Sensor Network Gossiping or How to Break the Broadcast Lower Bound
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1 Sensor Network Gossiping or How to Break the Broadcast Lower Bound Martín Farach-Colton 1 Miguel A. Mosteiro 1,2 1 Department of Computer Science Rutgers University 2 LADyR (Distributed Algorithms and Networks Lab) Universidad Rey Juan Carlos ISAAC 2007 M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 1/11
2 Introduction Information Dissemination in Radio Networks Radio Network = abstraction of a radio communication network k nodes hold a piece of information to diseminate. k = 1 Broadcast [BGI 92,KM 98] k = n: Gossiping [CGLP 01,LP 02] k arbitrary: k-selection [K 05] We study Gossiping in Sensor Networks M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 2/11
3 Introduction Information Dissemination in Radio Networks Radio Network = abstraction of a radio communication network k nodes hold a piece of information to diseminate. k = 1 Broadcast [BGI 92,KM 98] k = n: Gossiping [CGLP 01,LP 02] k arbitrary: k-selection [K 05] We study Gossiping in Sensor Networks M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 2/11
4 Introduction Information Dissemination in Radio Networks Radio Network = abstraction of a radio communication network k nodes hold a piece of information to diseminate. k = 1 Broadcast [BGI 92,KM 98] k = n: Gossiping [CGLP 01,LP 02] k arbitrary: k-selection [K 05] We study Gossiping in Sensor Networks M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 2/11
5 Introduction Information Dissemination in Radio Networks Radio Network = abstraction of a radio communication network k nodes hold a piece of information to diseminate. k = 1 Broadcast [BGI 92,KM 98] k = n: Gossiping [CGLP 01,LP 02] k arbitrary: k-selection [K 05] We study Gossiping in Sensor Networks M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 2/11
6 Introduction Information Dissemination in Radio Networks Radio Network = abstraction of a radio communication network k nodes hold a piece of information to diseminate. k = 1 Broadcast [BGI 92,KM 98] k = n: Gossiping [CGLP 01,LP 02] k arbitrary: k-selection [K 05] We study Gossiping in Sensor Networks M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 2/11
7 Introduction Information Dissemination in Radio Networks Radio Network = abstraction of a radio communication network k nodes hold a piece of information to diseminate. k = 1 Broadcast [BGI 92,KM 98] k = n: Gossiping [CGLP 01,LP 02] k arbitrary: k-selection [K 05] We study Gossiping in Sensor Networks M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 2/11
8 Introduction Information Dissemination in Radio Networks Radio Network = abstraction of a radio communication network k nodes hold a piece of information to diseminate. k = 1 Broadcast [BGI 92,KM 98] k = n: Gossiping [CGLP 01,LP 02] k arbitrary: k-selection [K 05] We study Gossiping in Sensor Networks M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 2/11
9 Introduction Information Dissemination in Radio Networks Radio Network = abstraction of a radio communication network k nodes hold a piece of information to diseminate. k = 1 Broadcast [BGI 92,KM 98] k = n: Gossiping [CGLP 01,LP 02] k arbitrary: k-selection [K 05] We study Gossiping in Sensor Networks M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 2/11
10 Introduction Information Dissemination in Radio Networks Radio Network = abstraction of a radio communication network k nodes hold a piece of information to diseminate. k = 1 Broadcast [BGI 92,KM 98] k = n: Gossiping [CGLP 01,LP 02] k arbitrary: k-selection [K 05] We study Gossiping in Sensor Networks M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 2/11
11 A Sensor Network Introduction Sensor Node Capabilities processing sensing communication Sensor Node Limitations range memory life cycle M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 3/11
12 A Sensor Network Introduction Sensor Node Capabilities processing sensing communication Sensor Node Limitations range memory life cycle M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 3/11
13 A Sensor Network Introduction Sensor Node Capabilities processing sensing communication Sensor Node Limitations range memory life cycle M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 3/11
14 A Sensor Network Introduction Sensor Node Capabilities processing sensing communication Sensor Node Limitations range memory life cycle M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 3/11
