M. Luby, M. Mitzenmacher and A. Shokrollahi. Analysis of Random Processes via And-Or Tree Evaluation. Proc. 9th Annual ACM-SIAM Symposium on Discrete Algorithms, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....to reconstruct the file. As files grow in size, the power of this application will be immense; hence the need to correct the error of the 1997 Reed Solomon coding tutorial. There are other erasure coding techniques in addition to the one which this tutorial addresses. Examples are Tornado codes [13, 14], Cauchy Reed Solomon codes [1] and other parity based schemes [6] Of these, Tornado codes are worth special mention, as they form the backbone of the Digital Fountain content dispersal system [2] Tornado codes have a randomized structure so that with the addition of m extra parity blocks, a ....

M. Luby, M. Mitzenmacher, and A. Shokrollahi. Analysis of random processes via and-or tree evaluation. In 9th Annual ACM-SIAM Symposium on Discrete Algorithms, January 1998.

....number of decoded message blocks continues to increase, more and more higher degree check and auxiliary blocks will become usable. Note that one can execute this decoding procedure on the fly, as check blocks arrive. To see that the decoding process takes linear time, following the approach of [5, 6], we think of the composite message blocks and the check blocks as the left and right vertices, respectively, of a bipartite graph G. A check block has edges to and only to the message blocks that comprise it in terms of the XOR. We say that an edge has left (respectively right) degree d if the ....

M. Luby, M. Mitzenmacher, and A. Shokrollahi. Analysis of Random Processes via And-Or Tree Evaluation. In SODA, 1998.

....methods on random CNFs. This question has been extensively researched empirically, and random CNFs are commonly used as test cases for analysis and comparison of SAT solvers. From a theoretical point of view, several lower and upper bounds were given for DLL type algorithms under this distribution [3, 8, 21]. Upper bounds for algorithms imply lower bounds on the satis ability threshold, and in fact, all lower bounds on the threshold so far, have come from analyzing speci c SAT solving algorithms. Most of the algorithms for which average case analysis has been applied so far are classi ed as myopic ....

....ed by RWalkSAT, if the higher layers have already been satis ed. The expected running time is exponential in the number of layers, and so we get our polynomial upper bound. At such low clause density, even the simple pure literal heuristic nds with high probability a satisfying assignment [8, 21]. Indeed, our proof uses heavily elements of the original proof for the pure literal heuristic, given by [8] and all the layers but the last are sets of clauses removed by a single application of the pure literal rule. 3.1 The Pure Literal Rule A literal in C is called pure if it appears ....

Luby M. , M. Mitzenmacher, and A. Shokrollahi. Analysis of Random Processes via AndOr Tree Evaluation. In Proceeding of ACM-SIAM Symposium on Discrete Algorithms, 1998.

....of high data availability in a partitionable ad hoc network, so that each mobile node can access a copy of the data it needs with high probability. However, it does not address the issue of dynamic data dissemination, especially in a peer to peer fashion. Byers et al. first propose Tornado coding [1, 2, 3, 7]. Their scheme involves the encoding of a large file (of k packets) to n encoded packets (where n = stretch factor k) The original file can then be recovered by decoding # arbitrary but distinct encoded packets. They apply Tornado coding to parallel file downloading from multiple static ....

M. Luby, M. Mitzenmacher, and A. Shokrollahi. Analysis of random processes via and-or tree evaluation. ACM/SIAM Symposium on Discrete Algorithms (SODA'98), 1998.

....large n, there exists a code with decoding ineciency and d growing logarithmically in 1= 3 Assumptions and Basic Scenarios 3.1 One Sided Degree Distributions The heuristics of this paper work to optimize only the output symbol degree distribution. Later analysis of Tornado codes such as [3] also optimized relative input symbol probabilities. Instead, we assume input symbols have equal probabilities when constructing output symbols. An important side e ect of this choice is that all output symbols of the same degree have equal probability. 3.2 Client Server Model In our model, the ....

M. Luby, M. Mitzenmacher, and A. Shokrollahi. Analysis of random processes via and-or tree evaluation. In 9th Annual ACM-SIAM Symposium on Discrete Algorithms, January 1998.

