Data Collection in Population Protocols with Non-uniformly Random Scheduler

Transcription

1 Data Collection in Population Protocols with Non-uniformly Random Scheduler Or: How to work less and get done more Joffroy Beauquier Janna Burman Shay Kutten Thomas Nowak Chuan Xu September 8, 2017 Data Collection in Population Protocols with Non-uniformly Random Scheduler 1/15

2 Overview 1 Network model and motivation 2 Non uniformly random scheduler 3 Data collection in Population Protocols Data Collection in Population Protocols with Non-uniformly Random Scheduler 2/15

3 Overview 1 Network model and motivation 2 Non uniformly random scheduler 3 Data collection in Population Protocols Lower bounds on expected convergence time Analytical results on time complexities of data collection protocols Energy-efficient protocol Numerical results Data Collection in Population Protocols with Non-uniformly Random Scheduler 2/15

4 Model Population Protocols 1 Anonymous agents 2 Interaction in pairs: 3 Asymmetric: initiator, responder 4 Scheduler: order of interaction Data Collection in Population Protocols with Non-uniformly Random Scheduler 3/15

5 Model Population Protocols 1 Anonymous agents 2 Interaction in pairs: 3 Asymmetric: initiator, responder 4 Scheduler: order of interaction Examples of passively Mobile Sensor Network ZebraNet (wildlife tracking) EMMA (pollution monitoring) Data Collection in Population Protocols with Non-uniformly Random Scheduler 3/15

6 Enhanced Population Protocols: Non uniformly random scheduler S(P), P R n n Uniform random scheduler: P i,j = 1/n(n 1) Non-uniform random scheduler: general probability distribution P i,j Motivation: differing mobility patterns, differing speeds Data Collection in Population Protocols with Non-uniformly Random Scheduler 4/15

7 Data Collection Every agents starts with an initial value. Data collection is complete when the base station has all values. Values can be transfered from agent to agent. Data Collection in Population Protocols with Non-uniformly Random Scheduler 5/15

8 Lower bounds on the expected covergence time Theorem The expected convergence time of any protocol solving data collection with non-uniformly random scheduler is Ω(n log n). Theorem The expected convergence time of any protocol solving data collection is Ω(max 1 n i j=1 (P i,j +P j,i ) ). Data Collection in Population Protocols with Non-uniformly Random Scheduler 6/15

9 TTF Protocol: Transfer to the Faster [Beauquier et al, PODC 10] Transfer all values from j to i if i is faster than j i: j: Data Collection in Population Protocols with Non-uniformly Random Scheduler 7/15

10 TTF Protocol: Transfer to the Faster [Beauquier et al, PODC 10] Transfer all values from j to i if i is faster than j i: j: faster = smaller cover time (= time to meet all agents) Data Collection in Population Protocols with Non-uniformly Random Scheduler 7/15

11 TTF Protocol: Transfer to the Faster [Beauquier et al, PODC 10] Transfer all values from j to i if i is faster than j i: j: faster = smaller cover time (= time to meet all agents) define x i (t) = number of data held by agent i at step t Data Collection in Population Protocols with Non-uniformly Random Scheduler 7/15

12 TTF Protocol: Transfer to the Faster [Beauquier et al, PODC 10] Transfer all values from j to i if i is faster than j i: j: faster = smaller cover time (= time to meet all agents) define x i (t) = number of data held by agent i at step t then x(t) = W (t) x(t 1) where W (t) = Data Collection in Population Protocols with Non-uniformly Random Scheduler 7/15

13 TTF Protocol Transfer all values from j to i if i is faster than j i: j: then x(t) = W (t) W (1) x(0) convergence speed of matrix product W (t)... W (1) depends on the second eigenvalues of the W (τ) Theorem The ( expected convergence time of the TTF protocol is n log n O where W is the expected value w.r.t. P i,j of a log λ 2 ( W ) 1 ) matrix associated to the matrices W (τ). Data Collection in Population Protocols with Non-uniformly Random Scheduler 8/15

14 TTF Protocol this upper bound on the data collection time of TTF is quite loose however, Data Collection in Population Protocols with Non-uniformly Random Scheduler 9/15

15 TTF Protocol this upper bound on the data collection time of TTF is quite loose however, Data Collection in Population Protocols with Non-uniformly Random Scheduler 9/15

16 Lazy TTF (p) During an interaction as initiator, do nothing with probability p i, otherwise execute TTF. Decide Transfer whether if i istofaster execute thanttf j i: j: initiator responder with probability p i Data Collection in Population Protocols with Non-uniformly Random Scheduler 10/15

17 Lazy TTF (p) During an interaction as initiator, do nothing with probability p i, otherwise execute TTF. Decide Transfer whether if i istofaster execute thanttf j i: j: initiator responder with probability p i Data Collection in Population Protocols with Non-uniformly Random Scheduler 10/15

