Set It and Forget It: Approximating the Set Once Strip Cover Problem
Set It and Forget It: Approximating the Set Once Strip Cover Problem Amotz Bar-Noy
Ben Baumer
Dror Rawitz
CUNY
Smith College
Tel Aviv University
1/6
Set Once Strip Cover
Maximizing barrier coverage lifetime using static sensors
2/6
Set Once Strip Cover
Maximizing barrier coverage lifetime using static sensors
Input: (x, b) – xi - location of sensor i – bi - battery of sensor i
0
x1 x2
x3 x4
x5
1
2/6
Set Once Strip Cover
Maximizing barrier coverage lifetime using static sensors
Input: (x, b) – xi - location of sensor i – bi - battery of sensor i
Solution: (ρ, τ ) – – – –
ρi - radius of sensor i τi - activation time of sensor i i stays active for bi /ρi time i covers the [xi − ρi , xi + ρi ] × [τi , bi /ρi ]
0
x1 x2
x3 x4
x5
1
2/6
Set Once Strip Cover
Maximizing barrier coverage lifetime using static sensors
Input: (x, b) – xi - location of sensor i – bi - battery of sensor i
Solution: (ρ, τ ) – – – –
T
ρi - radius of sensor i τi - activation time of sensor i i stays active for bi /ρi time i covers the [xi − ρi , xi + ρi ] × [τi , bi /ρi ]
0
x1 x2
x3 x4
x5
1
Goal: Maximize network lifetime – max T such that [0, 1] × [0, T ] is covered 2/6
Our Results
Set Once Strip Cover is NP-hard – Reduction from Partition
3/6
Our Results
Set Once Strip Cover is NP-hard – Reduction from Partition
3 2 -approximation
algorithm
– Round Robin: sensors cover the barrier one by one
3/6
Our Results
Set Once Strip Cover is NP-hard – Reduction from Partition
3 2 -approximation
algorithm
– Round Robin: sensors cover the barrier one by one – Duty cycle algorithms cannot do better
3/6
Our Results
Set Once Strip Cover is NP-hard – Reduction from Partition
3 2 -approximation
algorithm
– Round Robin: sensors cover the barrier one by one – Duty cycle algorithms cannot do better
Strip Cover – Sensors may change their radii finitely many times – [Bar-Noy & Baumer 11]: –
Assuming unit batteries, ratio of Round Robin is in [ 32 , 1.82] We close gap by showing that 32 ratio applies
3/6
Analysis of Round Robin: Proof Outline
Cut optimal schedule into strips – –
P Pj OPTj = OPT j RRj = RR
0
1
4/6
Analysis of Round Robin: Proof Outline
Cut optimal schedule into strips – –
P Pj OPTj = OPT j RRj = RR
0
1
Transform to unit batteries and stretch – OPTj is unchanged – RRj ≥ RR′j and RR′j may decrease
4/6
Analysis of Round Robin: Proof Outline
Cut optimal schedule into strips – –
P Pj OPTj = OPT j RRj = RR
0
1
Transform to unit batteries and stretch – OPTj is unchanged – RRj ≥ RR′j and RR′j may decrease
Prove
OPT RR′
≤
3 2
for stretched instances with unit batteries
4/6
Additional Result Set Radius Strip Cover
All sensor are activated at 0 (∀i, τi = 0)
T
0
1
5/6
Additional Result Set Radius Strip Cover
All sensor are activated at 0 (∀i, τi = 0)
T
0
1
0
1
Observations: – ∃ optimal schedule
T
where ρi = 0 or bi /ρi = T , for all i
– ∃ pair (i, k) such that xi + ρi = xk − ρk
5/6
Additional Result Set Radius Strip Cover
All sensor are activated at 0 (∀i, τi = 0)
T
0
1
0
1
Observations: – ∃ optimal schedule
T
where ρi = 0 or bi /ρi = T , for all i
– ∃ pair (i, k) such that xi + ρi = xk − ρk
O(n2 ) lifetime candidates
O(n2 log n) time algorithm
5/6
Open Problems
FPTAS for Set Once Strip Cover? – We have eliminated duty cycle algorithms
6/6
Open Problems
FPTAS for Set Once Strip Cover? – We have eliminated duty cycle algorithms
Is Strip Cover NP-hard?
6/6
Open Problems
FPTAS for Set Once Strip Cover? – We have eliminated duty cycle algorithms
Is Strip Cover NP-hard?
Energy is drained in proportion to riα , for α > 1 – Our algorithm for Set Radius Strip Cover extends to this case – Mobile Sensors [Bar-Noy Rawitz Terlecky 13]
6/6
Open Problems
FPTAS for Set Once Strip Cover? – We have eliminated duty cycle algorithms
Is Strip Cover NP-hard?
Energy is drained in proportion to riα , for α > 1 – Our algorithm for Set Radius Strip Cover extends to this case – Mobile Sensors [Bar-Noy Rawitz Terlecky 13]
1.5 dimensions – Covering the line with sensors located on the plane – Covering a region with sensors located on a line
6/6
Open Problems
FPTAS for Set Once Strip Cover? – We have eliminated duty cycle algorithms
Is Strip Cover NP-hard?
