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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
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
[back]
[back]
[back]
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
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
[back]
[back]
[back]
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
[back]
[back]
[back]
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
[back]
[back]
[back]
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
[back]
[back]
[back]
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
[back]
[back]
[back]
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
[back]
[back]
[back]
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
[back]
[back]
[back]
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
[back]
[back]
[back]
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
[back]
[back]
[back]
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
[back]
[back]
[back]
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
[back]
[back]
[back]
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
[back]
[back]
[back]
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]
[back]
[back]
[back]
~(n3)
where
n= N(N+M)W
[back]
[back]
[back]
[back]
[back]
[back]
[back]
[back]
[back]