Minimum Delay Data Gathering in Radio Networks Jean-Claude Bermond, - - PowerPoint PPT Presentation

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Minimum Delay Data Gathering in Radio Networks Jean-Claude Bermond, - - PowerPoint PPT Presentation

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Minimum Delay Data Gathering Minimum Delay Data Gathering in Radio Networks Jean-Claude Bermond, Nicolas Nisse, Patricio Reyes , Herv e Rivano Projet MASCOTTE - INRIA/I3S(CNRS-UNSA) Algotel. June 17, 2009 Algotel 09 Minimum Delay Data

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Minimum Delay Data Gathering

Minimum Delay Data Gathering in Radio Networks

Jean-Claude Bermond, Nicolas Nisse, Patricio Reyes, Herv´ e Rivano

Projet MASCOTTE - INRIA/I3S(CNRS-UNSA)

  • Algotel. June 17, 2009

Algotel 09

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Minimum Delay Data Gathering

Motivation

Sensor Network

Algotel 09

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SLIDE 3

Minimum Delay Data Gathering

Motivation

Sensor Network

Algotel 09

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SLIDE 4

Minimum Delay Data Gathering

Motivation

Sensor Network

Algotel 09

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SLIDE 5

Minimum Delay Data Gathering

Motivation

Sensor Network

Algotel 09

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SLIDE 6

Minimum Delay Data Gathering

Motivation

Sensor Network

? ?

Algotel 09

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Minimum Delay Data Gathering

Gathering Problem

The nodes have messages There is a special node called BS or gateway. Messages must be collected by the BS. Avoid interferences Time:

synchronous discrete: time-slots t = 1, 2, 3, . . .

Goal: Minimize the gathering time → # time-slots

Algotel 09

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Minimum Delay Data Gathering

Transmission & Interference model

Binary models

the sender sends a msg, then the recevier:

1

receive the (entire) message

2

no info is received

Transmission distance

u is able to transmit to v if dG(u, v) ≤ 1

Call u → v

1 time-slot 1 message

Round: set of non interfering calls ↔ simultaneous calls Idea: Good rounds ↔ time-slot

Algotel 09

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Minimum Delay Data Gathering

Motivation

Revah and Segal’07 Sensor Networks square grid with BS in (0, 0) set of M messages Interference ↔ matching Node cannot both receive and send at the same time-slot Node cannot receive more than one message at the same time-slot new constraint: no-buffering ↔ hot-potato routing

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

R&S Algo: *1.5-approximation

Algotel 09

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Minimum Delay Data Gathering

Gathering and Personalized Broadcasting

Gathering t = 0 BS

m1 m2 m3

Algotel 09

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Minimum Delay Data Gathering

Gathering and Personalized Broadcasting

Gathering t = 1 BS

m1 m2 m3

Algotel 09

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Minimum Delay Data Gathering

Gathering and Personalized Broadcasting

Gathering t = 2 BS

m1 m2 m3

Algotel 09

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Minimum Delay Data Gathering

Gathering and Personalized Broadcasting

Gathering t = 3 BS

m1 m2 m3

Algotel 09

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Minimum Delay Data Gathering

Gathering and Personalized Broadcasting

Gathering t = 4 BS

m1 m2 m3

Algotel 09

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Minimum Delay Data Gathering

Gathering and Personalized Broadcasting

Gathering t = 5 BS

m1 m2 m3

Algotel 09

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Minimum Delay Data Gathering

Gathering and Personalized Broadcasting

Personalized Broadcasting t = 0 BS

m1 m2 m3

Algotel 09

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Minimum Delay Data Gathering

Gathering and Personalized Broadcasting

Personalized Broadcasting t = 1 BS

m1 m2 m3

Algotel 09

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Minimum Delay Data Gathering

Gathering and Personalized Broadcasting

Personalized Broadcasting t = 2 BS

m1 m2 m3

Algotel 09

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Minimum Delay Data Gathering

Gathering and Personalized Broadcasting

Personalized Broadcasting t = 3 BS

m1 m2 m3

Algotel 09

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Minimum Delay Data Gathering

Gathering and Personalized Broadcasting

Personalized Broadcasting t = 4 BS

m1 m2 m3

Algotel 09

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Minimum Delay Data Gathering

Gathering and Personalized Broadcasting

Personalized Broadcasting t = 5 BS

m1 m2 m3

Algotel 09

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Minimum Delay Data Gathering

Gathering and Personalized Broadcasting

Personalized Broadcasting Sequence S = (m1, m2, m3) BS

m1 m2 m3 1 3 2

Algotel 09

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Minimum Delay Data Gathering

Interference

t = 0 m’ BS m

Algotel 09

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Minimum Delay Data Gathering

Interference

t = 1 m’ BS m

Algotel 09

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Minimum Delay Data Gathering

Interference

t = 2 m’ BS m

Algotel 09

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Minimum Delay Data Gathering

Interference

t = 3 m’ BS m

Algotel 09

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Minimum Delay Data Gathering

Interference

t = 4 m’ BS m

Algotel 09

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Minimum Delay Data Gathering

Interference

Two consecutive msgs ↔ disjoints paths m’ BS m

Algotel 09

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Minimum Delay Data Gathering

