Algorithms for Multi-Channel Aggregated Convergecast in...
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IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
Algorithms for Multi-Channel Aggregated Convergecast in Sensor
Networks
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3Bhaskar Krishnamachari†
†Dept. of Electrical Engineering, University of Southern California‡Dept. of Computer Science, University of Twente, Netherlands
3Bio-Informatics Institute, Dept. of Computer Science, Virginia Tech{amitabhg, bkrishna}@usc.edu, [email protected],
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 1/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
Motivation
A Data Aggregation Network
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 2/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
Motivation
A Data Aggregation Network
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 3/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
Motivation
An Illustration
a b c
d e f g
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f1 f1f1
f1Setting
(1) Half-duplex single transceiver
(2) Nodes aggregate pkts from childrenand transmit only one pkt
(3) TDMA
(4) Interference causes pkt loss
(5) Receiver-based frequencyassignment
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 4/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
Motivation
An Illustration
a b c
d e f g
s
1 3
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2 1
56
f1 f1f1
f1Setting
(1) Half-duplex single transceiver
(2) Nodes aggregate pkts from childrenand transmit only one pkt
(3) TDMA
(4) Interference causes pkt loss
(5) Receiver-based frequencyassignment
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 5/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
Motivation
An Illustration
a b c
d e f g
s
1 3
4
2 1
56
f1 f1f1
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a b c
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Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 6/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
Motivation
An Illustration
a b c
d e f g
s
1 3
4
2 1
56
f1 f1f1
f1
Question
What is the fastest rate at whichaggregated data can be collected fromthe network? (minimizing the schedulelength)
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Frame 1 Frame 2
Receiver Slot 1 Slot 2 Slot 3 Slot 1 Slot 2 Slot 3
s c a,d b,e,f c,g a,d b,e,f
abc
de f
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de f
g
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 7/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
MFAPAn Upper BoundBFS Time Slot Assignment
Minimum Frequency Assignment Problem
MFAP
Given a spanning tree T on an arbitrary graph G = (V , E), find the minimumnumber of frequencies that can be assigned to the receivers of T such that allthe interfering link constraints are removed.
e1 e
2
f1
f2
1 1
Figure: Interfering edge structure
Theorem
MFAP is NP-complete.Proof: Reduction from Vertex Color.
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 8/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
MFAPAn Upper BoundBFS Time Slot Assignment
An Upper Bound
Lemma
Construct a constraint graph GC = (VC , EC ) from G = (V , E) as follows:For each receiver in G, create a vertex in GC . Create an edge between two suchvertices in GC if their corresponding receivers in G are part of an interferingedge structure.Then, the number of frequencies required to remove all the interfering linkconstraints is: Kmax ≤ ∆(GC ) + 1, where ∆(GC ): max degree in GC .
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Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 9/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
MFAPAn Upper BoundBFS Time Slot Assignment
Evaluation: Frequency Bounds
Largest Degree First (LDF)
1. while VC 6= φ do2. u ← max degree in VC
3. Assign the first availablefrequency to u different fromits neighbors4. VC ← VC \ {u}
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
5
10
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Density
Num
ber
of fr
eque
ncie
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LargestDegreeFirst∆(G
C) + 1
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 10/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
MFAPAn Upper BoundBFS Time Slot Assignment
BFS Time Slot Assignment
Algorithm
Input: T = (V , ET )while ET 6= φ
e ← next edge from T in BFS orderAssign minimum time slot to eET ← ET \ {e}
Theorem
AlgorithmBFS-TimeSlotAssignment on atree gives the minimum schedule lengthequal to ∆(T ), where ∆(T ) is themaximum node degree in T .
Proof by induction on i:N(i + 1) = max{N(i), N(i) + 1}
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Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 11/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Multiple-Frequency Minimum Time Scheduling Problem
MFMTSP
Given a spanning tree T on an arbitrary graph G = (V , E) and q frequencies,find an assignment of frequencies and time slots such that the schedule lengthis minimized.
Theorem
MFMTSP is NP-complete.Proof: Reduction from Vertex Color.
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 12/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Multiple-Frequency Minimum Time Scheduling Problem
Reduction from Vertex Color.