15 A Sensor Network Introduction Sensor Node Capabilities processing sensing communication Sensor Node Limitations range memory life cycle M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 3/11
16 A Sensor Network Introduction Sensor Node Capabilities processing sensing communication Sensor Node Limitations range memory life cycle M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 3/11
17 Related Work Upper Bounds Introduction Symmetric Radio Networks: BII 93 O(n log 2 n) expected (BFS tree). CGLP 01 same, w.h.p. Asymmetric connected Radio Networks: CGR 01 O(n log 3 nlog(n/ɛ)) with prob 1 ɛ and O(n log 4 n) expected (limited broadcast doubles message copies per phase). LP 02 same, reduced by a log factor (limited broadcast is randomized). CR 03 O(n log 2 n) w.h.p. (linear randomized broadcast by special distribution). CGR 00 O(n 3/2 log 2 n) (deterministic, selecting sequences). ALL: globally synchronous, and Ω(nm) memory size, all but first: Ω(nm) message size. Sensor Networks R 07 O( nlog n) w.h.p. in RGGs (claimed optimal using KM s lower bound, but includes pre-coloring). M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 4/11
18 Related Work Lower Bounds Introduction Gossiping: CGLP 01 deterministic oblivious (no history): n 2 /2 n/2 + 1 fair (same p trans) protocols: n q n 2 /2, asymmetric network s.t. Ω(q) expected. GP 02 Ω(n 2 ) asymmetric networks Ω(n log n) symmetric networks not embeddable in GG. Broadcast (no preprocessing): BDP 97 Ω(D log n) globally synchronous, nodes know message history. CMS 01 Ω(n log D) symmetric networks, nodes are not synchronized. KP 04 Ω(n 1/4 ), diameter 4. KM 98 Ω(D log(n/d)) expected (best, more on this...) M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 5/11
19 Related Work Broadcast Lower Bound Introduction [KM 98] proved Ω(D log(n/d)) expected, showing a layered structure Crucial assumption:...any other processor is inactive until receiving a message for the first time. Crucial in proof: all layer nodes run same uniform protocol, upon receiving the broadcast message. M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 6/11
20 Related Work Broadcast Lower Bound Introduction [KM 98] proved Ω(D log(n/d)) expected, showing a layered structure Crucial assumption:...any other processor is inactive until receiving a message for the first time. Crucial in proof: all layer nodes run same uniform protocol, upon receiving the broadcast message. M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 6/11
21 Related Work Broadcast Lower Bound Introduction [KM 98] proved Ω(D log(n/d)) expected, showing a layered structure Crucial assumption:...any other processor is inactive until receiving a message for the first time. Crucial in proof: all layer nodes run same uniform protocol, upon receiving the broadcast message. M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 6/11
22 Introduction Node Constraints Model Sensor Networks The Weak Sensor Model [BGI 92, FCFM 05] Local synchronism. Adversarial wake-up schedule. Low-info channel contention: Radio tx on a shared channel. No collision detection. Non-simultaneous rx and tx. Constant memory size. Limited life cycle. Short transmission range. Discrete tx power range. One channel of communication. No position information. Unreliability. tx = transmission. rx = reception. M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 7/11
23 Introduction Node Constraints Model Sensor Networks The Weak Sensor Model [BGI 92, FCFM 05] Local synchronism. Adversarial wake-up schedule. Low-info channel contention: Radio tx on a shared channel. No collision detection. Non-simultaneous rx and tx. Constant memory size. Limited life cycle. Short transmission range. Discrete tx power range. One channel of communication. No position information. Unreliability. tx = transmission. rx = reception. M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 7/11
24 Our Results Introduction Sensor Network: n nodes range of transmission r diameter D max degree nodes only know n. all nodes hold message of size m to disseminate. O(nm) message and memory size. Gossiping algorithm: O( + D) w.h.p. relaxed-wsm-compatible Ω(D) and Ω( ) are lower bounds optimal. Observations: time improvement with no global synchronism (exploits geometry) classical broadcast lower bound of KM can be broken (by pre-processing) M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 8/11
25 Our Results Introduction Sensor Network: n nodes range of transmission r diameter D max degree nodes only know n. all nodes hold message of size m to disseminate. O(nm) message and memory size. Gossiping algorithm: O( + D) w.h.p. relaxed-wsm-compatible Ω(D) and Ω( ) are lower bounds optimal. Observations: time improvement with no global synchronism (exploits geometry) classical broadcast lower bound of KM can be broken (by pre-processing) M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 8/11