18 Lazy TTF (p) During an interaction as initiator, do nothing with probability p i, otherwise execute TTF. Decide Transfer whether if i istofaster execute thanttf j i: j: initiator responder with probability p i Data Collection in Population Protocols with Non-uniformly Random Scheduler 10/15

19 Lazy TTF (p) During an interaction as initiator, do nothing with probability p i, otherwise execute TTF. Decide Transfer whether if i istofaster execute thanttf j i: j: initiator responder with probability p i Data Collection in Population Protocols with Non-uniformly Random Scheduler 10/15

20 Lazy TTF (p) During an interaction as initiator, do nothing with probability p i, otherwise execute TTF. Decide Transfer whether if i istofaster execute thanttf j i: j: initiator responder with probability p i 1 When p is a vector of all ones, lazy TTF(p) = TTF 2 When p is a vector of all zeros, infinite time but zero energy consumption 3 Energy/Time trade off Data Collection in Population Protocols with Non-uniformly Random Scheduler 10/15

21 Lazy TTF (p) During an interaction as initiator, do nothing with probability p i, otherwise execute TTF. Theorem Decide Transfer whether if i istofaster execute thanttf j The ( expected convergence time of the TTF protocol is O. ) n log n log λ 2 ( W p) 1 i: j: initiator responder with probability p i Data Collection in Population Protocols with Non-uniformly Random Scheduler 10/15

22 Lazy TTF (p) During an interaction as initiator, do nothing with probability p i, otherwise execute TTF. Theorem Decide Transfer whether if i istofaster execute thanttf j The ( expected convergence time of the TTF protocol is O. ) n log n log λ 2 ( W p) 1 Choose i: p: initiator optimize the upper bound on j: the gathering responder time with probability p i Data Collection in Population Protocols with Non-uniformly Random Scheduler 10/15

23 Lazy TTF( ˆp) OP 1 : min λ 2 ( W ) p R n s.t Eq. (1) 0 p i 1 i {1,..., n} equivalent to OP 2 (convex) : min p R n,s s s.t si W 0 Eq. (1) 0 p i 1 i {1,..., n} Solving OP 2 ˆp. Data Collection in Population Protocols with Non-uniformly Random Scheduler 11/15

24 Numerical results: Gaps on Time and Energy between TTF and Lazy TTF(ˆp) For small systems, the expected convergence time T E (TTF) and T E (lazy TTF(ˆp)) can be calculated directly via the Markov chain. E: Total energy consumption of a protocol E(TTF) = 2T E (TTF) E wkp E(lazyTTF(ˆp)) = 2T E (lazy TTF(ˆp)) i (P i,j ˆp i + P j,i ˆp j ) E wkp. j Data Collection in Population Protocols with Non-uniformly Random Scheduler 12/15

25 Numerical results: Gaps on Time and Energy between TTF and Lazy TTF(ˆp) For small systems, the expected convergence time T E (TTF) and T E (lazy TTF(ˆp)) can be calculated directly via the Markov chain. E: Total energy consumption of a protocol E(TTF) = 2T E (TTF) E wkp E(lazyTTF(ˆp)) = 2T E (lazy TTF(ˆp)) i (P i,j ˆp i + P j,i ˆp j ) E wkp. j Each system of size n, S(n): schedulers randomly generated Gap(T E, n) = TE s(lazy TTF(ˆps )) TE s(ttf) TE s(ttf) /10000 and s S(n) Gap(E, n) = s S(n) E s (lazy TTF(ˆp s )) E s (TTF) E s / (TTF) Data Collection in Population Protocols with Non-uniformly Random Scheduler 12/15

26 Size n Gap(T E, n) Gap(E, n) % % % % % % % % % % Table: Gaps on time and energy. Data Collection in Population Protocols with Non-uniformly Random Scheduler 13/15

27 Conclusions: Initiate the study of non uniformly random scheduler in the context of population protocols Data Collection in Population Protocols with Non-uniformly Random Scheduler 14/15

28 Conclusions: Initiate the study of non uniformly random scheduler in the context of population protocols Give explicit lower bounds on expected convergence time of any data collection protocol Give analytical results for two distributed data collection protocols (a known TTF and a new parametrized energy efficient protocol) Data Collection in Population Protocols with Non-uniformly Random Scheduler 14/15

29 Conclusions: Initiate the study of non uniformly random scheduler in the context of population protocols Give explicit lower bounds on expected convergence time of any data collection protocol Give analytical results for two distributed data collection protocols (a known TTF and a new parametrized energy efficient protocol) Present numerical results to show the efficiency of the new protocol Data Collection in Population Protocols with Non-uniformly Random Scheduler 14/15

30 Thanks for your attention! Data Collection in Population Protocols with Non-uniformly Random Scheduler 15/15