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269
Design of capacityapproaching irregular lowdensity paritycheck codes
 IEEE TRANS. INFORM. THEORY
, 2001
"... We design lowdensity paritycheck (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on [1]. Assuming that the unde ..."
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Cited by 526 (7 self)
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We design lowdensity paritycheck (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on [1]. Assuming that the underlying communication channel is symmetric, we prove that the probability densities at the message nodes of the graph possess a certain symmetry. Using this symmetry property we then show that, under the assumption of no cycles, the message densities always converge as the number of iterations tends to infinity. Furthermore, we prove a stability condition which implies an upper bound on the fraction of errors that a beliefpropagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution. Our codes are found by optimizing the degree structure of the underlying graphs. We develop several strategies to perform this optimization. We also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds.
LT Codes
, 2002
"... We introduce LT codes, the first rateless erasure codes that are very efficient as the data length grows. ..."
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Cited by 466 (2 self)
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We introduce LT codes, the first rateless erasure codes that are very efficient as the data length grows.
A Digital Fountain Approach to Reliable Distribution of Bulk Data
 In Proc. of ACM SIGCOMM ’98
, 1998
"... The proliferation of applications that must reliably distribute bulk data to a large number of autonomous clients motivates the design of new multicast and broadcast prot.ocols. We describe an ideal, fully scalable protocol for these applications that we call a digital fountain. A digital fountain a ..."
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Cited by 462 (19 self)
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The proliferation of applications that must reliably distribute bulk data to a large number of autonomous clients motivates the design of new multicast and broadcast prot.ocols. We describe an ideal, fully scalable protocol for these applications that we call a digital fountain. A digital fountain allows any number of heterogeneous clients to acquire bulk data with optimal efficiency at times of their choosing. Moreover, no feedback channels are needed to ensure reliable delivery, even in the face of high loss rates. We develop a protocol that closely approximates a digital fountain using a new class of erasure codes that for large block sizes are orders of magnitude faster than standard erasure codes. We provide performance measurements that demonstrate the feasibility of our approach and discuss the design, implementation and performance of an experimental system. 1
Bullet: High Bandwidth Data Dissemination Using an Overlay Mesh
, 2003
"... In recent years, overlay networks have become an effective alternative to IP multicast for efficient point to multipoint communication across the Internet. Typically, nodes selforganize with the goal of forming an efficient overlay tree, one that meets performance targets without placing undue burd ..."
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Cited by 409 (23 self)
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In recent years, overlay networks have become an effective alternative to IP multicast for efficient point to multipoint communication across the Internet. Typically, nodes selforganize with the goal of forming an efficient overlay tree, one that meets performance targets without placing undue burden on the underlying network. In this paper, we target highbandwidth data distribution from a single source to a large number of receivers. Applications include largefile transfers and realtime multimedia streaming. For these applications, we argue that an overlay mesh, rather than a tree, can deliver fundamentally higher bandwidth and reliability relative to typical tree structures. This paper presents Bullet, a scalable and distributed algorithm that enables nodes spread across the Internet to selforganize into a high bandwidth overlay mesh. We construct Bullet around the insight that data should be distributed in a disjoint manner to strategic points in the network. Individual Bullet receivers are then responsible for locating and retrieving the data from multiple points in parallel. Key contributions of this work include: i) an algorithm that sends data to di#erent points in the overlay such that any data object is equally likely to appear at any node, ii) a scalable and decentralized algorithm that allows nodes to locate and recover missing data items, and iii) a complete implementation and evaluation of Bullet running across the Internet and in a largescale emulation environment reveals up to a factor two bandwidth improvements under a variety of circumstances. In addition, we find that, relative to treebased solutions, Bullet reduces the need to perform expensive bandwidth probing.
Tcplike congestion control for layered multicast data transfer
, 1998
"... Abstract—We present a novel congestion control algorithm suitable for use with cumulative, layered data streams in the MBone. Our algorithm behaves similarly to TCP congestion control algorithms, and shares bandwidth fairly with other instances of the protocol and with TCP flows. It is entirely rece ..."
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Cited by 384 (12 self)
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Abstract—We present a novel congestion control algorithm suitable for use with cumulative, layered data streams in the MBone. Our algorithm behaves similarly to TCP congestion control algorithms, and shares bandwidth fairly with other instances of the protocol and with TCP flows. It is entirely receiver driven and requires no perreceiver status at the sender, in order to scale to large numbers of receivers. It relies on standard functionalities of multicast routers, and is suitable for continuous stream and reliable bulk data transfer. In the paper we illustrate the algorithm, characterize its response to losses both analytically and by simulations, and analyse its behaviour using simulations and experiments in real networks. We also show how error recovery can be dealt with independently from congestion control by using FEC techniques, so as to provide reliable bulk data transfer.
Efficient erasure correcting codes
 IEEE Transactions on Information Theory
, 2001
"... Abstract—We introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discretetime random process. As a result, we obtain a simple criterion involving the fractions of nodes of different degrees on ..."
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Cited by 308 (22 self)
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Abstract—We introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discretetime random process. As a result, we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability. By carefully designing these graphs we can construct for any given rate and any given real number a family of linear codes of rate which can be encoded in time proportional to ��@I A times their block length. Furthermore, a codeword can be recovered with high probability from a portion of its entries of length @IC A or more. The recovery algorithm also runs in time proportional to ��@I A. Our algorithms have been implemented and work well in practice; various implementation issues are discussed. Index Terms—Erasure channel, large deviation analysis, lowdensity paritycheck codes. I.