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v1 v2
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ui1 ui2
vi1 vi2
q(q-1)/2 links
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f1 f1 f2 f1
2 1 13
4 5
f1f1f1
f1
f1
v12
u11 u12
v22
u21 u22
v32
u31 u32
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Tb1
Tb2 Tb
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s
v11
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 13/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Multiple-Frequency Minimum Time Scheduling Problem
Reduction from Vertex Color.
1
2
v1 v2
v3
vi
ui1 ui2
vi1 vi2
q(q-1)/2 links
v31v21v12
u11 u12
v22
u21 u22
v32
u31 u32
v11
q2 interfering links
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 14/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Multiple-Frequency Minimum Time Scheduling Problem
Reduction from Vertex Color.
1
2
v1 v2
v3
vi
ui1 ui2
vi1 vi2
q(q-1)/2 links
v31v21v12
u11 u12
v22
u21 u22
v32
u31 u32
Tb1
Tb2 Tb
3
s
v11
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 15/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Multiple-Frequency Minimum Time Scheduling Problem
Reduction from Vertex Color.
1
2
v1 v2
v3
vi
ui1 ui2
vi1 vi2
q(q-1)/2 links
v31v21
f1 f2 f1 f2 f2f1
1 1
2
22
2 1 13
4 5
f1f1f1
f1
f1
2
v12
u11 u12
v22
u21 u22
v32
u31 u32
3 3
4
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Tb1
Tb2 Tb
3
s
v11
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 16/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Time Slot Assignment on UDG
Lemma
Suppose γc denote the set of time slots to schedule the edges in cell c. Then,the minimum schedule length Γ for the whole network is:
Γ ≤ 4 ·maxc {|γc |}, ∀α ≥ 2.
e1e2 e
3e4
a
f fff Corner
Edge
Interior
a C1 C2 C3 C4
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 17/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Frequency Assignment on UDG
Load-Balanced Frequency Assignment
R = {v1, . . . , vn}: receivers on Tm : R → {f1, . . . , fK}
If m(vj) = fk , then children of vj
transmit on fk
Define load on fk under m as:lm(fk) =
∑
m(vj )=fk
deg in(vj)
Then, a load-balanced frequencyassignment m∗ is:
m∗ = arg minm
maxfk
{lm(fk)}
f2 f1
v3v2 v1
Figure: l(f1) = 5, l(f2) = 5
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 18/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Frequency Assignment on UDG
Load-Balanced Frequency Assignment
R = {v1, . . . , vn}: receivers on Tm : R → {f1, . . . , fK}
If m(vj) = fk , then children of vj
transmit on fk
Define load on fk under m as:lm(fk) =
∑
m(vj )=fk
deg in(vj)
Then, a load-balanced frequencyassignment m∗ is:
m∗ = arg minm
maxfk
{lm(fk)}
f2 f1
v3v2 v1
Figure: l(f1) = 5, l(f2) = 5
Lemma
Load-balanced frequency assignment isNP-complete.OPT (min-max) load is Lm∗
.
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 19/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Frequency Assignment on UDG
Algorithm FrequencyGreedy (φ)
(1) Sort the receivers in non-increasing order of in-degrees.deg in(v1) ≥ deg in(v2) ≥ . . . ≥ deg in(vn).
(2) Starting from v1, assign each successive node a frequency from {f1, . . . , fK}that has the least load, breaking ties arbitrarily.
f2 f1
v3v2 v1
Figure: l(f1) = 5, l(f2) = 5
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 20/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Frequency Assignment on UDG
Algorithm FrequencyGreedy (φ)
(1) Sort the receivers in non-increasing order of in-degrees.deg in(v1) ≥ deg in(v2) ≥ . . . ≥ deg in(vn).
(2) Starting from v1, assign each successive node a frequency from {f1, . . . , fK}that has the least load, breaking ties arbitrarily.
f2 f1
v3v2 v1
Figure: l(f1) = 5, l(f2) = 5
Lemma
Algorithm FrequencyGreedy givesa (4/3− 1/3K ) approximation on Lm∗
.
Proof: Follows from Graham’salgorithm for scheduling jobs onidentical parallel machines according tolongest processing time first (LPT).