Methodology

BS sends 1 msg per time-slot t 1 2 3 m1 m2 m3 Only HV and VH paths t 1 2 3 VH HV VH BS

m1 m2 m3 1 3 2

Algotel 09

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Minimum Delay Data Gathering

Methodology

gathering ↔ personalized broadcasting BS sends 1 msg at each time-slot two consecutive msgs ↔ disjoint paths M = {m1, . . . , mM} such that d(m1) ≥ d(m2) ≥ d(m3) ≥ . . . ≥ d(mM), with d(v) = dG(dest(mi), BS) Goal: provide BS with a good delivery order S = (s1, . . . , sM), si ∈ M, si = sj

Algotel 09

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Minimum Delay Data Gathering

Lower Bound

without interferences

  • rder (m1, . . . , mM)

LB = maxi≤M d(mi) + i − 1 LB is not always optimal

BS m2 m3 m1 1 2 3

(m1, m2, m3), LB = 3

BS m2 m3 m1 1 2 3

(m1, m3, m2), 4 time-slots

Algotel 09

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Minimum Delay Data Gathering

Next results

Recall: LB = maxi≤M d(mi) + i − 1 with (m1, . . . , mM) +2-approximation algorithm

Protocol which attains the LB + 2 i-th msg (distance order) ↔ time-slot i ± 2

+1-approximation algorithm

Protocol which attains the LB + 1 i-th msg (distance order) ↔ time-slot i ± 1

Algotel 09

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Minimum Delay Data Gathering

+2 Approx Algorithm

Recall: (m1, . . . , mM) ordered by distance Induction including pair of nodes. (m1, m2) (s1, s2) Sequence (s1, . . . , sM−2), satisfying the following:

(i) it broadcasts the messages without interferences, sending (arbitrarily) the last msg vertically (VH) (ii) si ∈ {mi−2, mi−1, mi, mi+1, mi+2}, i < M and sM−2 ∈ {mM−3, mM−2}

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

Two new messages {mM−1, mM} must be sent.

BS sM−2

Algotel 09

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Minimum Delay Data Gathering

+2-approx algorithm

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

BS r p q

M − 2 M − 1 M

t · · · M − 2 M − 1 M p → mM−2, mM−3 q → mM−1/mM r → mM/mM−1 Properties (i) and (ii) !!

Algotel 09

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Minimum Delay Data Gathering

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

BS

M − 2

mM−1 p mM

(a) before

BS mM

M − 2

mM−1 p

M − 1 M

(b) after

t · · · M − 3 M − 2 M − 1 M sM−3 p − − sM−3 mM−1 p mM Properties (i) and (ii) !!

Algotel 09

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Minimum Delay Data Gathering

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

BS

M − 2

mM−1 p mM

(c) before

BS mM

M − 2

mM−1 p

M − 1 M

(d) after

t · · · M − 3 M − 2 M − 1 M mM−2 mM−3 − − mM−2 mM−1 mM−3 mM Properties (i) and (ii) !! ↔ +2-approx

Algotel 09

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Minimum Delay Data Gathering

+1-approx Algorithm

(i) it broadcasts the messages without interferences, sending the last msg vertically (iii) si ∈ {mi−1, mi, mi+1}, i < M and sM ∈ {mM−1, mM}

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

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

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

Algotel 09

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Minimum Delay Data Gathering

2

m7 m5 m3 m2 m6 m4 m1 m8 4 6 5 3 BS 1

Figure: Before msgs m7 and m8

t 1 2 3 4 5 6 7 8 m1 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) !!

Algotel 09

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Minimum Delay Data Gathering

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

Figure: non valid sched, s4, s5 interfer

t 1 2 3 4 5 6 7 8 m1 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) !!

Algotel 09

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Minimum Delay Data Gathering

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

Figure: non valid sched, s2, s3 interfer

t 1 2 3 4 5 6 7 8 m1 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) !!

Algotel 09

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Minimum Delay Data Gathering

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

Figure: Final valid schedule

t 1 2 3 4 5 6 7 8 m1 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) !! ↔ +1-approx

Algotel 09

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Minimum Delay Data Gathering

Complexity

M number of messages +2 Approximation O(M) +1 Approximation O(M) ← − we don’t have to change ALL the sequence

Algotel 09

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Minimum Delay Data Gathering

Conclusions

Results +2-approx distributed +2-approx +1-approx (Revah and Segal 07: *1.5-approx) no-buffering Further work

  • nline version?

different interference models

Algotel 09

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Minimum Delay Data Gathering

Minimum Delay Data Gathering in Radio Networks

Jean-Claude Bermond, Nicolas Nisse, Patricio Reyes, Herv´ e Rivano

Projet MASCOTTE - INRIA/I3S(CNRS-UNSA)

  • Algotel. June 17, 2009

Algotel 09

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Minimum Delay Data Gathering

Complexity

i j mj mj mj mj i + 2 × × × × × × mi−1 × × mj · · · × × × × mi+2 mi+2 mi+2 mi+2 mi+2 mi+2 mi+2 × × ml−1 mi+2 l l + 2 ml+2 ml+2 ml+2 ml+2

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

Algotel 09