Data Gathering in Wireless Networks

Gathering Problem

Data Gathering in Radio Networks

Patricio Reyes

Advisors: Jean-Claude Bermond - Herve RivanoMASCOTTE Project - INRIA/I3S(CNRS-UNSA)

GTEC – Universidad de CorunaOctober 27, 2009

P. Reyes Data Gathering in Radio Networks 1/119

Gathering Problem

Table of Contents

Part I: Gathering

2 Gathering

3 Minimum Data Gathering in Sensor Networks

Part II: Round Weighting

4 Round Weighting

5 Round Weighting with Symmetrical Interference


GatheringMinimum Data Gathering in Sensor Networks

Part I

Gathering



MotivationMain Results

Internet in the villages

France Telecom R&D (Orange Labs)

Optimization: Throughput






Optimization: Throughput (Delay)




Delay

Sensor Network

Optimization: Minimum completion timeP. Reyes Data Gathering in Radio Networks 14/119



Delay

Sensor Network




Delay

Sensor Network

?

?




Model

Network topology / village ←→ graph

Messages must be collected by the BS or gateway

Avoid interferences

Time:

synchronousdiscrete: time-slots t = 1, 2, 3, . . .

Call u → v within transmission range

1 time-slot1 message

Round: set of non interfering calls during 1 time-slot

Goal: Minimize the gathering time → # time-slots




Interference and Transmission model

2 parameteres: dI , dT

dTdI

BLUE

bla bla

bla bla

bla bla

??? ???

??? ???

??? ???




Graph metric

Figure: Euclidean-metric and Graph-metric model

If dT = 1 ←→ transmission graph





1 Asymmetrical Model

One node transmits

Interference distance dI

u → v v ′ ← u′ Interfere if d(u, v ′) ≤ dI or d(u′, v) ≤ dI

(a) Compati-ble calls withdT = dI = 1

(b) Interfer-ing calls withdT = dI = 1





2 Symmetrical Model

Two node transmits

Interference distance d sI

Motivation: msg u → v ACK u ← vu ↔ v v ′ ↔ u′ Interfere if minx∈{u,v};y∈{u′,v ′} dG (x , y) < d s

I

Figure: Compatible calls for symmetrical interference with d sI = dT = 1.




Known Results

General GraphsAsymmetrical model: Arbitrary dI ≥ dT

Minimum Time Gathering is NP-Hard [BGK+06]

unitary demand ←→ NP-Hard [BGK+06, Kor08]

4-approximation [BGK+06]

Best result using shortest paths [Kor08]

Paths

unitary demand

BS an end-vertex

optimal for dT = 1, any dI ≥ dT [BCY06, BCY09]

BS arbitrary

optimal for dT = 1, any 1 ≤ dI ≤ 4 [BCY06, BCY09]




Known Results

Trees

dI = dt = 1 optimal algorithm considering buffering [BY08]and no-buffering [BGR08]

d sI = dT = 1 optimal algorithm, general graph, unitary

demand [GR06, GR09]

Grids

BS in the center: Optimal algorithm for dT = 1, anydI ≥ dT [BP05, BP09]




Our Contribution

Linear Programming: Integer (Binary) Program... our bestresult: 3x3 grid

Incremental Protocols in the Pathjoint work with J-C. Bermond, R. Klasing, N. Morales, S. Perennes

Asymmetrical Interference ModelUnitary demandBS in an arbitrary vertex

1+-approximation

BS in an end-vertex

optimal for dT = 2, 3, 5 and arbitrary dI ≥ dT

Conjecture: Polynomial in the length of the path

Minimum Data Gathering in Sensor Networksjoint work with J-C. Bermond, N. Nisse, H. Rivano

Algotel’09, AdHocNow’09



Model+2-approximation

Table of Contents

2 GatheringMotivationMain Results

3 Minimum Data Gathering in Sensor NetworksModel+2-approximation




Motivation

Random Sensor Network behaves as a grid(Klasing, Lotkin, Navarra, Perennes ’05 [KLNP05])




