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 Gathering Motivation Sensor Network Algotel 09
Minimum Delay Data Gathering Motivation Sensor Network Algotel 09
Minimum Delay Data Gathering Motivation Sensor Network Algotel 09
Minimum Delay Data Gathering Motivation Sensor Network Algotel 09
Minimum Delay Data Gathering Motivation Sensor Network ? ? Algotel 09
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
Minimum Delay Data Gathering Transmission & Interference model Binary models the sender sends a msg, then the recevier: receive the (entire) message 1 no info is received 2 Transmission distance u is able to transmit to v if d G ( 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
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
Minimum Delay Data Gathering Gathering and Personalized Broadcasting Gathering t = 0 m 1 m 2 m 3 BS Algotel 09
Minimum Delay Data Gathering Gathering and Personalized Broadcasting Gathering t = 1 m 1 m 2 m 3 BS Algotel 09
Minimum Delay Data Gathering Gathering and Personalized Broadcasting Gathering t = 2 m 1 m 2 m 3 BS Algotel 09
Minimum Delay Data Gathering Gathering and Personalized Broadcasting Gathering t = 3 m 1 m 2 m 3 BS Algotel 09
Minimum Delay Data Gathering Gathering and Personalized Broadcasting Gathering t = 4 m 1 m 2 m 3 BS Algotel 09
Minimum Delay Data Gathering Gathering and Personalized Broadcasting Gathering t = 5 m 1 m 2 m 3 BS Algotel 09
Minimum Delay Data Gathering Gathering and Personalized Broadcasting Personalized Broadcasting t = 0 m 1 m 2 m 3 BS Algotel 09
Minimum Delay Data Gathering Gathering and Personalized Broadcasting Personalized Broadcasting t = 1 m 1 m 2 m 3 BS Algotel 09
Minimum Delay Data Gathering Gathering and Personalized Broadcasting Personalized Broadcasting t = 2 m 1 m 2 m 3 BS Algotel 09
Minimum Delay Data Gathering Gathering and Personalized Broadcasting Personalized Broadcasting t = 3 m 1 m 2 m 3 BS Algotel 09
Minimum Delay Data Gathering Gathering and Personalized Broadcasting Personalized Broadcasting t = 4 m 1 m 2 m 3 BS Algotel 09
Minimum Delay Data Gathering Gathering and Personalized Broadcasting Personalized Broadcasting t = 5 m 1 m 2 m 3 BS Algotel 09
Minimum Delay Data Gathering Gathering and Personalized Broadcasting Personalized Broadcasting Sequence S = ( m 1 , m 2 , m 3 ) m 1 1 m 2 m 3 3 2 BS Algotel 09
Minimum Delay Data Gathering Interference t = 0 m’ m BS Algotel 09
Minimum Delay Data Gathering Interference t = 1 m’ m BS Algotel 09
Minimum Delay Data Gathering Interference t = 2 m’ m BS Algotel 09
Minimum Delay Data Gathering Interference t = 3 m’ m BS Algotel 09
Minimum Delay Data Gathering Interference t = 4 m’ m BS Algotel 09
Minimum Delay Data Gathering Interference Two consecutive msgs ↔ disjoints paths m’ m BS Algotel 09
Minimum Delay Data Gathering Methodology BS sends 1 msg per time-slot t 1 2 3 m 1 m 2 m 3 m 1 1 m 2 Only HV and VH paths m 3 t 1 2 3 3 2 VH HV VH BS Algotel 09
Minimum Delay Data Gathering Methodology gathering ↔ personalized broadcasting BS sends 1 msg at each time-slot two consecutive msgs ↔ disjoint paths M = { m 1 , . . . , m M } such that d ( m 1 ) ≥ d ( m 2 ) ≥ d ( m 3 ) ≥ . . . ≥ d ( m M ), with d ( v ) = d G ( dest ( m i ) , BS) Goal: provide BS with a good delivery order S = ( s 1 , . . . , s M ), s i ∈ M , s i � = s j Algotel 09
Minimum Delay Data Gathering Lower Bound without interferences order ( m 1 , . . . , m M ) LB = max i ≤ M d ( m i ) + i − 1 LB is not always optimal m 1 m 1 1 1 3 m 2 2 m 2 2 m 3 3 BS m 3 BS ( m 1 , m 3 , m 2 ), ( m 1 , m 2 , m 3 ), LB = 3 4 time-slots Algotel 09
