computing functions over wireless networks

8/6/2019 Computing Functions Over Wireless Networks

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June 26, 2009 , P. R. Kumar

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


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Examples of Tasks Environment monitoring

Determine the Average temperature: (x1 +x2 + +xn)/n

Alarm networks Determine the Max temperature: Maxxi

Sensor networks are not just data networks with sensor measurementsreplacing 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(x1,x2, ,xn) E.g., Average, Mode, Median, Percentile, Max

Determined by Histogram or Types

How should information be processed in the networkto compute suchfunctions?

This can also be regarded as network coding for sensor networks

Sensor networks


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Model of problem:Protocol model,

Non-information theoretic

(Giridhar & K 05)


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Model of problem

Multi-hop model for wireless communication (Giridhar & K 05)

Collocated network All nodes within range of all

Multi-hop random network (Penrose 1997, Gupta & K 98)

Critical common range for connectivity of random graph

where

At time t, sensor i takes a measurementxi(t) {1, 2,,D}

Fusion node needs to calculate F(x1,,xn) exactly

Non-information theoretic formulation

cn +. (Take r(n) =

2logn

n

)


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Model of problem

Protocol Model for wirelesscommunication

Receiver should be outside othertransmitters interference footprints

Communicate at rate Wbits/sec (TakeW=1 bit/sec wlog)

At time t, sensor i takes a measurementxi(t) {1, 2,,D}

No probability distribution onxi(t) s

Fusion node needs to calculateF(x1,,xn) exactly

Non-information theoretic formulation

r2r1(1+) r1

(1+)r2

Two types of networks

Collocated network

Large range so all nodes

can hear each other

Random network

n nodes randomly distributed

Need range at no less than

for network to be connected


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Block coding allowed

Definition of Computational RateRmax

(n)

Nmeasurements of node 1:x1

Nmeasurements of node 2 :x2

Nmeasurements of node n :xn

If allNfunctions computed in time T

Then Computational Rate

Best Rate over all Strategies Sand block lengthsN:

(Giridhar & K 05)


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Sample of results: Average vs. Max


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The Averageversus Max

Theorem (Giridhar & K 05): The rate at whichthe Averagecan be harvested is

Strategy

Tessellate

Fuse locally

Compute along a rooted tree of cells

Strategy: Take advantage of Block Coding

Theorem (Giridhar & K 05): The rate at which

the Maxcan be harvested is

1

loglogn

First node announces times of max values: ( 1 1 1 )

Second node announces times of additional max values: ( 1 1 )

Third node announces of yet more max values: ( 1 )


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Summary: Order of difficulty ofcomputations

Random Multi-hopnetwork

Max

(1/loglog n)

Collocated network:Max

Random Multi-hopnetwork:

Average, Mode, Type

(1/log n)

Collocated network:Average, Mode, Type(1/n) Datadownloading

(Giridhar & K 05)


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More details of results(Giridhar & K 05)


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Results: A classification of functions

Divisible functions

Amenable to divide and conquer

if deg(Gn) = O(log |(Fn)|)

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 05)


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Examples (Giridhar & K 05)

Data download problem: Fn(x1,,xn) = (x1,,xn):

In collocated or random networks:

Histogram of frequencies or Type: Fn(x1,,xn) = (z1,z2,,zD)

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:


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Block coding allowed

Compute allNfunctions

If computed in time T

Then Rate

Best Rate over all Strategies Sand block lengthsN:

Bound onRmax:

Definition of RateRmax

(n)

Nmeasurements of node 1:x1 Nmeasurements of node 2 :x2

Nmeasurements of node n :xn

(Giridhar & K 05)


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Divisible functions:

There exists FS(xi: iS) for every subset S{1,2,,n}

With |(FS)| |(Fn)|

for partition {S1,, Sm} of S

Theorem: if deg(Gn) = O(log |(Fn)|)

Special cases

Data Downloading: deg(Gn)O(log |

(Fn)|) = O(logD

n

) = O(n)

So

Histogram:

So for Random networks

Hence for Random networks

Divisible functions

FSFS1FS2

FS5FS3

FS4FS6

(Giridhar& K 05)

