper part 01-1I2 denote the rows in the lower part 01Jx represent the columns that correspond to the xki in the original (IP)Jy represent the columns that correspond to the ykj in the original (IP)

Simulation StudyAs the number of targets increase from 10 to 100 the number of sensors also increase from 10 to 100 and bandwidth increase from 2 to 20It verifies that higher f leads to higher breach rate

Conclusion and Extensions

To improve sensor coverage deploying more sensors must be accompanied by increasing bandwidth otherwise the coverage may be decreased as a result

To minimize the maximal breach is also an NP-complete problem that requires efficient approximation algorithm

[back]

~(n3)

where

n= N(N+M)W

Integer Programming Formulation of the Minimum Breach Problem

Integer Programming Formulation of the Minimum Breach Problem

Heuristic I RELAXATION

First Step the Integer Programming problem is relaxed to a Linear Programming problem and an optimal solution for LP is computed

Second Step a greedy algorithm is employed to find an integer solution based on the optimal solution obtained at the first set1048708Third Step the solution from problem is used to construct the subsets

Heuristic II MINBREACH

x1

x2

x3

xn

x = =

yki

xki

Heuristic II MINBREACHI1 denote the rows in the upper part 01-1I2 denote the rows in the lower part 01Jx represent the columns that correspond to the xki in the original (IP)Jy represent the columns that correspond to the ykj in the original (IP)

Simulation StudyAs the number of targets increase from 10 to 100 the number of sensors also increase from 10 to 100 and bandwidth increase from 2 to 20It verifies that higher f leads to higher breach rate

Conclusion and Extensions

To improve sensor coverage deploying more sensors must be accompanied by increasing bandwidth otherwise the coverage may be decreased as a result

To minimize the maximal breach is also an NP-complete problem that requires efficient approximation algorithm

[back]

~(n3)

where

n= N(N+M)W

Integer Programming Formulation of the Minimum Breach Problem

Heuristic I RELAXATION

First Step the Integer Programming problem is relaxed to a Linear Programming problem and an optimal solution for LP is computed

Second Step a greedy algorithm is employed to find an integer solution based on the optimal solution obtained at the first set1048708Third Step the solution from problem is used to construct the subsets

Heuristic II MINBREACH

x1

x2

x3

xn

x = =

yki

xki

Heuristic II MINBREACHI1 denote the rows in the upper part 01-1I2 denote the rows in the lower part 01Jx represent the columns that correspond to the xki in the original (IP)Jy represent the columns that correspond to the ykj in the original (IP)

Simulation StudyAs the number of targets increase from 10 to 100 the number of sensors also increase from 10 to 100 and bandwidth increase from 2 to 20It verifies that higher f leads to higher breach rate

Conclusion and Extensions

To improve sensor coverage deploying more sensors must be accompanied by increasing bandwidth otherwise the coverage may be decreased as a result

To minimize the maximal breach is also an NP-complete problem that requires efficient approximation algorithm

[back]

~(n3)

where

n= N(N+M)W

Heuristic I RELAXATION

First Step the Integer Programming problem is relaxed to a Linear Programming problem and an optimal solution for LP is computed

Second Step a greedy algorithm is employed to find an integer solution based on the optimal solution obtained at the first set1048708Third Step the solution from problem is used to construct the subsets

Heuristic II MINBREACH

x1

x2

x3

xn

x = =

yki

xki

Heuristic II MINBREACHI1 denote the rows in the upper part 01-1I2 denote the rows in the lower part 01Jx represent the columns that correspond to the xki in the original (IP)Jy represent the columns that correspond to the ykj in the original (IP)

Simulation StudyAs the number of targets increase from 10 to 100 the number of sensors also increase from 10 to 100 and bandwidth increase from 2 to 20It verifies that higher f leads to higher breach rate

Conclusion and Extensions

To improve sensor coverage deploying more sensors must be accompanied by increasing bandwidth otherwise the coverage may be decreased as a result

To minimize the maximal breach is also an NP-complete problem that requires efficient approximation algorithm

[back]

~(n3)

where

n= N(N+M)W

Heuristic II MINBREACH

x1

x2

x3

xn

x = =

yki

xki

Heuristic II MINBREACHI1 denote the rows in the upper part 01-1I2 denote the rows in the lower part 01Jx represent the columns that correspond to the xki in the original (IP)Jy represent the columns that correspond to the ykj in the original (IP)

Simulation StudyAs the number of targets increase from 10 to 100 the number of sensors also increase from 10 to 100 and bandwidth increase from 2 to 20It verifies that higher f leads to higher breach rate

Conclusion and Extensions

To improve sensor coverage deploying more sensors must be accompanied by increasing bandwidth otherwise the coverage may be decreased as a result

To minimize the maximal breach is also an NP-complete problem that requires efficient approximation algorithm

[back]

~(n3)

where

n= N(N+M)W

Heuristic II MINBREACHI1 denote the rows in the upper part 01-1I2 denote the rows in the lower part 01Jx represent the columns that correspond to the xki in the original (IP)Jy represent the columns that correspond to the ykj in the original (IP)

Simulation StudyAs the number of targets increase from 10 to 100 the number of sensors also increase from 10 to 100 and bandwidth increase from 2 to 20It verifies that higher f leads to higher breach rate

Conclusion and Extensions

To improve sensor coverage deploying more sensors must be accompanied by increasing bandwidth otherwise the coverage may be decreased as a result

To minimize the maximal breach is also an NP-complete problem that requires efficient approximation algorithm

[back]

~(n3)

where

n= N(N+M)W

Simulation StudyAs the number of targets increase from 10 to 100 the number of sensors also increase from 10 to 100 and bandwidth increase from 2 to 20It verifies that higher f leads to higher breach rate

Conclusion and Extensions

To improve sensor coverage deploying more sensors must be accompanied by increasing bandwidth otherwise the coverage may be decreased as a result

To minimize the maximal breach is also an NP-complete problem that requires efficient approximation algorithm

[back]

~(n3)

where

n= N(N+M)W

Conclusion and Extensions

To improve sensor coverage deploying more sensors must be accompanied by increasing bandwidth otherwise the coverage may be decreased as a result

To minimize the maximal breach is also an NP-complete problem that requires efficient approximation algorithm

[back]

~(n3)

where

n= N(N+M)W

[back]

~(n3)

where

n= N(N+M)W

[back]

[back]

[back]