Minimum Delay Data Gathering Next results Recall: LB = max i ≤ M d ( m i ) + i − 1 with ( m 1 , . . . , m M ) +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
Minimum Delay Data Gathering +2 Approx Algorithm Recall: ( m 1 , . . . , m M ) ordered by distance Induction including pair of nodes. ( m 1 , m 2 ) � ( s 1 , s 2 ) Sequence ( s 1 , . . . , s M − 2 ), satisfying the following: (i) it broadcasts the messages without interferences, sending (arbitrarily) the last msg vertically (VH) (ii) s i ∈ { m i − 2 , m i − 1 , m i , m i +1 , m i +2 } , i < M and s M − 2 ∈ { m M − 3 , m M − 2 } t i M − 2 · · · · · · m i − 2 , m i − 1 , m i m M − 3 , m M − 2 m i +1 , m i +2 Two new messages { m M − 1 , m M } must be sent. s M − 2 BS Algotel 09
Minimum Delay Data Gathering +2-approx algorithm Notation: q , r ∈ { m M − 1 , m M } , q lower than r , and p = s M − 2 Case 1 q lower than p r M p M − 2 M − 1 q BS t · · · M − 2 M − 1 M p → m M − 2 , m M − 3 q → m M − 1 / m M r → m M / m M − 1 Properties (i) and (ii) !! Algotel 09
Minimum Delay Data Gathering Notation: q , r ∈ { m M − 1 , m M } , q lower than r , and p = s M − 2 Case 2 q higher than p m M − 1 m M − 1 M − 2 m M m M M p p M − 2 M − 1 BS BS (a) before (b) after t · · · M − 3 M − 2 M − 1 M s M − 3 p − − s M − 3 m M − 1 p m M Properties (i) and (ii) !! Algotel 09
Minimum Delay Data Gathering Notation: q , r ∈ { m M − 1 , m M } , q lower than r , and p = s M − 2 Case 2 q higher than p m M − 1 m M − 1 M − 2 m M m M M p p M − 2 M − 1 BS BS (c) before (d) after t · · · M − 3 M − 2 M − 1 M m M − 2 m M − 3 − − m M − 2 m M − 1 m M − 3 m M Properties (i) and (ii) !! ↔ +2-approx Algotel 09
Minimum Delay Data Gathering +1-approx Algorithm (i) it broadcasts the messages without interferences, sending the last msg vertically (iii) s i ∈ { m i − 1 , m i , m i +1 } , i < M and s M ∈ { m M − 1 , m M } · · · · · · M − 2 t i m i − 1 , m i , m i +1 m M − 3 , m M − 2 Use +2-approx but fixing cases s i ∈ { m i − 2 , m i +2 } +2-approx, except special case: s M − 2 = m M − 3 Algotel 09
Minimum Delay Data Gathering m 7 m 8 m 5 m 3 m 2 2 4 m 6 m 4 m 1 6 5 3 1 BS Figure: Before msgs m 7 and m 8 t 1 2 3 4 5 6 7 8 m 1 m 2 m 4 m 3 m 6 m 5 − − m 1 m 2 m 4 m 3 m 5 m 7 m 6 m 8 m 1 m 2 m 3 m 5 m 4 m 7 m 6 m 8 m 1 m 3 m 2 m 5 m 4 m 7 m 6 m 8 Properties (i) and (iii) !! Algotel 09
Minimum Delay Data Gathering m 7 6 m 8 8 m 5 m 3 m 2 2 4 m 6 m 4 m 1 5 7 3 1 BS Figure: non valid sched, s 4 , s 5 interfer t 1 2 3 4 5 6 7 8 m 1 m 2 m 4 m 3 m 6 m 5 − − m 1 m 2 m 4 m 3 m 5 m 7 m 6 m 8 m 1 m 2 m 3 m 5 m 4 m 7 m 6 m 8 m 1 m 3 m 2 m 5 m 4 m 7 m 6 m 8 Properties (i) and (iii) !! Algotel 09
Minimum Delay Data Gathering m 7 6 m 8 8 m 5 m 3 m 2 2 4 3 m 6 m 4 m 1 5 7 1 BS Figure: non valid sched, s 2 , s 3 interfer t 1 2 3 4 5 6 7 8 m 1 m 2 m 4 m 3 m 6 m 5 − − m 1 m 2 m 4 m 3 m 5 m 7 m 6 m 8 m 1 m 2 m 3 m 5 m 4 m 7 m 6 m 8 m 1 m 3 m 2 m 5 m 4 m 7 m 6 m 8 Properties (i) and (iii) !! Algotel 09
Minimum Delay Data Gathering m 7 6 m 8 8 m 5 m 3 m 2 4 2 m 6 m 4 m 1 7 3 5 1 BS Figure: Final valid schedule t 1 2 3 4 5 6 7 8 m 1 m 2 m 4 m 3 m 6 m 5 − − m 1 m 2 m 4 m 3 m 5 m 7 m 6 m 8 m 1 m 2 m 3 m 5 m 4 m 7 m 6 m 8 m 1 m 3 m 2 m 5 m 4 m 7 m 6 m 8 Properties (i) and (iii) !! ↔ +1-approx Algotel 09
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
Minimum Delay Data Gathering Conclusions Results +2-approx distributed +2-approx +1-approx (Revah and Segal 07: *1.5-approx) no-buffering Further work online version? different interference models Algotel 09
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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