Model

Revah and Segal’07

Sensor Networks

square grid with BS in (0, 0)

set of M messages

technical detail: nodes in the border do not have msgs

Interference

Primary node interference model ←→ dT = d sI = 1

simultaneous transmissions are a matching

R&S Algo: *1.5-approximation




Our Results

+2-approx (distributed version)

+1-approx

example:Optimal solution: 100 time-slotsR&S: 150 time-slotsOur algo: 101 time-slots

Time-complexity: linear w.r.t number of messages

BONUS: no-buffering ↔ hot-potato routing

node v receives a msg at time-slot tnode v sends the msg at time-slot t + 1




Revah & Segal’s Methodology

Gathering starting with the closest messages

BS receives two consecutive messages

next time step: BS does not receive message

Gathering

t = 0

BS

1

2

1








Gathering

t = 0

BS

1

2

1




Gathering and Personalized Broadcasting

Choose the good approach!!!Gathering ←→ Personalized Broadcasting

Gathering

t = 0

BS

m1

m2

m3






Gathering

t = 1

BS

m1

m2

m3






Gathering

t = 2

BS

m1

m2

m3






Gathering

t = 3

BS

m1

m2

m3






Gathering

t = 4

BS

m1

m2

m3






Gathering

t = 5

BS

m1

m2

m3






Personalized Broadcasting: inverse time

t = 0

BS

m1

m2

m3







t = 1

BS

m1

m2

m3







t = 2

BS

m1

m2

m3







t = 3

BS

m1

m2

m3







t = 4

BS

m1

m2

m3







t = 5

BS

m1

m2

m3







Sequence S = (m1,m2,m3)

BS

m1

m2

m3

1

3 2




Methodology

Protocols:

BS sends 1 msg pertime-slott 1 2 3

m1 m2 m3

Horizontal-Vertical(HV) and VH pathst 1 2 3

VH HV VH BS

m1

m2

m3

1

3 2




Interference

Primary Interference model ←→ matching

Only consecutive messages may interfer

t = 0

m’

BS

m




Interference



t = 1

m’

BS

m




Interference



t = 2

m’

BS

m




Interference



t = 3

m’

BS

m




Interference



t = 4

m’

BS

m




Interference



Two consecutive msgs ↔ disjoints paths

m’

BS

m




Lower Bound

Same problem BUT without interferences

order (m1, . . . ,mM) decreasing distance w.r.t BS

LB = maxi≤M d(mi ) + i − 1

LB is not tight

BS3

1

2

m1

m2

m3

(m1,m2,m3), LB = 4

BS2

1

3

m1

m2

m3

(m1,m3,m2),5 time-slots




Methodology

gathering ↔ personalized broadcasting

BS sends 1 msg at each time-slot

two consecutive msgs ↔ disjoint paths

M = {m1, . . . ,mM} ordered by decreasing distance to BS

Goal: Interference-free scheduling. Minimize the delay foreach msg.

NEXT Result: +2-approximation algorithm

Protocol which attains the LB + 2

i -th msg (distance order) ↔ time-slot i ± 2




+2 Approx Algorithm

Modify the sequence (m1, . . . ,mM)

Induction: Sequence (s1, . . . , sM−2) satisfying:

(i) interference-freesM−2 sent VH and sM−2 ∈ {mM−3, mM−2}

(ii) si ∈ {mi−2, mi−1, mi , mi+1, mi+2}, i < M − 2

t · · · i · · · M − 2mi−2, mi−1, mi mM−3, mM−2

mi+1, mi+2

Append {mM−1,mM}

BS

sM−2




+2-approx algorithm

Notation: q, r ∈ {mM−1,mM}, q lower than r , and p = sM−2

Case 1 q lower than p

r

p

M − 2

M − 1

M

BS

q

t · · · M − 2 M − 1 M

p → mM−2,mM−3 q → mM−1/mM r → mM/mM−1

Properties (i) and (ii) !!