....redundant packets are generated by taking the exclusive or of packets in the previous stage. Assume c 1 in Figure 1 is generated from x 1 ; x 2 , and x 3 . If x 3 is lost during transmission, the receiver can recover x 3 upon receiving x 1 ; x 2 , and c 1 . The technique is described in detail in [4, 5]. In [6] Byers, Luby and Mitzenmacher propose the use of Digital Fountains a reliable multicast protocol that uses Tornado codes. In [6] the sender (server) initially chooses a stretch factor, N=K. It generates an encoding of the original data, where K denotes the number of original packets ....

M. Luby, M. Mitzenmacher, A. Shokrollahi, \Analysis of Random Processes via And-Or Tree Evaluation", Proceedings of the 9 th Annual ACM-SIAM Symposium on Discrete Algorithms, January, 1998.

....but exhibit additional key properties which provide a better approximation to an idealized digital fountain. We rst outline how the theoretical basis for these codes di ers from the traditional Reed Solomon (RS) erasure codes. Then we give a speci c example of a Tornado code based on [18] [19] and compare its performance to a standard RS code. For the rest of the discussion, we will consider erasure codes that take a set of k source data packets and produce a set of redundant packets for a total of n = k encoding packets all of a xed length P . A. Theory We begin by providing ....

....graph l a h g f e d c b a b f a b c d g c e g h c d e f h denotes exclusive or Fig. 1. Structure of Tornado Codes the system of equations so that the number of additional packets and the coding times are simultaneously small is a dicult challenge [18] [19]. For both Tornado and LT codes, the linear equations have the form y 3 = x 1 x 4 x 7 , where is bitwise exclusive or. Tornado codes also use equations of the form y 53 = y 3 y 7 y 13 ; that is, redundant packets may be derived from other redundant packets, and in general there may be ....

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M. Luby, M. Mitzenmacher, and A. Shokrollahi, \Analysis of Random Processes via And-Or Tree Evaluation." In Proceedings of the 9 th Annual ACM-SIAM Symposium on Discrete Algorithms, January 1998, pp. 364-73.

....efficient as a function of the data length. Thus, LT codes are universal in the sense that they are simultaneously near optimal for every erasure channel and they are very efficient as the data length grows. The analysis of LT codes is quite different than the analysis of Tornado codes [16] 17] [15]. In particular, the Tornado codes analysis is only applicable to graphs with constant maximum degree, and LT codes use graphs of logarithmic density, and thus the Tornado codes analysis does not apply. Furthermore, the Tornado codes analysis relies on techniques that lead to a reception overhead ....

M. Luby, M. Mitzenmacher, A. Shokrollahi. Analysis of Random Processes via And-Or Tree Evaluation. Proceedings of 9th Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, California, January 25--27, 1998.

....neighborhood around each node. Analysis in this case is greatly simplified since random variables that correspond to messages in our message passing algorithms can be treated as independent. The martingale arguments relating the idealized tree model and actual graphs was first applied to coding in [9] and is now standard; see for example [9, 10, 11, 16] 3 Symmetric q ary channels 3.1 A simple decoding algorithm To begin our analysis, it is useful to consider the idealized case of a symmetric q ary channel. Recall that in a symmetric q ary channel the probability that a symbol is received ....

....this case is greatly simplified since random variables that correspond to messages in our message passing algorithms can be treated as independent. The martingale arguments relating the idealized tree model and actual graphs was first applied to coding in [9] and is now standard; see for example [9, 10, 11, 16]. 3 Symmetric q ary channels 3.1 A simple decoding algorithm To begin our analysis, it is useful to consider the idealized case of a symmetric q ary channel. Recall that in a symmetric q ary channel the probability that a symbol is received in error is p, and when an error occurs the received ....

M. Luby, M. Mitzenmacher, and M. A. Shokrollahi. Analysis of Random Processes via And- Or Tree Evaluation. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 364-373, 1998.

....around each node. Anal ysis in this case is greatl y simpl ified since random variabl es that correspond to messages in our message passing al gorithms can be treated as independent. The martingal e arguments rel ating the ideal ized tree model and actual graphs is now standard; see for exampl e [10, 11, 12, 17]. 3 Symmetric q ary channels 3.1 A simple decoding algorithm To begin our an al sis, it is useful to consider the idealF#5 case of a symmetric q ary . Recal that in a symmetric q ary the probabilH y that a symbol is received in error is p, and when an error occurs the received symbol is ....

