Wireless Networking and Communications Group 1/37 Capacities of Erasure Networks Qualification...
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Transcript of Wireless Networking and Communications Group 1/37 Capacities of Erasure Networks Qualification...
1/37
Wireless Networking and Communications Group
Capacities of Erasure Networks
Qualification Proposal
of Brian Smith
February 23rd, 2007
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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
Outline• Erasure Networks
– Unifying Theme
• Information Capacity– Prior Work– Multiple Access Constraints– General Model – Feedback– Gaussian Networks
• Transport Capacity– Upper bounds– Achievability Results
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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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 Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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
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13
23
34
24 12
13
23
34
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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
Outline• Erasure Networks
– Unifying Theme
• Information Capacity– Prior Work– Multiple Access Constraints– General Model – Feedback– Gaussian Networks
• Transport Capacity– Upper bounds– Achievability Results
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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
– Franceschetti, Dousse, Tse & Thiran [2004]
Achievability in Random Networks
Squares: Constant size C Slabs: Width k log √n
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Wireless Networking and Communications Group
– 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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
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
43/37
Wireless Networking and Communications Group
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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Wireless Networking and Communications Group
• 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
0110101010101010101
0111010101000101010