26 Our Results Introduction Sensor Network: n nodes range of transmission r diameter D max degree nodes only know n. all nodes hold message of size m to disseminate. O(nm) message and memory size. Gossiping algorithm: O( + D) w.h.p. relaxed-wsm-compatible Ω(D) and Ω( ) are lower bounds optimal. Observations: time improvement with no global synchronism (exploits geometry) classical broadcast lower bound of KM can be broken (by pre-processing) M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 8/11
27 Gossiping Algorithm The Algorithm Partition nodes in masters and slaves every slave is at d ar from some master (0 < a < 1/3) every pair of masters are at d > ar MIS(ar) Every master reserves blocks of time steps for local use master and slaves communicate without collisions within r Coloring(r), using counter (achives local synch. and coll. detection) M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 9/11
28 Gossiping Algorithm The Algorithm Partition nodes in masters and slaves every slave is at d ar from some master (0 < a < 1/3) every pair of masters are at d > ar MIS(ar) Every master reserves blocks of time steps for local use master and slaves communicate without collisions within r Coloring(r), using counter (achives local synch. and coll. detection) M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 9/11
29 Gossiping Algorithm The Algorithm Partition nodes in masters and slaves every slave is at d ar from some master (0 < a < 1/3) every pair of masters are at d > ar MIS(ar) Every master reserves blocks of time steps for local use master and slaves communicate without collisions within r Coloring(r), using counter (achives local synch. and coll. detection) M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 9/11
30 Gossiping Algorithm The Algorithm Partition nodes in masters and slaves every slave is at d ar from some master (0 < a < 1/3) every pair of masters are at d > ar MIS(ar) Every master reserves blocks of time steps for local use master and slaves communicate without collisions within r Coloring(r), using counter (achives local synch. and coll. detection) slave trans. master ack. beacon master trans. b O(1) b O(1) β O(1) b O(1) b O(1) M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 9/11
31 Gossiping Algorithm The Algorithm Partition nodes in masters and slaves every slave is at d ar from some master (0 < a < 1/3) every pair of masters are at d > ar MIS(ar) Every master reserves blocks of time steps for local use master and slaves communicate without collisions within r Coloring(r), using counter (achives local synch. and coll. detection) Every master maintains set of messages received initially set contains own message only slaves pass message to master (using reserved blocks and radius ar) master adds messages to set window back-on/back-off + O(log 2 n) times p trans = 1/ log n Every master disseminates local set (using reserved blocks) masters deterministically pass set to neighboring masters (radius r) masters add messages received from other masters to local set flooding among masters M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 9/11
32 Gossiping Algorithm The Algorithm Partition nodes in masters and slaves every slave is at d ar from some master (0 < a < 1/3) every pair of masters are at d > ar MIS(ar) Every master reserves blocks of time steps for local use master and slaves communicate without collisions within r Coloring(r), using counter (achives local synch. and coll. detection) Every master maintains set of messages received initially set contains own message only slaves pass message to master (using reserved blocks and radius ar) master adds messages to set window back-on/back-off + O(log 2 n) times p trans = 1/ log n Every master disseminates local set (using reserved blocks) masters deterministically pass set to neighboring masters (radius r) masters add messages received from other masters to local set flooding among masters M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 9/11