Case 2 q higher than p

M − 2

mM−1

p

mM

BS

(a) before

mM

M − 2

mM−1

p

M − 1

M

BS

(b) after

By property (ii): sM−2 ∈ {mM−3,mM−2}t · · · M− 3 M − 2 M − 1 M

before . . . sM−3 p − −

after . . . sM−3 mM−1 p mM

Properties (i) and (ii) !!P. Reyes Data Gathering in Radio Networks 63/119




Case 2 q higher than p

M − 2

mM−1

p

mM

BS

(c) before

mM

M − 2

mM−1

p

M − 1

M

BS

(d) after

By property (ii): sM−2 ∈ {mM−3,mM−2}t · · · M− 3 M − 2 M − 1 M

before . . . mM−2 mM−3 − −

after . . . mM−2 mM−1 mM−3 mM

Properties (i) and (ii) !! ↔ +2-approxP. Reyes Data Gathering in Radio Networks 64/119



Future Work

Conjecture: +1-approx algorithm is optimal

Complexity?

Complexity for general graphs with no-buffering? [BGR08]

online version?Gathering < Personalized Broadcasting

Messages in the border of the grid? Work in progress...

Different interference models?


Round WeightingRW with Symmetrical Interference

Part II

Round Weighting



IntroductionMain Results

Table of Contents

4 Round WeightingIntroductionMain Results

5 Round Weighting with Symmetrical InterferenceIntroductionLB using Call-cliquesResults




Introduction

Gathering with a permanent demand?

BS

round1

Round Weighting [KMP08]relaxation of Gathering problem

order of rounds does not matterrelative importance of the rounds matter

Allocate bandwidth in a periodic way




Introduction


BS

round1







Round Weighting for gathering instances

Round weighting for gathering instances

a graph G = (V ,E ), a base station BS ∈ V , a set of possiblerounds

demand function from v ∈ V to the BS, b : V → R+

Solution:

A round weight function w over the rounds

induces a capacity over the edges: cw : E −→ R+

admissible if there exists a flow φ satisfying

the demandsuch that φ(e) ≤ cw (e) ∀e

Goal:

Minimize the overall weight of w , i.e. W =∑

R∈R w(R)

Warning

exponential # of rounds w.r.t. the number of edges




Idea

BS

1




Idea

BS

1cw (e) =P

e∈R w(R)




Idea

BS

1φ(e) ≤ cw (e) =P

e∈R w(R)




Example: One path

path ←→ no routing problem...

One unit of demand: b(v)

dT = d sI = 1.

round 1

BS = 0 b(v)

Solution w : R −→ R+, w(R1) = w(R2) = b(v) such that

satisfies the traffic demand b(v , BS)

there exists a flow φ transmitting the demand

respects the capacity induced




Example: One path

Example:


dT = d sI = 1.

round 1

BS = 0 b(v)R1 R1 R1 R1 R1








Example: One path

Example:


dT = d sI = 1.

round 1

BS = 0 b(v)R1 R1 R1 R1 R1R2 R2 R2








Using two paths

Case 1: Two paths ←→ even cycle

2

11

1

2 2BS

w(R1) = w(R2) = 1/2, total weight W = 1 ←→ optimal solution




Using two paths

Case 2: Two paths ←→ odd cycle

2

BS

1 1

2 3

w(R1) = w(R2) = w(R3) = 1/2total weight W = 3/2.




Using two paths

Case: Two paths ←→ odd cycle

2,6

2,4,5 1,3,6

4,51,3

BS

w(R1) = w(R2) = w(R3) = w(R4) = w(R5) = w(R6) = 1/5total weight W = 6/5.

In a general (odd) cycle?




Using two paths

BS

p

q

Result: routing via an odd cycle with p > q can be made withtotal weight W = 2p

2p−1

Best solution? YES

dual formulationoptimal dual solution W = 2p

2p−1

joint work with J.-C. Bermond, H. Rivano and J. Yu




Known Results

Based on [KMP08]

Lower bounds by duality

NP-hard even for unitary demand (gathering instances)

4-approximation for general graphs

Polynomial in the path

Open question: NP-Hard for grids? (gathering instances)




Our Contribution

RW in Primary Node Interference Modeljoint work with J.-C. Bermond, H. Rivano and J. Yu

Symmetrical Interference, d sI = dT = 1

General Demand

Routing via (odd) cycles: optimal solutionRouting via cycles with ears: optimal solutionsUpper Bound for the general RW problem in a 2-connectedgraph

RW in wireless networks with symmetrical interferencejoint work with J.-C. Bermond, C. Gomes and H. Rivano

Symmetrical Interference, dT = 1, any d sI ≥ dT



IntroductionLB using Call-cliquesResults

Table of Contents

4 Round WeightingIntroductionMain Results

5 Round Weighting with Symmetrical InterferenceIntroductionLB using Call-cliquesResults




Call-clique

Call-clique: set of edges pairwise interfering.