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 21/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
An Upper Bound on γc
Lemma
If Lφc denote the load on the maximally loaded frequency in cell c achieved by
FrequencyGreedy, then any greedy time slot assignment can schedule allthe edges in c within 2Lφ
c time slots, i.e.,
|γc | ≤ 2Lφc .
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 22/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
A Constant Factor Approximation
Theorem
Given a tree T on a UDG G, and K frequencies, there exists an algorithm Gthat achieves a constant factor 8µα (4/3− 1/3K ) approximation on theoptimal schedule length, where µα > 0 is a constant for a given cell size α ≥ 2,i.e.,
ΓG = O(ΓOPT )
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 23/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Proof Sketch
G consists of 2 phases:
(1) Run FrequencyGreedy in each cell ci
(2) Run any greedy time slot assignment schemee.g., schedule a maximal number of edges in each slot
Lower bound on OPT: ΓOPT ≥ maxc
{
Lm∗
c
}
/µα
ΓG ≤ 4 ·maxc{|γc |} ≤ 8 ·max
c{Lφ
c }
≤ 8 ·maxc{(4/3− 1/3K ) · Lm∗
c }
≤ 8µα (4/3− 1/3K ) · ΓOPT
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 24/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Arbitrary Trees Under UDG
0 50 100 150 2000
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x−axis
y−ax
is
Figure: Shortest Path Tree
0 50 100 150 2000
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y−ax
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Figure: Minimum Spanning Tree
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 25/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Arbitrary Trees Under UDG
Theorem
Given a UDG G and K frequencies, there exists an algorithm H that achieves aconstant factor 8µα∆C approximation on the optimal schedule length, whereµα > 0, ∆C > 0 are constants for a given cell size α ≥ 2, i.e.,
ΓH = O(ΓOPT )
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 26/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Proof Sketch
Vc : set of nodes in cell cRc = {v1, . . . , vn}: set of receivers in c on arbitrary T∆in(T ): max in-degree in T
Lower bound on OPT: ΓOPT ≥ maxc
{⌈
|Vc |K
⌉}
/µα
Then
ΓH ≤ 4 ·maxc{|γc |} ≤ 8 ·max
c{Lφ
c }
≤ 8 ·maxc{⌈|Vc |/K⌉} ·∆in(T )
≤ 8µα∆in(T ) · ΓOPT
Bounded-degree spanning tree always exists on UDG. Therefore, proved.
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 27/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Schedule Length
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Sch
edul
e le
ngth
K = 1K = 2K = 3K = 4K = 5
Number of nodes
Figure: Algorithm G on Shortest Path Tree for Different Network Sizes
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 28/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
On Arbitrary GraphsAssignment on Unit Disk Graphs: Approximation AlgorithmsAssignment on Arbitrary Trees Under UDGEvaluation
Schedule Length
100 200 300 400 500 600 700 8005
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15
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40
45
50
Number of nodes
Sch
edul
e le
ngth
SPT, K = 1MIT, K = 1MIT, K = 3
Figure: Algorithm G on Minimum Interference Tree for Different Network Sizes
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 29/ 30
IntroductionOptimal Frequency Assignment
Minimizing Schedule LengthConclusion
Conclusions
Summary
(1) Addressed a scheduling problem under aggregated convergecast usingmultiple channels.
(2) Proved a NP-completeness result on finding the minimum number ofchannels required to remove all the interfering links in an arbitrary wirelessnetwork.
(3) Proved a NP-completeness result on minimizing the schedule length for agiven number of channels in an arbitrary wireless networks.
(4) Proposed an optimal time slot scheduling scheme when enough frequenciesare available.
(5) Proposed constant factor approximation algorithms to minimize theschedule length on unit disk graphs.
(6) Evaluated algorithms using simulations: most of the times 3 to 4frequencies are enough all the interfering links.
Amitabha Ghosh† Ozlem D. Incel‡ V.S. Anil Kumar3 Bhaskar Krishnamachari† †Dept. of Electrical Engineering, University of Southern CaliforniaAlgorithms for Multi-Channel Aggregated Convergecast in Sensor Networks 30/ 30