33 Gossiping Algorithm The Algorithm Partition nodes in masters and slaves every slave is at d ar from some master (0 < a < 1/3) every pair of masters are at d > ar MIS(ar) Every master reserves blocks of time steps for local use master and slaves communicate without collisions within r Coloring(r), using counter (achives local synch. and coll. detection) Every master maintains set of messages received initially set contains own message only slaves pass message to master (using reserved blocks and radius ar) master adds messages to set window back-on/back-off + O(log 2 n) times p trans = 1/ log n Every master disseminates local set (using reserved blocks) masters deterministically pass set to neighboring masters (radius r) masters add messages received from other masters to local set flooding among masters M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 9/11
34 Gossiping Algorithm The Algorithm Partition nodes in masters and slaves every slave is at d ar from some master (0 < a < 1/3) every pair of masters are at d > ar MIS(ar) Every master reserves blocks of time steps for local use master and slaves communicate without collisions within r Coloring(r), using counter (achives local synch. and coll. detection) Every master maintains set of messages received initially set contains own message only slaves pass message to master (using reserved blocks and radius ar) master adds messages to set window back-on/back-off + O(log 2 n) times p trans = 1/ log n Every master disseminates local set (using reserved blocks) masters deterministically pass set to neighboring masters (radius r) masters add messages received from other masters to local set flooding among masters M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 9/11
35 Gossiping Algorithm Time efficiency The Algorithm Assume phase synchronism 1 Partition nodes in masters and slaves MIS O(log 2 n) 2 Every master reserves blocks of time steps for local use Coloring O(log n) 3 Every master maintains set of messages received window back-on/back-off O( + log 2 n log ) 4 Every master disseminates local set flooding among masters O(D) Overall: O(log 2 n + log n + + log 2 n log + D) O( + D) even without synch. M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 10/11
36 Gossiping Algorithm Time efficiency The Algorithm Assume phase synchronism 1 Partition nodes in masters and slaves MIS O(log 2 n) 2 Every master reserves blocks of time steps for local use Coloring O(log n) 3 Every master maintains set of messages received window back-on/back-off O( + log 2 n log ) 4 Every master disseminates local set flooding among masters O(D) Overall: O(log 2 n + log n + + log 2 n log + D) O( + D) even without synch. M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 10/11
37 Gossiping Algorithm Time efficiency The Algorithm Assume phase synchronism 1 Partition nodes in masters and slaves MIS O(log 2 n) 2 Every master reserves blocks of time steps for local use Coloring O(log n) 3 Every master maintains set of messages received window back-on/back-off O( + log 2 n log ) 4 Every master disseminates local set flooding among masters O(D) Overall: O(log 2 n + log n + + log 2 n log + D) O( + D) even without synch. M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 10/11
38 Gossiping Algorithm Time efficiency The Algorithm Assume phase synchronism 1 Partition nodes in masters and slaves MIS O(log 2 n) 2 Every master reserves blocks of time steps for local use Coloring O(log n) 3 Every master maintains set of messages received window back-on/back-off O( + log 2 n log ) 4 Every master disseminates local set flooding among masters O(D) Overall: O(log 2 n + log n + + log 2 n log + D) O( + D) even without synch. M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 10/11
39 Gossiping Algorithm Time efficiency The Algorithm Assume phase synchronism 1 Partition nodes in masters and slaves MIS O(log 2 n) 2 Every master reserves blocks of time steps for local use Coloring O(log n) 3 Every master maintains set of messages received window back-on/back-off O( + log 2 n log ) 4 Every master disseminates local set flooding among masters O(D) Overall: O(log 2 n + log n + + log 2 n log + D) O( + D) even without synch. M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 10/11
40 Gossiping Algorithm Time efficiency The Algorithm Assume phase synchronism 1 Partition nodes in masters and slaves MIS O(log 2 n) 2 Every master reserves blocks of time steps for local use Coloring O(log n) 3 Every master maintains set of messages received window back-on/back-off O( + log 2 n log ) 4 Every master disseminates local set flooding among masters O(D) Overall: O(log 2 n + log n + + log 2 n log + D) O( + D) even without synch. M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 10/11
41 Thank you M. Farach-Colton, M. A. Mosteiro Sensor Network Gossiping 11/11
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