Computing functions over wireless networks

Transcription

1 This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License. Based on a work at decision.csl.illinois.edu See last page and Computing functions over wireless networks P. R. Kumar Dept. of Electrical and Computer Engineering, and Coordinated Science Lab University of Illinois, Urbana-Champaign prkumar@illinois.edu Web: 1/37

2 How to process information in the network in sensor networks? (Or how to do data fusion over a sensor network? Or how to compute a function of data over a sensor network? Or how to perform in-network information processing in a sensor network?) 2/37

3 Outline Difference between sensor networks and data networks 4 Model of problem: Protocol model, Non-information theoretic model 5 Sample of results: Average vs. Max 9 More details of results 12 Some information theoretic results for sensor networks 27 References 34 3/37

4 Sensor networks Examples of Tasks Environment monitoring» Determine the Average temperature: (x 1 + x x n )/n Alarm networks» Determine the Max temperature: Max x i Sensor networks are not just data networks with sensor measurements replacing files They are application specific Nodes need not just relay packets» They can discard, combine, process packets» Combination of computing and communication More generally: Consider a symmetric function F(x 1, x 2,, x n ) E.g., Average, Mode, Median, Percentile, Max Determined by Histogram or Types How should information be processed in the network to compute such functions? This can also be regarded as network coding for sensor networks 4/37

5 Model of problem: Protocol model, Non-information theoretic (Giridhar & K ʻ05) 5/37

6 Model of problem Multi-hop model for wireless communication (Giridhar & K ʼ05) Collocated network» All nodes within range of all Multi-hop random network (Penrose 1997, Gupta & K ʼ98)» Critical common range for connectivity of random graph» where c n +. (Take r(n) = 2log n ) n At time t, sensor i takes a measurement x i (t) {1, 2,,D} Fusion node needs to calculate F(x 1,, x n ) exactly Non-information theoretic formulation 6/37

7 Model of problem Protocol Model for wireless communication Receiver should be outside other transmittersʼ interference footprints Two types of networks Collocated network Large range so all nodes can hear each other r 1 (1+Δ) r 1 r 2 (1+Δ)r 2 Random network Communicate at rate W bits/sec (Take W =1 bit/sec wlog) At time t, sensor i takes a measurement x i (t) {1, 2,,D} No probability distribution on x i (t) ʼs Fusion node needs to calculate F(x 1,, x n ) exactly Non-information theoretic formulation n nodes randomly distributed Need range at no less than for network to be connected 7/37

8 Definition of Computational Rate R max (n) Block coding allowed N measurements of node 1: x 1 N measurements of node 2 : x 2 N measurements of node n : x n If all N functions computed in time T Then Computational Rate Best Rate over all Strategies S and block lengths N: (Giridhar & K ʻ05) 8/37

9 Sample of results: Average vs. Max 9/37

10 The Average versus Max Theorem (Giridhar & K ʻ05): The rate at which the Average can be harvested is Strategy» Tessellate» Fuse locally» Compute along a rooted tree of cells Theorem (Giridhar & K ʻ05): The rate at which the Max can be harvested is 1 Θ loglogn Strategy: Take advantage of Block Coding» First node announces times of max values: ( )» Second node announces times of additional max values: ( 1 1 )» Third node announces of yet more max values: ( 1 ) 10/37

11 Summary: Order of difficulty of computations June 26, 2009, P. R. Kumar (1/n) Collocated network: Average, Mode, Type Data downloading (1/log n) Random Multi-hop network: Average, Mode, Type Collocated network: Max (1/loglog n) Random Multi-hop network Max (Giridhar & K ʻ05) 11/37

12 More details of results (Giridhar & K ʻ05) 12/37

13 Results: A classification of functions Divisible functions Amenable to divide and conquer if deg(g n ) = O(log R(F n ) ) Symmetric functions Data centric paradigm: Identity of node is not important, only its value Type-sensitive functions Hard to compute in collocated case, and random case Type-threshold functions collocated case, and random case (Giridhar & K ʻ05) 13/37

14 Examples (Giridhar & K ʼ05) Data download problem: F n (x 1,, x n ) = (x 1,, x n ): In collocated or random networks: Histogram of frequencies or Type : F n (x 1,, x n ) = (z 1, z 2,, z D ) Collocated case: Random networks: Special case: Any symmetric function Mean, Mode, Median, Majority: Collocated case: Random case: Max, Min, Range, Occurrence of a value: Collocated case: Random case: 14/37

15 Definition of Rate R max (n) Block coding allowed N measurements of node 1: x 1 N measurements of node 2 : x 2 N measurements of node n : x n If computed in time T Then Rate Compute all N functions Best Rate over all Strategies S and block lengths N: Bound on R max : (Giridhar & K ʻ05) 15/37