(Giridhar & K 05)


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Tessellate into square cells of area r2/2

Neighboring occupied cells can communicatewith 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(Gn) nodes within cell

Collect functional value of log|(Fn)| bits from

bounded number of child cells Pass on functional value of log|(Fn)| bits to

parent cell

All operations can be performed in time

So

Constructive strategy

Proof of forDivisible Functions


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Symmetric functions

Symmetric functions depend only on type

where

Type-sensitive functions

There is a 0 < c < 1 such that a fraction c of valuesis never enough to pin down the value of the function Fn

Examples: Mean, Median, Mode, Majority


Only want to know whether eachzi exceeds a thresholdzi*

There is a threshold vector such that

Examples: Max, Min, Range, Occurrence of value (Giridhar & K 05)


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Collision-free strategies in collocatedcase

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 owninformation

Node g1 transmits packet P1(xg1)

Node g2(P1(xg1)) transmits packet P2(P1(xg1),xg2)

Node g3(P1(xg1), P2(P1(xg1),xg2)) transmits packet P3(P1(xg1), P2 (xg2),xg3)

Note: We are not allowing information transmission to occur through collisions

(Giridhar & K 05)


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for Type-sensitivefunctions in collocated case

Wlog supposeD=2

Initially,xg1 is in the set S0

g1 with cardinality |S0

g1| = 2N

After first transmission,xg1 can be in one of two sets depending on whether ittransmits 0 or 1

Let the transmission correspond to the larger set, call it be S1g1 |S1g1| 1/2 |S

0g1|

After t-th transmission of node k, letxk lie in Stkwith | S

tk| 1/2 | S

t-1k|

So at the end, uncertainty set is:

Thus at least nN-Tplaces in the nNvalues (x1,x2 ,,xn) are undetermined

However to compute Fn(x(1),x(2),,x(N)), at least cnNvalues are needed

So nN-T (1-c)nN

So TcnN

Hence

Thus for collocated case (Giridhar & K 05)


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for Type-thresholdfunctions in collocated case

Consider Max function (argument can be generalized)

Threshold vector = (1,1,,1)

Lower Bound

Take block length

Node 1 transmits its locations of theN1 1s in

Node 2 transmits theN2 new 1s in its list

Node 3 transmits theN3 new 1s in its list

To describeNi takes logNbits

To describe the locations ofNi 1s requires So

Maximized whenNi =N/n. Use

Thus:

(Giridhar & K 05)


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Upper boundin collocated case .

TakeN> 2n. Consider

Claim: Each suchx produces a unique set of transmissions P1,P2,,PT Suppose not. Then there are two:x andy which produce same transmissions They differ in somexkyk

Then also produces same transmissionssince node khears the same under xkorykand so reacts the same

But this has different Max values fromx

Thus Max functions are not determined from transmissions

ExactlyN/2n 1s inx1 ExactlyN/2n 1s inx2

ExactlyN/2n 1s inxn

At most one 1 ...At most one 1

(Giridhar & K 05)


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Finishing the proof for the Maxfunction

Number of such vectorsx =

So 2T>

So T> Nlog(n-1)

So

Hence

This proves (Giridhar & K 05)


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Generalizing to any Type-thresholdfunction for collocated case

Feasibility of

Node i sends only list of 1s for values which threshold has notbeen attained

Upper bound of

There exist a threshold vector such that

Now consider anx which has z1-1 vectors of the form (1,1,,1)

z2 vectors of the form (2,2,,2)

z3 vectors of the form (2,2,,2)

Remaining have 1s or 2s only

Now problem is reduced to a Max (Giridhar & K 05)


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Theorem (Giridhar & K 05)

Type-sensitive functions:

Proof

Tessellate unit area domain into squares of areaA = (r)2/2

Transmissions are local within square

Assume Genie communicates all messages instantaneously to all nodes

We know at least cnNtransmissions are needed

At least one square has greater than cnNA receptions

However only one node can receive at a time in a square

So TcnNA = cnN(r)2/2

But for connectivity

So TcNlog n

Random networks: Type sensitivenetworks


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Theorem (Giridhar & K 05)

Type-threshold functions:

