algorithms for multi-channel aggregated convergecast in...

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],

[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



Motivation

A Data Aggregation Network




Motivation

An Illustration

a b c

d e f g

s

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




Motivation

An Illustration

a b c

d e f g

s

1 3

4

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




Motivation

An Illustration

a b c

d e f g

s

1 3

4

2 1

56

f1 f1f1

f1

a b c

d e f g

s

f1 f1f2

f1




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)

a b c

d e f g

s

1 1

1

2 2

23

f1 f1f2

f1

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

g

de f

g




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.





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 .

s

a b

c d e

f g h

i

j k

l m n o p

s

a b

c d e

f g h

f3

f3

f1

f1

f1

f2

f2 f

2

f1





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

15

20

25

Density

Num

ber

of fr

eque

ncie

s

LargestDegreeFirst∆(G

C) + 1





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}

f2

f1

f3

f1

f3

f1

1 2

1

1

1

1

2

2

2

2

2 33

2

33

s

a b

c d e

f g h

i

j k

l m n o p

f1

f2

f2




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.






Reduction from Vertex Color.

1

2

v1 v2

v3

vi

ui1 ui2

vi1 vi2

q(q-1)/2 links

f2f2

v31v21

f1 f1 f2 f1

2 1 13

4 5

f1f1f1

f1

f1

v12

u11 u12

v22

u21 u22

v32

u31 u32

3 3

4

6

Tb1

Tb2 Tb

3

s

v11







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







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







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

6

Tb1

Tb2 Tb

3

s

v11





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





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






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∗

.






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






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).





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 .





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 )





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





Arbitrary Trees Under UDG

0 50 100 150 2000

20

40

60

80

100

120

140

160

180

200

x−axis

y−ax

is

Figure: Shortest Path Tree

0 50 100 150 2000

20

40

60

80

100

120

140

160

180

200

x−axis

y−ax

is

Figure: Minimum Spanning Tree





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 )





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.





Schedule Length

100 200 300 400 500 600 700 8005

10

15

20

25

30

35

40

45

50

55

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





Schedule Length

100 200 300 400 500 600 700 8005

10

15

20

25

30

35

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




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.


algorithms for multi-channel aggregated convergecast in...

Documents