BS

1

Figure: d sI = 3, dT = 1

w(R1) = w(R2) = w(R3) = w(R4) = 1/2. Total weight W = 2No matter the paths used, W ≥ 2




Call-clique


BS

1


w(R1) = w(R2) = 1/2. Total weight W = 2No matter the paths used, W ≥ 2




Call-clique


BS

R2R1 1


w(R1) = w(R2) = 1/2. Total weight W = 2No matter the paths used, W ≥ 2




Call-clique


BS

1






Call-clique


BS

R2R1 1R3

R4






Lower Bound using a call-clique

Call-clique K0: set of edges at dist at most⌈

dsI2

⌉

from BS.

BS

1

l

dsI2

m

Lemma (LB)

W ≥∑

v∈VK0d(v ,BS)b(v) +

⌈

dsI2

⌉

∑

v /∈VK0b(v)

LB is tight for d sI odd, BS in the center




More than one call-clique

Grid with the BS in the corner

BS1

Figure: dT = 1, d sI = 2

Overlapped call-cliques ←→ cover more edges around BSLB ←→ combination of call-cliques






BS1






Quality of call-cliques depends on the demand

How to find good call-cliques?Demand concentrated in one nodeExample: node (3, 2)

dsI = 4

BS

(a) call-clique K1

dsI = 4

BS

(b) call-clique K2

(repeated 2x)

dsI = 4

BS

(c) call-clique K3.

Figure: Example with d sI = 4 and the demand is concentrated in node

(3, 2). Four call-cliques are needed to obtain a tight lower bound of114 b((3, 2)) which is higher than 5

2b((3, 2)).P. Reyes Data Gathering in Radio Networks 104/119



Relationship with Duality

Duality of Round Weighting [KMP08]

finding a metric m : E → R+

maximizing the total distance that the traffic needs to travel(W =

∑

v∈V dm(BS, v)b(v))

Constraint: maximum length of a round is 1:∑

e∈R m(e) ≤ 1

=12

+12

1

1 111

11/2

1/2




Next Result...

Grid

BS in the corner

uniform demand

Result

Optimal Round Weighting Protocol

Methodology

LB: What are the good call-cliques?

UB: What is the good protocol to route?




Lower Bound for Grids

Case d sI odd

v∗(0, k)

(k, 0)BS

l

dsI2

m

Figure: Call-clique Kmax for d sI odd with BS at the corner. In this

scheme, d sI = 9. The call-clique K0 consists in all the wide edges.




Lower Bound for Grids

Case d sI even

v∗

(k, 0)

(0, k)

BS

l

dsI2

m

Figure: Two overlapped cliques for d sI even with BS at the corner. In this

scheme, d sI = 8.




Upper Bound for Grids: Protocol

Optimal for uniform demand

Demand is independently routed by means of cycles

except some critical nodes

Routing is combined with other demanding nodes




Routing Protocol

dsI − 1

BS

shortest path

two cycles

v∗

one cycle

one cycle

Figure: v∗ = (⌈

d2

⌉

,⌈

d2

⌉

)




Routing in grids

1 cycle

dsI − 1

BS

shortest path

two cycles

v∗

one cycle

one cycle

Figure: v∗ = (⌈

d2

⌉

,⌈

d2

⌉

)




Routing in grids

1 cycle

combiningnodes

dsI − 1

BS

shortest path

two cycles

v∗

one cycle

one cycle

Figure: v∗ = (⌈

d2

⌉

,⌈

d2

⌉

)