Informed content delivery across adaptive overlay networks
 In Proceedings of ACM SIGCOMM
, 2002
"... Abstract—Overlay networks have emerged as a powerful and highly flexible method for delivering content. We study how to optimize throughput of large transfers across richly connected, adaptive overlay networks, focusing on the potential of collaborative transfers between peers to supplement ongoing ..."
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Cited by 236 (9 self)
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Abstract—Overlay networks have emerged as a powerful and highly flexible method for delivering content. We study how to optimize throughput of large transfers across richly connected, adaptive overlay networks, focusing on the potential of collaborative transfers between peers to supplement ongoing downloads. First, we make the case for an erasureresilient encoding of the content. Using the digital fountain encoding approach, end hosts can efficiently reconstruct the original content of size from a subset of any symbols drawn from a large universe of encoding symbols. Such an approach affords reliability and a substantial degree of applicationlevel flexibility, as it seamlessly accommodates connection migration and parallel transfers while providing resilience to packet loss. However, since the sets of encoding symbols acquired by peers during downloads may overlap substantially, care must be taken to enable them to collaborate effectively. Our main contribution is a collection of useful algorithmic tools for efficient summarization and approximate reconciliation of sets of symbols between pairs of collaborating peers, all of which keep message complexity and computation to a minimum. Through simulations and experiments on a prototype implementation, we demonstrate the performance benefits of our informed contentdelivery mechanisms and how they complement existing overlay network architectures. Index Terms—Bloom filter, content delivery, digital fountain, erasure code, minwise sketch, overlay, peertopeer, reconciliation. I.
Analysis of sumproduct decoding of lowdensity paritycheck codes using a Gaussian approximation
 IEEE TRANS. INFORM. THEORY
, 2001
"... Density evolution is an algorithm for computing the capacity of lowdensity paritycheck (LDPC) codes under messagepassing decoding. For memoryless binaryinput continuousoutput additive white Gaussian noise (AWGN) channels and sumproduct decoders, we use a Gaussian approximation for message densi ..."
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Cited by 209 (2 self)
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Density evolution is an algorithm for computing the capacity of lowdensity paritycheck (LDPC) codes under messagepassing decoding. For memoryless binaryinput continuousoutput additive white Gaussian noise (AWGN) channels and sumproduct decoders, we use a Gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm. We convert the infinitedimensional problem of iteratively calculating message densities, which is needed to find the exact threshold, to a onedimensional problem of updating means of Gaussian densities. This simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better, but also makes it easier to design good irregular LDPC codes for AWGN channels. For various regular LDPC codes we have examined, thresholds can be estimated within 0.1 dB of the exact value. For rates between 0.5 and 0.9, codes designed using the Gaussian approximation perform within 0.02 dB of the best performing codes found so far by using density evolution when the maximum variable degree is IH. We show that by using the Gaussian approximation, we can visualize the sumproduct decoding algorithm. We also show that the optimization of degree distributions can be understood and done graphically using the visualization.
Improved lowdensity paritycheck codes using irregular graphs
 IEEE Trans. Inform. Theory
, 2001
"... Abstract—We construct new families of errorcorrecting codes based on Gallager’s lowdensity paritycheck codes. We improve on Gallager’s results by introducing irregular paritycheck matrices and a new rigorous analysis of harddecision decoding of these codes. We also provide efficient methods for ..."
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Cited by 187 (16 self)
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Abstract—We construct new families of errorcorrecting codes based on Gallager’s lowdensity paritycheck codes. We improve on Gallager’s results by introducing irregular paritycheck matrices and a new rigorous analysis of harddecision decoding of these codes. We also provide efficient methods for finding good irregular structures for such decoding algorithms. Our rigorous analysis based on martingales, our methodology for constructing good irregular codes, and the demonstration that irregular structure improves performance constitute key points of our contribution. We also consider irregular codes under belief propagation. We report the results of experiments testing the efficacy of irregular codes on both binarysymmetric and Gaussian channels. For example, using belief propagation, for rate I R codes on 16 000 bits over a binarysymmetric channel, previous lowdensity paritycheck codes can correct up to approximately 16 % errors, while our codes correct over 17%. In some cases our results come very close to reported results for turbo codes, suggesting that variations of irregular low density paritycheck codes may be able to match or beat turbo code performance. Index Terms—Belief propagation, concentration theorem, Gallager codes, irregular codes, lowdensity paritycheck codes.
Accessing multiple mirror sites in parallel: using tornado codes tospeed up downloads
 in INFOCOM ’99.Eighteenth AnnualJoint Conference oftheIEEEComputer andCommunications Societies. Proceedings. IEEE,vol.1,NewYork,NY,USA,Mar. 21–25,1999,pp.275–283
"... Abstracr Mirror sites enable client requests to be serviced by any of a number of servers, reducing load at individual servers and dispersing network load. Typically, a client requests service from a single mirror site. We consider enabling a client to access a file from multiple mirror sites in pa ..."
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Cited by 182 (19 self)
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Abstracr Mirror sites enable client requests to be serviced by any of a number of servers, reducing load at individual servers and dispersing network load. Typically, a client requests service from a single mirror site. We consider enabling a client to access a file from multiple mirror sites in parallel to speed up the download. To eliminate complex clientserver negotiations that a straightforward implementation of this approach would require, we develop a feedbackfree protocol based on erasure codes. We demonstrate that a protocol using fast Tornado codes can deliver dramatic speedups at the expense of transmitting a moderate number of additional packets into the network. Our scalable solution extends naturally to allow multiple clients to access data from multiple mirror sites simultaneously. Our approach applies naturally to wireless networks and satellite networks as well. I.