16 Divisible functions Divisible functions: There exists F S (x i : i S) for every subset S {1,2,,n} With R(F S ) R(F n ) F S2 F S3 F S1 F S F S6 F S5 F S4 for partition {S 1,, S m } of S Theorem: if deg(g n ) = O(log R(F n ) ) (Giridhar & K ʻ05) Special cases Data Downloading: deg(g n ) O(log R(F n ) ) = O(log D n ) = O(n) So Histogram: So for Random networks Hence for Random networks (Giridhar & K ʻ05) 16/37

17 Proof of for Divisible Functions Tessellate into square cells of area r 2 /2 Neighboring occupied cells can communicate with each other Form a tree rooted at fusion center out of occupied connected cells Choose a relay node in each occupied cell and a parent in the next cell towards the root Locally compute and pass on along tree to root Collect data from deg(g n ) nodes within cell Collect functional value of log R(F n ) bits from bounded number of child cells Pass on functional value of log R(F n ) bits to parent cell All operations can be performed in time So Constructive strategy 17/37

18 Symmetric functions Symmetric functions depend only on type where Type-sensitive functions There is a 0 < c < 1 such that a fraction c of values is never enough to pin down the value of the function F n Examples: Mean, Median, Mode, Majority Type-threshold functions Only want to know whether each z i exceeds a threshold z i* There is a threshold vector such that Examples: Max, Min, Range, Occurrence of value (Giridhar & K ʻ05) 18/37

19 Collision-free strategies in collocated case June 26, 2009, P. R. Kumar Every node knows when to transmit based on what it hears on channel The content of the packet it transmits depends on what it heard, as well as its own information Node g 1 transmits packet P 1 (x g1 ) Node g 2 (P 1 (x g1 )) transmits packet P 2 (P 1 (x g1 ), x g2 ) Node g 3 (P 1 (x g1 ), P 2 (P 1 (x g1 ), x g2 )) transmits packet P 3 (P 1 (x g1 ), P 2 (x g2 ), x g3 ) Note: We are not allowing information transmission to occur through collisions (Giridhar & K ʻ05) 19/37

20 for Type-sensitive functions in collocated case June 26, 2009, P. R. Kumar Wlog suppose D=2 Initially, x g1 is in the set S 0 g1 with cardinality S0 g1 = 2N After first transmission, x g1 can be in one of two sets depending on whether it transmits 0 or 1 Let the transmission correspond to the larger set, call it be S 1 g1 S 1 g1 1/2 S0 g1 After t-th transmission of node k, let x k lie in S t k with St k 1/2 St-1 k So at the end, uncertainty set is: Thus at least nn-t places in the nn values (x 1, x 2,, x n ) are undetermined However to compute F n (x(1),x(2),,x(n)), at least cnn values are needed So nn-t (1-c)nN So T cnn Hence Thus for collocated case (Giridhar & K ʻ05) 20/37

21 for Type-threshold functions in collocated case June 26, 2009, P. R. Kumar Consider Max function (argument can be generalized) Threshold vector = (1,1,,1) Lower Bound Take block length Node 1 transmits its locations of the N 1 1ʼs in Node 2 transmits the N 2 new 1ʼs in its list Node 3 transmits the N 3 new 1ʼs in its list To describe N i takes log N bits To describe the locations of N i 1ʼs requires So Maximized when N i = N/n. Use Thus: (Giridhar & K ʻ05) 21/37

22 Upper bound in collocated case. June 26, 2009, P. R. Kumar Take N > 2n. Consider Exactly N/2n 1ʼs in x 1 Exactly N/2n 1ʼs in x 2 Exactly N/2n 1ʼs in x n At most one 1...At most one 1 Claim: Each such x produces a unique set of transmissions P 1,P 2,,P T Suppose not. Then there are two: x and y which produce same transmissions They differ in some x k y k Then also produces same transmissions since node k hears the same under x k or y k and so reacts the same But this has different Max values from x Thus Max functions are not determined from transmissions (Giridhar & K ʻ05) 22/37

23 Finishing the proof for the Max function June 26, 2009, P. R. Kumar Number of such vectors x = So 2 T > So T > N log(n-1) So Hence This proves (Giridhar & K ʻ05) 23/37

24 Generalizing to any Type-threshold function for collocated case June 26, 2009, P. R. Kumar Feasibility of Node i sends only list of 1ʼs for values which threshold has not been attained Upper bound of There exist a threshold vector such that Now consider an x which has» z 1-1 vectors of the form (1,1,,1)» z 2 vectors of the form (2,2,,2)» z 3 vectors of the form (2,2,,2)» Remaining have 1ʼs or 2ʼs only Now problem is reduced to a Max (Giridhar & K ʻ05) 24/37

25 Random networks: Type sensitive networks June 26, 2009, P. R. Kumar Theorem (Giridhar & K ʼ05) Type-sensitive functions: Proof Tessellate unit area domain into squares of area A = (Δr) 2 /2 Transmissions are local within square Assume Genie communicates all messages instantaneously to all nodes We know at least cnn transmissions are needed At least one square has greater than cnna receptions However only one node can receive at a time in a square So T cnna = cnn(δr) 2 /2 But for connectivity So T c N log n 25/37

