Wireless Networking and Communications Group 1/37 Capacities of Erasure Networks Qualification Proposal of Brian Smith February 23 rd, 2007

Wireless Networking and Communications Group 2/37 Outline Erasure Networks –Unifying Theme Information Capacity –Multiple Access Constraints –General Model –Feedback –Gaussian Networks Transport Capacity –Upper bounds –Achievability Results

Wireless Networking and Communications Group 3/37 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

Wireless Networking and Communications Group 4/37 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

Wireless Networking and Communications Group 5/37 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 XOutput Y E 1 1-    Memoryless Symmetric Binary Erasure Channel

Wireless Networking and Communications Group 6/37 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

Wireless Networking and Communications Group 7/37 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

Wireless Networking and Communications Group 8/37 Outline Erasure Networks –Unifying Theme Information Capacity –Prior Work –Multiple Access Constraints –General Model –Feedback –Gaussian Networks Transport Capacity –Upper bounds –Achievability Results

Wireless Networking and Communications Group 9/37 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 Julian’s Memoryless Erasure Channel

Wireless Networking and Communications Group 10/37 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

Wireless Networking and Communications Group 11/37 Wireless Erasure Network Modified Cut-Set Bound Modified Cut-Set Bound Example Bound Evaluation

Wireless Networking and Communications Group 12/37 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!

Wireless Networking and Communications Group 13/37 Capacity Achieving Modified Cut-Set Bound Example Bound Evaluation No Interference Broadcast Constraint Multiple-Access Constraint

Wireless Networking and Communications Group 14/37 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

Wireless Networking and Communications Group 15/37 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

Wireless Networking and Communications Group 16/37 Multiple-Access Constraint Proof Idea For 2 nRB 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 E S b –In time block b, all nodes in the cut (labeled by S) receive “identical” inputs –Challenge: Events labeled by E S b and E S’ b+1 not independent L=6 SS’

Wireless Networking and Communications Group 17/37 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

Wireless Networking and Communications Group 18/37 General Erasure Interference Network Model Our contribution: A new model Allow each node to have more than one input and output

Wireless Networking and Communications Group 19/37 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

Wireless Networking and Communications Group 20/37 Feedback in Erasure Networks Joint 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: ???

Wireless Networking and Communications Group 21/37 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

Wireless Networking and Communications Group 22/37 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

Wireless Networking and Communications Group 23/37 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 s d1d1 d2d2

Wireless Networking and Communications Group 24/37 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

Wireless Networking and Communications Group 25/37 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

Wireless Networking and Communications Group 26/37 Outline Erasure Networks –Unifying Theme Information Capacity –Prior Work –Multiple Access Constraints –General Model –Feedback –Gaussian Networks Transport Capacity –Upper bounds –Achievability Results

Wireless Networking and Communications Group 27/37 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

Wireless Networking and Communications Group 28/37 Transport Capacity Joint work with P. Gupta –Erasure Probability as Function of Geographic Distance –Minimum Node Separation Constraint: d>d min –Threshold Model  (d)=0, d  d*; 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 x5x5 x4x4 x3x3 x2x2 x1x1 y6y6 y8y8 y7y7 y9y9

Wireless Networking and Communications Group 29/37 Models: Broadcast Constraint and Single Antenna Our Result: Transport Capacity ≤ κ n for both cases Constant depends only on d min, d* in exponential decay mode –Some Notation: d1hd1h d2hd2h d 34 Cut 1 Node 1

Wireless Networking and Communications Group 30/37 Proof Sketch –Broadcast Constraint, Only –Single Antenna Expected Value of the Number of ‘1’ Entries in H m

Wireless Networking and Communications Group 31/37 Squish dihdih d i+1 h Node i Node m d m-1 h Move all nodes with index greater than ‘m’ in as close as possible, within minimum distance constraint, to bound summation

Wireless Networking and Communications Group 32/37 –Franceschetti, Dousse, Tse & Thiran [2004] Achievability in Random Networks Squares: Constant size CSlabs: Width k log √n

Wireless Networking and Communications Group 33/37 –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

Wireless Networking and Communications Group 34/37 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 information d* ln n

Wireless Networking and Communications Group 35/37 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

Wireless Networking and Communications Group 36/37 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

Wireless Networking and Communications Group 37/37 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

Wireless Networking and Communications Group 38/37 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

Wireless Networking and Communications Group 39/37 Major Coursework EE381JProbability and Stochastic ProcessesdeVecianaA EE381K-2Digital CommunicationsAndrewsA EE381K-7Information TheoryVishwanathA EE381K-9Advanced Signal ProcessingHeathA EE381VChannel CodingVishwanathA EE380NOptimization in Engineering SystemsBaldickA EE381K-13Communication Networks: Analysis and DesignShakkottaiA EE381K-5Advanced Communication NetworksShakkottaiA EE381K-8Digital Signal ProcessingBovikA

Wireless Networking and Communications Group 40/37 Supporting Coursework CS388GAlgorithms: Technique and TheoryPlaxtonA M381CReal AnalysisBecknerA CS388GCombinatorics and Graph TheoryZuckermanB+ M385CTheory of ProbabilityZitkovicCR M393CStatistical PhysicsRadinA

Wireless Networking and Communications Group 41/37 Conference Publications “Transport Capacity of Wireless Erasure Networks,” B.Smith, S. Vishwanath. In Proceedings of the 44 th Allerton Conference on Communication, Control, and Computing, Monticello, IL, Sep “Network Coding in Interference Networks,” B. Smith, S. Vishwanath. In Proceedings of 2005 Conference on Information Sciences and Systems, Baltimore, MD, Mar “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 42 nd Allerton Conference on Communication, Control, and Computing, Monticello, IL, Oct

Wireless Networking and Communications Group 42/37 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

Wireless Networking and Communications Group 43/37 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

Wireless Networking and Communications Group 44/37 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 2 nR different random strings? –The probability that it matches any string is less than 2 nR 2 -(1-  )n which is arbitrarily small for large enough n and R<(1-  ). Basic Proof Idea for Erasure Channel