wireless networking and communications group 1/37 capacities of erasure networks qualification...

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Wireless Networking and Communications Group

Capacities of Erasure Networks

Qualification Proposal

of Brian Smith

February 23rd, 2007

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Outline

• Erasure Networks– Unifying Theme

• Information Capacity– Multiple Access Constraints– General Model – Feedback– Gaussian Networks

• Transport Capacity– Upper bounds– Achievability Results

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Erasure Networks in Practice

• Network layer viewpoint– Underlying physical layer coding scheme on packets– Or, packets dropped due to queuing buffer overflows

• Networks where links are dynamically created and destroyed– Network with some fixed and some mobile nodes– Or, nodes in a changing environment

• Urban or Battlefield

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Erasure Networks in Theory

• Network Capacity Calculation Tractability– Very few multi-terminal networks where exact capacity is known

• Multiple Access Channel

• Degraded Broadcast Channel

• Physically Degraded Relay Channel

• Balance of Analyzability and Practical Use• Tool: Cut-set Bound

– Complete Cooperation by two sets of nodes

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Universal Theme for Erasure Networks

• Packets either dropped or correctly received– Transmitted symbol never

mistaken for an alternative

• Network made up of conglomerations of erasure channels– Directed graph model– {0,1} alphabet for simplicity

Input X Output Y

0

1

0

E

1

1-

1-

Memoryless Symmetric Binary Erasure Channel

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Interference in Erasure Networks

• Network Coding in Wireline Networks– “Network Information Flow,” [Ahlswede, Cai, Li, Yeung 2000]

– Consider edges in graph completely independently• No interference

• Wireless Erasure Networks– “Capacity of …,” [Dana, Gowaiker, Palanki, Hassibi, Effros 2006]

– Broadcast constraint to model wireless medium– Result: Cut-set multi-cast capacity achievable

• With some side-information known to destinations

– “On network coding for interference networks,” [Bhadra, Gupta, Shakkottai 2006]

• Capacity asymptotically achievable in field size

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Broad Research Goals

• Insight in Erasure Networks with Interference– Transmitter side and receiver side interference

• Information Capacity– Traditional notion of capacity– Unicast and multi-cast

• Transport Capacity– Asymptotic order-bounds– Multiple uni-casts

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Outline• Erasure Networks

– Unifying Theme

• Information Capacity– Prior Work– Multiple Access Constraints– General Model – Feedback– Gaussian Networks


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Prior Work• “Capacity of Erasure

Networks,” David Julian thesis, 2003

– Main idea: Erasure overlay• Study channels of which

capacity is known

• Add an erasure process (with or without memory) to the underlying channel

– Applied to DMC, MAC, degraded broadcast channel, among others

– Lower bound on general multi-terminal networks

Ey

yyxypxyp

~

~),|()|~(

Julian’s Memoryless Erasure Channel

);(~

YXIC

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Prior Work• “Capacity of Wireless Erasure Networks,”

– Dana, Gowaikar, Hassibi & Effros [2006]

• Model– Broadcast requirement on directed graph

– Links are independent erasure channels

– No receiver interference • Receiver gets vector

• Result– Multicasting from single source to multiple receivers can be performed at

generalized min-cut max-flow rate

12

13

23

34

24

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Wireless Erasure Network Modified Cut-Set Bound

Si Sjij

C

ε1

Modified Cut-Set Bound

242313 11

Example Bound Evaluation

12

13

23

34

24

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Multiple-Access Constraint Networks• Model

– No broadcast requirement– Receiver interference

• Receiver gets finite-field sum of unerased symbols

• Observation:– Swap source and destination– Reverse direction of all edges– Upper-bound on capacity of network with

multiple-access constraint same as original network• Achievability?

– Proof gains a new layer of complexity because of mixing– Our Result: Yes!

12

13

23

34

24

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Capacity Achieving

CSj Siijε1

Si Sjij

C

ε1

Modified Cut-Set Bound

242313 11

Example Bound Evaluation

No Interference

Broadcast Constraint

Multiple-Access Constraint

CSjSi

ijε1,

242313 111

242313 11

12

13

23

34

24

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Proof for Erasure Networks

• Random Block Coding– Given locations of all erasures, simulate network for

all possible input codewords– Error event: There exists a codeword (other than the

correct codeword) which produces identical output at final destination

• Broadcast Constraint Case– Wait to perform encoding for a block until all inputs

related to that block have arrived• Different delays for different paths from source to a node

• But, multiple-access constraint introduces mixing

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Proof for Erasure Networks• Random Block Coding

– Given erasure locations, simulate network for all input codewords

– Error event: There exists more than one codeword which produces received output at final destination

• Broadcast Constraint Case– Wait to perform encoding for a block

until all inputs related to that block have arrived

• Different delays for different paths from source to a node

• But, multiple-access constraint introduces mixing

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Multiple-Access Constraint Proof Idea• For 2nRB messages,

– Generate source codewords of length n(B+L), uniformly from {0,1}

• For each possible input sequence at each relay node– Generate a length n codeword