26 Type-threshold functions Theorem (Giridhar & K ʼ05) Type-threshold functions: Proof Consider Max for simplicity Tessellate unit area domain into squares of area A = (Δr) 2 /2 Some square has greater than na nodes Suppose all nodes outside this square have value 0 Then we need to compute Max of na nodes We need cn log (na) transmissions Only one node can receive at any given time So T cn log (na) But is needed for connectivity So So T N Ω(log log n) So Achievability can be proved by using tree gathering 26/37

27 Some information theoretic results for sensor networks 27/37

28 Complexities of function computation over wireless networks June 26, 2009, P. R. Kumar Slepian-Wolf Theorem (ʻ73) Total information fusion over wires from correlated sources X R 1 =H(X) Several other complexities in sensor networks Y R 2 =H(Y X) Wireless nodes There are no independent links: Sources share channel Multiple access problem Also, sensors can communicate with each other and thus cooperate Source-channel separation does not hold Only a function needs to be computed, not all information Little is known in a pure information theoretic setting 28/37

29 Multiple relay network (Xie and K ʻ05) Consider a network p(y 2 (t), y 3 (t),..., y n (t) x 1 (t), x 2 (t),..., x n 1 (t)) A feasible rate Generalization of Cover and El Gamal ʻ79 29/37

30 Multiple access channel (Ahlswede ʻ71, Liao ʻ72) June 26, 2009, P. R. Kumar Consider the multiple access channel The capacity region for any p(x 1 )p(x 2 ) 30/37

31 A multiple source multiple relay network (Xie and K ʻ07) June 26, 2009, P. R. Kumar Consider the network A feasible rate vector is 31/37

32 A feasible rate for a sensor network with a sink June 26, 2009, P. R. Kumar Consider the network Feasible rate region for acyclic choice of routes Maximize over Based on combination of backward decoding for relay channel and multiple access channel (Xie and K ʻ07) 32/37

33 Exact capacity region for some Gaussian sensor networks with phase fading June 26, 2009, P. R. Kumar Kramer, Gastpar and Gupta ʻ03 have determined exact capacity region for relay channel with phase fading for some geometries The capacity region is Theorem (Xie and Kumar ʻ07) Phase fading unknown to transmitter Node 5 far away from other nodes 33/37

34 References-1 M.D. Penrose, The longest edge of the random minimal spanning tree, The Annals of Probability, vol. 7, no. 2, pp , Piyush Gupta and P. R. Kumar, Critical Power for Asymptotic Connectivity in Wireless Networks, in Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W. H. Fleming. Edited by W. M. McEneany, G. Yin, and Q. Zhang, Birkhauser, Boston, MA, pp , ISBN Arvind Giridhar and P. R. Kumar, Computing and Communicating Functions Over Sensor Networks, IEEE Journal on Selected Areas in Communications, pp , vol. 23, no. 4, April Arvind Giridhar and P. R. Kumar, Towards a Theory of In-Network Computation in Wireless Sensor Networks IEEE Communications Magazine, vol. 44, no. 4, pp , April Arvind Giridhar and P. R. Kumar, In-Network Information Processing in Wireless Sensor Networks. In Wireless Sensor Networks: Signal Processing and Communications Perspectives, A. Swami, Q. Zhao, Y.- W. Hong and L. Tong, Editors. pp , John Wiley, Chichester, England, /37

35 References-2 Slepian D and Wolf J 1973 Noiseless coding of correlated information sources. IEEE Transactions on Information Theory, 19(4), T. Cover and A. El Gamal, Capacity theorems for the relay channel, IEEE Trans. Inform. Theory, vol. 25, pp , R. Ahlswede, Multi-way communication channels, in Proceedings of the 2nd Int. Symp. Inform. Theory (Tsahkadsor, Armenian S.S.R.), (Prague), pp , Publishing House of the Hungarian Academy of Sciences, H. Liao, Multiple access channels. PhD thesis, Department of Electrical Engineering, University of Hawaii, Honolulu, HA, G. Kramer, M. Gastpar, and P. Gupta, Cooperative strategies and capacity theorems for relay networks, IEEE Trans. Inform. Theory, vol. 51, pp , September /37

36 References-3 Liang-Liang Xie and P. R. Kumar, An Achievable Rate for the Multiple- Level Relay Channel, IEEE Transactions on Information Theory, vol. 51, no. 4, pp , April Liang-Liang Xie and P. R. Kumar, Multisource, multidestination, multirelay wireless networks, IEEE Transactions on Information Theory, Special issue on Models, Theory and Codes for Relaying and Cooperation in Communication Networks, vol. 53, no. 10, pp , October Liang-Liang Xie and P. R. Kumar, Information-Theoretic Studies of Wireless Sensor Networks. In Handbook on Array Processing and Sensor Networks, Simon Haykin and K. J. Ray Liu, Editors. IEEE- Wiley, /37

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