M. Luby, M. Mitzenmacher, and M. A. Shokrol l ahi. Anal ysis of Random Processes via And-Or Tree Eval uation. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 364-373, 1998.

....section, we describe in some detail the construction of a specific Tornado code and explain some of the general principles behind Tornado codes. We first outline how these codes differ from traditional Reed Solomon erasure codes. Then we give a specific example of a Tornado code based on [11] [12] and compare its performance to a standard Reed Solomon code. For the rest of the discussion, we will consider erasure codes that take a set of # source data packets and produce a set of # redundant packets for a total of # # # # # encoding packets all of a fixed length # . A. Theory We begin by ....

....# packets no longer suffice to reconstruct the source data; instead slightly more than # packets are needed. In fact, designing the proper structure for the system of equations so that the number of additional packets and the coding times are simultaneously small is a difficult challenge [11] [12]. For Tornado codes, the equations have the form # # # # # # # # # # # , where # is bitwise exclusive or. Tornado codes also n Bipartite graph Bipartite graph = source data = redundancy k l a h g f e d c b a b f a b c d g c e g h c d e f h denotes ....

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M. Luby, M. Mitzenmacher, and A. Shokrollahi, "Analysis of Random Processes via And-Or Tree Evaluation." In Proceedings of the # ## Annual ACM-SIAM Symposium on Discrete Algorithms, January 1998.

....of codes on graphs by Wiberg et.al. 13] and Forney [14] MacKay [15] 16] showed that there exist Gallager codes that outperform turbo codes [17] A major breakthrough was the construction of irregular Gallager codes [18] and the development of a method to analyze them for erasure channels [9] [19]. These methods were adapted to memoryless channels with continuous output alphabets (additive white Gaussian noise channels, Laplace channels, etc. by Richardson and Urbanke [20] who also coined the term density evolution for a tool to analyze the asymptotic performance of Gallager and turbo ....

....then conclude that there exists at least one graph and one coset for which the decoding probability of error can be made arbitrarily small on an information sequence generated uniformly at random if the noise variance does not exceed a threshold. The proofs follow closely the ideas presented in [19], 20] for memoryless channels and rely heavily on results presented there. The main di erence is that the channel under consideration here has an input dependent memory. Therefore, we rst must prove a concentration statement for every possible input sequence, and then show that the average ....

M. Luby, M. Mitzenmacher, and M. A. Shokrollahi, \Analysis of random processes via and-or tree evaluation," in Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 364-373, 1998.

....length of the original message to encode and decode, making them unsuitable for many practical applications. For example, encoding and decoding an entire file could take orders of magnitude more time than the download. By making use of Tornado codes, a new class of erasure codes developed in [12] [13], we can develop practical schemes that employ encoding. Finally, because we expect our approach to be useful in various domains (including mobile wireless networks, satellite networks, and the Internet) we develop a general approach applicable to a variety 0 7803 5420 6 99 10.00 (c) 1999 IEEE 3 ....

....refer the reader to [6] for our justification of this choice. To keep the paper self contained, we briefly describe the relevant properties and the performance of Tornado erasure codes. A more detailed, technical description of Tornado codes and their theoretical properties is provided in [12] and [13]. A. Tornado Code Overview As with standard erasure codes, Tornado codes produce an n packet encoding from a k packet source. However, Tornado codes relax the decoding guarantee as follows: to reconstruct the source data, it is necessary to recover #k of the n encoding packets, where # 1. We ....

[Article contains additional citation context not shown here]

M. Luby, M. Mitzenmacher, and A. Shokrollahi, "Analysis of Random Processes via And-Or Tree Evaluation." In Proceedings of the 9 th Annual ACM-SIAM Symposium on Discrete Algorithms, January 1998.

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M. Luby, M. Mitzenmacher, and M.A. Shokrollahi. Analysis of random processes via and-or tree evaluation. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 364--373, 1998.

....the length of the original message to encode and decode, making them unsuitable for many practical applications. For example, encoding and decoding an entire file could take orders of magnitude more time than the download. By making use of Tornado codes, a new class of erasure codes developed in [12, 13], we can develop practical schemes that employ encoding. Finally, because we expect our approach to be useful in various domains (including mobile wireless networks, satellite networks, and the Internet) we develop a general approach applicable to a variety 3 of media and give detailed ....