Routing in grids

1 cycle

combiningnodes

two cyles

dsI − 1

BS

shortest path

two cycles

v∗

one cycle

one cycle

Figure: v∗ = (⌈

d2

⌉

,⌈

d2

⌉

)




Conclusions

Symmetrical Interference, d sI ≥ dT

Idea of Call-Clique

Lower Bounds for general demand using

One call-clique ←→ d sI odd

Multiple call-cliques ←→ d sI even

Optimal solution for uniform demand in grids with

BS in the middle of the gridBS in the corner of the grid

General demand in grid

Critical nodes

joint work with J.-C. Bermond, C. Gomes and H. Rivano


Main Results

Main Results

Interference Demand buffer problem topology result

asymmetric dI , dT = 2, 3, 5 unitary buffer MTG path BS at end optimalasymmetric dI , dT unitary buffer MTG path 1-approxsymmetric ds

I = dT = 1 general no-buffer MTG grid +1-approx,distributed+2-approx

symmetric dsI = dT = 1 general – RWP 2-connected near optimal

(routing)symmetric dT = 1, any ds

I uniform – RWP grid optimalgeneral – RWP grid approx

Table: Results of this thesis related to MTG and RWP.


Main Results

Summarizing...

Problems

Gathering

Round Weighting

Future Work

Gathering is polynomial in the path ? (BS in an end-vertex)

Data Gathering in Sensor Networks: Complexity in grids ?

online version?

RW is NP-Hard in grids?

Thanks a lot - Merci beaucoup - Muchas Gracias


Main Results

Data Gathering in Radio Networks

Patricio Reyes

Advisors: Jean-Claude Bermond - Herve RivanoMASCOTTE Project - INRIA/I3S(CNRS-UNSA)

GTEC – Universidad de CorunaOctober 27, 2009


Incremental Protocols in the pathMinimum Data Gathering in Sensor NetworksRW in the Primary Node Interference Model

IntroductionResults

Table of Contents

7 Incremental Protocols in the pathIntroductionResults

8 Minimum Data Gathering in Sensor NetworksIntroduction

9 Round Weighting in the Primary Node Interference ModelResults



IntroductionResults

Motivation

Gathering in the path

BS in an end-vertex

Uniform demand

Is (uniform) gathering in the path polynomial?

Optimal solutions for dT = 1 anddI = 1, 2, 3, 4 [BCY06, BCY09]



IntroductionResults

Example of Gathering in the path

1

2

3

4

5

6

7

21BS = 0 3 4 5

21BS = 0 3 4 5

21BS = 0 3 4 5

21BS = 0 3 4 5

21BS = 0 3 4 5

21BS = 0 3 4 5

8

21BS = 0 3 4 5

21BS = 0 3 4 5

921BS = 0 3 4 5

6

6

6

6

6

6

6

6

6

10

11

12

13

14

15

16

1821BS = 0 3 4 5

1721BS = 0 3 4 5

21BS = 0 3 4 5

21BS = 0 3 4 5

21BS = 0 3 4 5

21BS = 0 3 4 5

21BS = 0 3 4 5

21BS = 0 3 4 5

21BS = 0 3 4 5

6

6

6

6

6

6

6

6

6

Figure: A protocol for gathering on a path with 7 nodes whendI = 2, dT = 1. Nodes from 1 to 6 have one message each. The protocolgathers all the messages into BS in 18 time-steps.



IntroductionResults

Idea of the proof

Lemma

Given a simple protocol A for Pn → protocol B for Pn+1?

For dT > 0 and dI = pdT + q

|A| = |B|+

{

p + 1p + 2



IntroductionResults

Example incremental protocol

1 2 3 4 5 6 7 8 9 10 11 12 13 14t = 01 2 3 4 5 6 7 8 9 10 11 12 13t = 0

12345678910111213141516171819202122232425262728

10

10

11

11

11

11

12

12

1

2

3

4

4

5

5

6

6

7

7

7

8

9

10

9

8

10

12

12

9

8

13

13

13

13

13

1234567891011121314151617181920212223242526

10

10

11

11

11

11

12

12

1

2

3

4

4

5

5

6

6

7

7

7

8

9

10

9

8

10

12

12

9

8

13

13

13

13

13 14

14

14

14

14

Figure: Incremental Protocol for P15 starting from P14. dI = 4, dT = 3.p = 1



IntroductionResults

Conclusions

BS in an end-vertex

Optimal solutions for

dT = (1), 2, 3, 5, for any dI ≥ dT

dT = 4, for dI = pdT or dI = pdT + 2

BS in an arbitrary vertex

1+-approximation



Introduction

Table of Contents






Introduction

+1-approx Algorithm

(i) it broadcasts the messages without interferences, sending thelast msg vertically