Proof

Consider Max for simplicity

Tessellate unit area domain into squares of areaA = (r)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 cNlog (nA) transmissions

Only one node can receive at any given time

So TcNlog (nA) But is needed for connectivity

So

So TN(log log n)

So

Achievability can be proved by using tree gathering


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Some information theoretic resultsfor sensor networks


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Complexities of function computationover wireless networks

Slepian-Wolf Theorem (73) Total information fusion over wires from correlated sources

X

Y

R1=H(X)

R2=H(Y|X) Several other complexities in sensor networks

Wireless nodes There are no independent links: Sources share channel

Multiple access problem

Little is known in a pure information theoretic setting

Also, sensors can communicate with each other and thuscooperate

Source-channel separation does not hold

Only a function needs to be computed, not all information


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Multiple relay network (Xie and K 05)

Consider a network

Generalization of Cover and El Gamal 79

A feasible rate

p(y2 (t),y3(t),...,yn (t) | x1(t),x2 (t),...,xn1(t))


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Multiple access channel(Ahlswede 71, Liao 72)

The capacity region

for anyp(x1)p(x2)

Consider the multiple access channel


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A multiple source multiple relaynetwork (Xie and K 07)

A feasible rate vector is

Consider the network


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A feasible rate for a sensor networkwith a sink

Feasible rate region for acyclic choice of routes

Maximize over

Based on combination of backward decoding for relay channel

and multiple access channel

Consider the network

(Xie and K 07)


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Exact capacity region for some Gaussiansensor networks with phase fading

Kramer, Gastpar and Gupta 03have determined exact capacityregion for relay channel withphase fading for some geometries

Theorem (Xie and Kumar 07)

Phase fading unknown totransmitter

Node 5 far away from other nodes

The capacity region is


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References-1 M.D. Penrose, The longest edge of the random minimal spanning

tree, The Annals of Probability, vol. 7, no. 2, pp. 340-361, 1997.

Piyush Gupta and P. R. Kumar, Critical Power for AsymptoticConnectivity 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. 547566, 1998. ISBN 0-8176-4078-9

Arvind Giridhar and P. R. Kumar, Computing and CommunicatingFunctions Over Sensor Networks, IEEE Journal on Selected Areas in

Communications, pp. 755764, vol. 23, no. 4, April 2005. Arvind Giridhar and P. R. Kumar, Towards a Theory of In-Network

Computation in Wireless Sensor Networks IEEE CommunicationsMagazine, vol. 44, no. 4, pp. 98107, April 2006. Arvind Giridhar and P. R. Kumar, In-Network Information Processing in

Wireless Sensor Networks. In Wireless Sensor Networks: SignalProcessing and Communications Perspectives, A. Swami, Q. Zhao, Y.-

W. Hong and L. Tong, Editors. pp. 4367, John Wiley, Chichester,England, 2007.


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Slepian D and Wolf J 1973 Noiseless coding of correlated information

sources. IEEE Transactions on Information Theory, 19(4), 471480.

T. Cover and A. El Gamal, Capacity theorems for the relay channel,IEEE Trans. Inform. Theory, vol. 25, pp. 572-584, 1979.

R. Ahlswede, Multi-way communication channels, in Proceedings ofthe 2nd Int. Symp. Inform. Theory (Tsahkadsor, Armenian S.S.R.),(Prague), pp. 23-52, Publishing House of the Hungarian Academy of

Sciences, 1971.

H. Liao, Multiple access channels. PhD thesis, Department of ElectricalEngineering, University of Hawaii, Honolulu, HA, 1972.

G. Kramer, M. Gastpar, and P. Gupta, Cooperative strategies andcapacity theorems for relay networks, IEEE Trans. Inform. Theory, vol.51, pp. 3037-3063, September 2005.

References-2


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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. 13481358, April 2005.

Liang-Liang Xie and P. R. Kumar, Multisource, multidestination,

multirelay wireless networks, IEEE Transactions on InformationTheory, Special issue on Models, Theory and Codes for Relaying and

Cooperation in Communication Networks, vol. 53, no. 10, pp. 3586

3595, October 2007. 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, 2009.


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