....Codes In this section, we lay out the theoretical properties of Tornado codes in more detail. We outline how these codes differ from traditional Reed Solomon erasure codes, and focus in particular on performance comparisons. A detailed, technical description of these codes is provided in [12] and [13]. For the rest of the discussion, we will consider erasure codes that take a set of k packets and produce a set of redundant packets for a total of n = k encoding packets all of a fixed length P . Tornado codes have the following erasure property: to reconstruct the source data, it suffices ....

[Article contains additional citation context not shown here]

M. Luby, M. Mitzenmacher, and A. Shokrollahi, "Analysis of Random Processes via And-Or Tree Evaluation." In Proceedings of the 9 th Annual ACM-SIAM Symposium on Discrete Algorithms, January 1998.

....they performed poorly in practice. The work [10] had several major impacts on subsequent work on LDPC codes. First, it contained for the first time a rigorous analysis of a probabilistic decoding algorithm for LDPC codes. This analysis was later greatly simplified by Luby, Mitzenmacher, and myself [7], and this later method developed into the core of the analysis of LDPC codes under other, more complicated error models. Second, the paper [10] proves that highly irregular bipartite graphs perform much better than regular graphs (which were the method of choice up to then) if the particular ....

....high speed networks, but codes are typically used in situations where one does not know the positions of the errors. Here the problem is much harder. Based on L A T E X style file for Lecture Notes in Computer Science documentation 3 the approach in [10] and equipped with the new analysis in [7], Luby, Mitzenmacher, Spielman, and myself rigorously analyzed some of Gallager s original flipping decoders and invented methods to design appropriate degree distributions so that the corresponding graphs could recover from as many errors as possible [8] To obtain a better performance, the ....

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M. Luby, M. Mitzenmacher, and M.A. Shokrollahi. Analysis of random processes via and-or tree evaluation. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 364--373, 1998.

.... The main theorem of [8] states that if the graph is chosen at random with degree distribution ( ae) and if the erasures occur at random positions, then the above erasure recovery algorithm can correct a ffi fraction of losses if ae(1 Gamma ffi(x) 1 Gamma x for x 2 [0; 1] In a later paper [6], this condition was slightly relaxed to ffi (1 Gamma ae(1 Gamma x) x for x 2 (0; ffi] 1) The paper [8] further exhibited for any ffl 0 and any rate R an infinite sequence of degree distributions ( ae) giving rise to codes of rate at least R such that the above inequality is valid for ....

....from the one given in that paper. From a theoretical point of view, codes obtained from these sequences should perform better than the heavy tail Poisson distribution, since they allow for a larger loss fraction for a given rate and a given average left degree. Furthermore, the analysis given in [6] suggests that the actual performance of the code is related to how accurately the neighborhood of a message node is described by a tree given by the degree distributions. For instance, the performance of regular graphs is much more sharply concentrated around the value predicted by the ....

M. Luby, M. Mitzenmacher, and M.A. Shokrollahi. Analysis of random processes via and-or tree evaluation. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 364--373, 1998.

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M. Luby, M. Mitzenmacher and A. Shokrollahi. Analysis of Random Processes via And-Or Tree Evaluation. Proc. 9th Annual ACM-SIAM Symposium on Discrete Algorithms, 1998.

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M. Luby, M. Mitzenmacher, and A. Shokrollahi. Analysis of random processes via and-or tree evaluation. In 9th Annual ACM-SIAM Symposium on Discrete Algorithms, January 1998.

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M. Luby, M. Mitzenmacher, and A. Shokrollahi. Analysis of Random Processes via And-Or Tree Evaluation. In SODA, 1998.

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Luby M. , M. Mitzenmacher, and A. Shokrollahi. Analysis of Random Processes via And-Or Tree Evaluation. In Proceeding of ACM-SIAM Symposium on Discrete Algorithms, 1998.

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M. G. Luby, M. Mitzenmacher, and M. A. Shokrollahi. Analysis of random processes via and-or tree evaluation. In SODA: ACM-SIAM Symposium on Discrete Algorithms, pages 364--373, Jan. 1998.

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M. Luby, M. Mitzenmacher, A. Shokrollahi. Analysis of Random Processes via And-Or Tree Evaluation. In Proceedings of the 9 th Annual ACM-SIAM Symposium on Discrete Algorithms, January, 1998.

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