(iii) si ∈ {mi−1,mi ,mi+1}, i < Mand sM ∈ {mM−1,mM}

t · · · i · · · M − 2mi−1, mi , mi+1 mM−3, mM−2

Use +2-approx but fixing cases si ∈ {mi−2,mi+2}

+2-approx, exceptspecial case: sM−2 = mM−3



Introduction

2

m7

m5 m3 m2

m6 m4 m1

m8

5 3 1

46

BS

Figure: Before msgs m7 and m8

t 1 2 3 4 5 6 7 8m1 m2 m4 m3 m6 m5 − −m1 m2 m4 m3 m5 m7 m6 m8

m1 m2 m3 m5 m4 m7 m6 m8

m1 m3 m2 m5 m4 m7 m6 m8

Properties (i) and (iii) !!



Introduction

2

8

7

6

m7

m5 m3 m2

m6 m4 m1

m8

3 15

4

BS

Figure: non valid sched, s4, s5 interfer


m1 m2 m3 m5 m4 m7 m6 m8

m1 m3 m2 m5 m4 m7 m6 m8




Introduction

2

8

5

4

7

6

m7

m5 m3 m2

m6 m4 m1

m8

1

3

BS

Figure: non valid sched, s2, s3 interfer


m1 m2 m3 m5 m4 m7 m6 m8

m1 m3 m2 m5 m4 m7 m6 m8




Introduction

m7

m8

m5 m3

m6 m4 m1

4

7

8

2

m2

3 15

BS

6

Figure: Final valid schedule


m1 m2 m3 m5 m4 m7 m6 m8

m1 m3 m2 m5 m4 m7 m6 m8

Properties (i) and (iii) !! ↔ +1-approx



Introduction

2

m7

m5 m3 m2

m6 m4 m1

m8

5 3 1

46

BS

(a) Before msgs m7 and m8

2

8

7

6

m7

m5 m3 m2

m6 m4 m1

m8

3 15

4

BS

(b) non valid sched, s4, s5 in-terfer

2

8

5

4

7

6

m7

m5 m3 m2

m6 m4 m1

m8

1

3

BS

(c) non valid sched, s2, s3 in-terfere

m7

m8

m5 m3

m6 m4 m1

4

7

8

2

m2

3 15

BS

6

(d) Final valid sched.

t 1 2 3 4 5 6 7 8m1 m2 m4 m3 m6 m5 − −

m1 m2 m4 m3 m5 m7 m6 m8P. Reyes Data Gathering in Radio Networks 130/119


Introduction

2

m7

m5 m3 m2

m6 m4 m1

m8

5 3 1

46

BS

(e) Scheduling before including messages m7

and m8

2

8

7

6

m7

m5 m3 m2

m6 m4 m1

m8

3 15

4

BS P. Reyes Data Gathering in Radio Networks 131/119


Introduction

Complexity

M number of messages

+2 Approximation

O(M)



Introduction

Complexity

ij

mj

mj

mj

mj

i + 2

××

× ×

××

mi−1

× ×mj

· · ·

××

× ×

mi+2

mi+2

mi+2

mi+2

mi+2

mi+2

mi+2

× ×

ml−1mi+2

l l + 2

ml+2

ml+2

ml+2

ml+2

Time Complexity of +1-approx: O(M)



Results

Table of Contents






Results

Model

Interference

Symmetrical Interference

dT = 1

d sI = 1←→ each round is a matching

Demand

General demand



Results

Idea

Demand from v to BS.1. Using a path

1

R1

R2

w(R1) = w(R2) = 1, then W = w(R1) + w(R2) = 2

Question: Is it possible to route 1 unit of demand with costW = 1?