• Error event E– After all time blocks completed,

destination node receives inputs which could have been generated by two different messages

• Error event ESb

– In time block b, all nodes in the cut (labeled by S) receive “identical” inputs

– Challenge: Events labeled by ESb and

ES’b+1 not independent

L=6

S S’

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MAC-BC Duality• Our result: Duality in capacity of the networks• Claim: There also exists a duality in coding schemes

– Random linear coding is sufficient (in BC)– Show: It is possible to construct a MAC code with the same rate,

given a linear broadcast constraint code

12

13

23

34

24 12

13

23

34

24

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General Erasure Interference Network Model

32

22

12

11

3,21,5

2,21,5

2,22,4

1,11,4

2,21,3

1,21,3

1,11,3

15

24

14

13

00

000

000

0

x

x

x

x

y

y

y

y

• Our contribution: A new model

• Allow each node to have more than one input and output

31

2 4

5

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Work on Information Capacity

• Multiple-Access Finite-Field Sum Constraint Only– Devised new model– Converse and Achievability Complete– Multicast Achievability still open– Submitting to IT Workshop

• General Erasure Interference Network Model– Devised model– Converse Complete– Achievability: Must prove inequality concerning expected rank of

random matricies

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Feedback in Erasure NetworksJoint work with B. Hassibi

• Feedback in Erasure Channel– Eliminates need for coding

• Similar benefit to Erasure Networks?– What kinds of feedback? What kinds of networks?– No receiver or transmitter interference: Same result as channel– Broadcast Constraint:

• Our contribution: Proposed Algorithm

– Multiple Access Constraint: ???

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Proposed Feedback Scheme

• Details of Algorithm– Mark each packet with an identifying header– Each node randomly chooses one packet in queue to transmit– Only delete from queue when notified by final destination

• Benefits of Random Feedback Scheme– Capacity achieving without network coding– No knowledge of network topology required at relay nodes

• Will even self-adjust to new topology• Source only needs to know value of min-cut

– Only need to feedback packet number from destination to each node• As opposed to network coding, mechanism already exists

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Intuition • Intuition

– Seems wasteful, but – Queue lengths adjust to make repetition negligible

• Long queues build up to left of min-cut

• Throughput optimal

• Three node linear network: Say 2>> 1

Source

Destination

1=1-1 2=1-2

Virtual Source Queue

< 1

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Multicast Case with Feedback

• Without coding, multicast capacity cannot in general be achieved

• Simple Counter Example– One source, two multicast destinations – Without coding, all source can do is repeat– This requires an expected time of 8/3 > 2 to get a packet to both

receivers

• Acausal feedback is sufficient• What help is causal feedback?

=1/2

=1/2

sd1

d2

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Gaussian Networks

• Feedback problem– Networking tools to solve a problem in erasure network domain

• Gaussian Networks– Additive Gaussian noise network– Propose to use ideas from erasure network

• Multiple-access constraint, only

• Broadcast constraint, only

• Goal: Determine new bounds on capacity of these networks

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Expected Contributions on Information Capacity

• General Finite-Field Additive Interference Network– Multicast and Single-Source Single-Destination– Includes Multiple-Access Constraint Multicast

• Code Duality

• Benefits of feedback– Prove Broadcast Constraint, Single-Destination– Multicast case– Multiple-Access constraint case

• Gaussian Interference Networks

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Outline• Erasure Networks

– Unifying Theme

• Information Capacity– Prior Work– Multiple Access Constraints– General Model – Feedback– Gaussian Networks


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Transport Capacity

• Capacity of Multiple-Source Multiple-Destination Networks (Multiple Unicast)– Hard Problem– Transport Capacity: Convenient Scalar Description

• Distance Weighted Rate-Sum (bit-meters)

– Work by Gupta & Kumar[2000] and Xie & Kumar[2004]• Gaussian Interference Channel Model• Information Theoretic Linear Bound on Transport Capacity

Growth– Under high attenuation model

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Transport CapacityJoint work with P. Gupta

– Erasure Probability as Function of Geographic Distance

– Minimum Node Separation Constraint: d>dmin

– Threshold Model(d)=0, dd*; 1, d>d*

– Exponential Model(d)=1-e-d/d*

– Polynomial Decay Model(d)=1-1/(1+d), >3

– In all cases: (0)=0, (∞)=1

x5

x4

x3

x2x1

y6

y8

y7

y9

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Models: Broadcast Constraint and Single Antenna• Our Result: Transport Capacity ≤ κn for both cases

• Constant depends only on dmin, d* in exponential decay mode

– Some Notation:

d1h d2

h

d34

Cut 1

Node 1

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Proof Sketch

m

i

n

mj

ddn

m

hm

ijedT1 1

*/1

1

11

– Broadcast Constraint, Only

n

mj

ddm

i

n

m

hm

ijed1

*/

1

1

1

– Single Antenna

)]([1

1m

n

m

hm HrankEdT

)]1(1[1 1

)(1

1

m

i

n

mj

ijm

n

m

hm hEd

Expected Value of the Number of ‘1’ Entries in Hm

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Squish

dih di+1

h

Node i

min2min

21

)( dmjdddm

ik

hkij

Node m

dm-1h Move all nodes with

index greater than ‘m’ in as close as possible, within minimum distance constraint, to bound summation