Energy is drained in proportion to riα , for α > 1 – Our algorithm for Set Radius Strip Cover extends to this case – Mobile Sensors [Bar-Noy Rawitz Terlecky 13]
1.5 dimensions – Covering the line with sensors located on the plane – Covering a region with sensors located on a line
6/6
Ben Baumer
Dror Rawitz
CUNY
Smith College
Tel Aviv University
1/6
Set Once Strip Cover
Maximizing barrier coverage lifetime using static sensors
2/6
Set Once Strip Cover
Maximizing barrier coverage lifetime using static sensors
Input: (x, b) – xi - location of sensor i – bi - battery of sensor i
0
x1 x2
x3 x4
x5
1
2/6
Set Once Strip Cover
Maximizing barrier coverage lifetime using static sensors
Input: (x, b) – xi - location of sensor i – bi - battery of sensor i
Solution: (ρ, τ ) – – – –
ρi - radius of sensor i τi - activation time of sensor i i stays active for bi /ρi time i covers the [xi − ρi , xi + ρi ] × [τi , bi /ρi ]
0
x1 x2
x3 x4
x5
1
2/6
Set Once Strip Cover
Maximizing barrier coverage lifetime using static sensors
Input: (x, b) – xi - location of sensor i – bi - battery of sensor i
Solution: (ρ, τ ) – – – –
T
ρi - radius of sensor i τi - activation time of sensor i i stays active for bi /ρi time i covers the [xi − ρi , xi + ρi ] × [τi , bi /ρi ]
0
x1 x2
x3 x4
x5
1
Goal: Maximize network lifetime – max T such that [0, 1] × [0, T ] is covered 2/6
Our Results
Set Once Strip Cover is NP-hard – Reduction from Partition
3/6
Our Results
Set Once Strip Cover is NP-hard – Reduction from Partition
3 2 -approximation
algorithm
– Round Robin: sensors cover the barrier one by one
3/6
Our Results
Set Once Strip Cover is NP-hard – Reduction from Partition
3 2 -approximation
algorithm
– Round Robin: sensors cover the barrier one by one – Duty cycle algorithms cannot do better
3/6
Our Results
Set Once Strip Cover is NP-hard – Reduction from Partition
3 2 -approximation
algorithm
– Round Robin: sensors cover the barrier one by one – Duty cycle algorithms cannot do better
Strip Cover – Sensors may change their radii finitely many times – [Bar-Noy & Baumer 11]: –
Assuming unit batteries, ratio of Round Robin is in [ 32 , 1.82] We close gap by showing that 32 ratio applies
3/6
Analysis of Round Robin: Proof Outline
Cut optimal schedule into strips – –
P Pj OPTj = OPT j RRj = RR
0
1
4/6
Analysis of Round Robin: Proof Outline
Cut optimal schedule into strips – –
P Pj OPTj = OPT j RRj = RR
0
1
Transform to unit batteries and stretch – OPTj is unchanged – RRj ≥ RR′j and RR′j may decrease
4/6
Analysis of Round Robin: Proof Outline
Cut optimal schedule into strips – –
P Pj OPTj = OPT j RRj = RR
0
1
Transform to unit batteries and stretch – OPTj is unchanged – RRj ≥ RR′j and RR′j may decrease
Prove
OPT RR′
≤
3 2
for stretched instances with unit batteries
4/6
Additional Result Set Radius Strip Cover
All sensor are activated at 0 (∀i, τi = 0)
T
0
1
5/6
Additional Result Set Radius Strip Cover
All sensor are activated at 0 (∀i, τi = 0)
T
0
1
0
1
Observations: – ∃ optimal schedule
T
where ρi = 0 or bi /ρi = T , for all i
– ∃ pair (i, k) such that xi + ρi = xk − ρk
5/6
Additional Result Set Radius Strip Cover
All sensor are activated at 0 (∀i, τi = 0)
T
0
1
0
1
Observations: – ∃ optimal schedule
T
where ρi = 0 or bi /ρi = T , for all i
– ∃ pair (i, k) such that xi + ρi = xk − ρk
O(n2 ) lifetime candidates
O(n2 log n) time algorithm
5/6
Open Problems
FPTAS for Set Once Strip Cover? – We have eliminated duty cycle algorithms
6/6
Open Problems
FPTAS for Set Once Strip Cover? – We have eliminated duty cycle algorithms
Is Strip Cover NP-hard?
6/6
Open Problems
FPTAS for Set Once Strip Cover? – We have eliminated duty cycle algorithms
Is Strip Cover NP-hard?
Energy is drained in proportion to riα , for α > 1 – Our algorithm for Set Radius Strip Cover extends to this case – Mobile Sensors [Bar-Noy Rawitz Terlecky 13]
6/6
Open Problems
FPTAS for Set Once Strip Cover? – We have eliminated duty cycle algorithms
Is Strip Cover NP-hard?
Energy is drained in proportion to riα , for α > 1 – Our algorithm for Set Radius Strip Cover extends to this case – Mobile Sensors [Bar-Noy Rawitz Terlecky 13]
1.5 dimensions – Covering the line with sensors located on the plane – Covering a region with sensors located on a line
6/6
Open Problems
FPTAS for Set Once Strip Cover? – We have eliminated duty cycle algorithms
Is Strip Cover NP-hard?
Energy is drained in proportion to riα , for α > 1 – Our algorithm for Set Radius Strip Cover extends to this case – Mobile Sensors [Bar-Noy Rawitz Terlecky 13]
1.5 dimensions – Covering the line with sensors located on the plane – Covering a region with sensors located on a line
6/6