Results

Using an even cycle

2

11

1

2 2BS

w(R1) = w(R2) = 1/2, total weight W = 1



Results

Using an odd cycle

2,6

2,4,5 1,3,6

4,51,3

BS

w(R1) = w(R2) = w(R3) = w(R4) = w(R5) = w(R6) = 1/5total weight W = 6/5.

Question: In a general (odd) cycle?



Results

Odd Cycle

BS

p

q

Result: routing via an odd cycle with p > q can be made withcost W = 2p

2p−1

Best solution?

dual formulationoptimal dual solution W = 2p

2p−1



Results

Cycle with ears

BS

p

q

t

Result: Optimal routing via an odd cycle with an ear can bemade with cost W = 2pt

2pt−1



Results

Two connected graph

BS2

1

2

Result: There exists a solution with cost B + 1/5|2bmax − B |,with bmax = maxv b(v)



Results

Two connected graph

BS2

1

2




Results

Two connected graph

BS2

0

1




Results

Two connected graph

BS

0

0

1




Results

Conclusions

Symmetrical Interference, d sI = dT = 1

Routing via odd cycles: optimal solution

Routing via cycles with ears: optimal solutions

Upper Bound for the general RW problem in a 2-connectedgraph



Results

J.-C. Bermond, R. Correa, and M.-L. Yu.

Gathering algorithms on paths under interference constraints.In 6th Conference on Algorithms and Complexity, volume 3998 of Lecture Notes in Computer Science,pages 115–126, Roma, Italy, May 2006.

J.-C. Bermond, R. Correa, and M.-L. Yu.

Optimal gathering protocols on paths under interference constraints.Discrete Mathematics, 2009.To appear.

J.-C. Bermond, J. Galtier, R. Klasing, N. Morales, and S. Perennes.

Hardness and approximation of gathering in static radio networks.Parallel Processing Letters, 16(2):165–183, June 2006.

J-C. Bermond, L. Gargano, and A.A. Rescigno.

Gathering with minimum delay in tree sensor networks.In SIROCCO 2008, volume 5058 of Lecture Notes in Computer Science, pages 262–276, Villars-sur-Ollon,Switzerland, June 2008. Springer-Verlag.

J.-C. Bermond and J. Peters.

Efficient gathering in radio grids with interference.In Septiemes Rencontres Francophones sur les Aspects Algorithmiques des Telecommunications(AlgoTel’05), pages 103–106, Presqu’ıle de Giens, May 2005.

J.-C. Bermond and J. Peters.

Optimal gathering in radio grids with interference.2009.manuscript.

J.-C. Bermond and M.-L. Yu.

Optimal gathering algorithms in multi-hop radio tree networks with interferences.



Results

In ADHOC-NOW 2008, volume 5198 of Lecture Notes in Computer Science, pages 204–217.Springer-Verlag, September 2008.Full version to appear in AdHoc & Sensor Wireless Networks 2009.

L. Gargano and A. A. Rescigno.

Optimally fast data gathering in sensor networks.In Rastislav Kralovic and Pawel Urzyczyn, editors, MFCS, volume 4162 of Lecture Notes in ComputerScience, pages 399–411. Springer, 2006.

L. Gargano and A. A. Rescigno.

Collision-free path coloring with application to minimum-delay gathering in sensor networks.Discrete Applied Mathematics, 157(8):1858 – 1872, 2009.

R. Klasing, Z. Lotker, A. Navarra, and S. Perennes.

From balls and bins to points and vertices.In Proceedings of the 16th Annual International Symposium on Algorithms and Computation (ISAAC 2005),volume 3827 of Lecture Notes in Computer Science, pages 757–766. Springer Verlag, December 2005.

R. Klasing, N. Morales, and S. Perennes.

On the complexity of bandwidth allocation in radio networks.Theoretical Computer Science, 406(3):225–239, October 2008.

P. Korteweg.

Online gathering algorithms for wireless networks.PhD thesis, Technische Universiteit Eindhoven, 2008.


Data Gathering in Wireless Networks

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