10,1

Ck

k

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– Franceschetti, Dousse, Tse & Thiran [2004]

Achievability in Random Networks

Squares: Constant size C Slabs: Width k log √n

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– Draining Phase– Highway Phase– Distribution Phase

– Result: • Per node throughput

decays (1/√n)• Source-destination

distance increases (√n)

– No network coding required – routing is order-optimal (submitted ISIT 2007)

Achievability in Random Networks

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Super-linear Growth: Remove Minimum Distance– Place n nodes in two groups, spaced d*ln n apart

• Our result: n ln n growth• But, specifying erasure locations requires ln n extra

informationd* ln n

ne

neR

n

k

n

n

k

n

j

n

k

n

j

dnd

1

1

1 11 1

/ln

1

11111

**

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Completed Work on Transport Capacity

• Linear growth of transport capacity converse in extended network under a variety of models– Interference models– Erasure decay-with-distance models

• Achievability of linear growth in random model• Achievability of superlinear growth

– Removing minimum distance constraint

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Expected Contributions on Transport Capacity

• Two-dimensional networks with polynomial decay– Bound summation properly so that it converges

• Super-linear growth in Networks with Low Attenuation– 1-D bound: >3– Find an achievable super-linear scheme for <3?

• Ozgur, Leveque, Tse [2006] for Gaussian network

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Summary

• Network layer vs. Physical layer perspective– Assume underlying coding scheme

• Non-Traditional Application of Network Coding– Interference Models– Scaling– Forwarding/routing can be optimal (order or throughput)

• Research Goal: – Understand and create capacity results for a

common class of networks

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Plan of Work• Proposal: February 2007• Spring 2007

– Information Theory of Wireless Networks• Submit MAC case to IEEE Information Theory Workshop• Study relationship of expected rank of matricies as related to general case

• Summer 2007– Feeback Capacity: Travel to CalTech to work with Prof. Hassibi

– Transport Capacity• Bound 2-D summation tight enough for linear growth bound• Work achievablity of low-attenuation super-linear growth

• Fall 2007– Gaussian Interference Networks

• Spring 2007: Dissertation and Graduation

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Major Coursework

EE381J Probability and Stochastic Processes deVeciana A

EE381K-2 Digital Communications Andrews A

EE381K-7 Information Theory Vishwanath A

EE381K-9 Advanced Signal Processing Heath A

EE381V Channel Coding Vishwanath A

EE380N Optimization in Engineering Systems Baldick A

EE381K-13 Communication Networks: Analysis and Design Shakkottai A

EE381K-5 Advanced Communication Networks Shakkottai A

EE381K-8 Digital Signal Processing Bovik A

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Supporting Coursework

CS388G Algorithms: Technique and Theory Plaxton A

M381C Real Analysis Beckner A

CS388G Combinatorics and Graph Theory Zuckerman B+

M385C Theory of Probability Zitkovic CR

M393C Statistical Physics Radin A

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Conference Publications• “Transport Capacity of Wireless Erasure Networks,” B.Smith, S.

Vishwanath. In Proceedings of the 44th Allerton Conference on Communication, Control,

and Computing, Monticello, IL, Sep. 2006.

• “Network Coding in Interference Networks,” B. Smith, S. Vishwanath. In Proceedings of 2005 Conference on Information Sciences and Systems, Baltimore, MD, Mar. 2005.

• “Routing is Order-Optimal in Erasure Networks with Interference,” B. Smith, P. Gupta, S. Vishwanath. Submitted to 2007 IEEE ISIT.

• “Capacity of MAC Erasure Networks,” B. Smith, S. Vishwanath. Under preparation for submission.

• “Cooperative Communication in Sensor Networks: Relay Channel with

Correlated Sources,” B.Smith, S. Vishwanath. In Proceedings of the 42nd Allerton

Conference on Communication, Control, and Computing, Monticello, IL, Oct. 2004.

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Journal Publications• “Capacity Analysis of the Relay Channel with Correlated Sources,” in

revision, IEEE Transactions on Information Theory

• Asymptotic Transport Capacity of Wireless Networks,” under preparation for submission, IEEE Transactions on Information Theory

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Julian’s Erasure Network Examples• “Multi-terminal network with independent

links”– Cut-set sum of links scaled by erasures

• In general, multi-terminal capacity is no less than the capacity of underlying network scaled by erasures

CC ~

21

~

~

C

CC

}1,0{

}2,...,1,0{ 3lg

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• Take two random length-n bit strings• “Erase” n of the symbols

• What is the probability that the two new strings match? Ans: 2-(1-)n

• What if we compare the first string with 2nR different random strings? – The probability that it matches any string is less than 2nR2-(1-)n

which is arbitrarily small for large enough n and R<(1-).

Basic Proof Idea for Erasure Channel

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0111010101000101010

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