Efficient erasure correcting codes

We study the space complexity of refuting unsatisfiable random k-CNFs in the resolution proof system. We prove that for any large enough /spl Delta/, with high probability a random k-CNF over n variables and /spl Delta/n clauses requires resolution clause space of /spl Omega/(n/spl middot//spl Delta//sup -1+/spl epsiv//k-2-/spl epsiv//), for any 0 >/spl radic/n. This bound is nearly tight. Specifically, we show that with high probability, a random 3-CNF with /spl Delta/n clauses requires tree-like refutation size of exp(/spl Omega/(n//spl Delta//sup 1+/spl epsiv//1-/spl epsiv//)), for any 0</spl epsiv/<1/2. Our space lower bound is the consequence of three main contributions. 1. We introduce a 2-player matching game on bipartite graphs G to prove that there are no perfect matchings in G. 2. We reduce lower bounds for the clause space of a formula F in resolution to lower bounds for the complexity of the game played on the bipartite graph G(F) associated with F. 3. We prove that the complexity of the game is large whenever G is an expander graph. Finally, a simple probabilistic analysis shows that for a random formula F, with high probability G(F) is an expander. We also extend our result to the case of G-PHP, a generalization of the pigeonhole principle based on bipartite graphs G. We prove that the clause space for G-PHP can be reduced to the game complexity on G.

We consider the problem of testing bipartiteness in the adjacency matrix model. The best known algorithm, due to Alon and Krivelevich, distinguishes between bipartite graphs and graphs that are /spl epsi/-far from bipartite using 0(1//spl epsi//sup 2/) queries. We show that this is optimal for non-adaptive algorithms, up to polylogarithmic factors. We also show a lower bound of /spl Omega/(1//spl epsi//sup 3/2/) for adaptive algorithms.

We present a randomized parallel algorithm for the Voronoi diagram of line segments on coarse grained parallel machines, which, for any input of n line segments on P processors (n=/spl Omega/(P/sup 3+/spl epsiv//), for any /spl epsiv/>0) requires (with high probability) O([n log n/P]) local operations, O(n/P) global-operations (or messages) per processor, O(1) levels of global data dependency (or O(1) communication phases). This generic parallel algorithm can be transformed to the CRCW PRAM model in O(log n) time with O(n) processors and thus provides a different solution from, and conceptually much simpler than, that of (Rajasekaran, 1994).

The relationship between weak convergence of channel probability measures, channel capacity, and error probability of block codes is examined for memoryless channels with general input and output alphabets. It is shown that channel capacity is a lower semi-continuous function and that every block code with maximal probability of error /spl delta/ for a nominal channel for any /spl epsiv/>0 can be modified such that the modification has a probability of error less than /spl delta/+/spl epsiv/ for all channels in a sufficiently small neighborhood of the nominal channel.

All algorithms below are optimal alphabet-independent parallel CRCW PRAM algorithms. In one dimension: Given a pattern string of length m for the string-matching problem, we design an algorithm that computes a deterministic sample of a sufficiently long substring in constant time. This problem used to be a bottleneck in the pattern preprocessing for one- and two-dimensional pattern matching. The best previous time bound was O(log/sup 2/ m/log log m). We use this algorithm to obtain the following results. 1. Improving the preprocessing of the constant-time text search algorithm from O(log/sup 2/ m/log log m) to n(log log m), which is now best possible. 2. A constant-time deterministic string-matching algorithm in the case that the text length n satisfies n=/spl Omega/(m/sup 1+/spl epsiv//) for a constant /spl epsiv/>0. 3. A simple probabilistic string-matching algorithm that has constant time with high probability for random input. 4. A constant expected time Las-Vegas algorithm for computing the period of the pattern and all witnesses and thus string matching itself, solving the main open problem remaining in string matching. >

One way to detect intruders on the Internet is to compare the similarity of two thumbprints. A thumbprint is a summary of a connection that characterizes the connection. The packet gap thumbprint consists of sequences of non-negative real number representing the time gaps between "send" packets. This paper formalized definitions of similarity between two non-negative real number sequences, by introducing /spl epsiv/-similarity, partial sum and longest /spl epsiv/-similar subsequence (LSS). Length of LSS is a measurement of similarity between two sequences. The longest /spl epsiv/-similar subsequence (LSS) problem is a generalization of the well known longest common subsequence (LCS) problem. The goal of this paper is to find an optimal solution to the LSS problem. We analyzed the property of partial sums and proposed to focus on the minimum matched partial sum which leads to an optimal solution to LSS while reduce the problem space. As the LSS problem has optimal structure, we proposed an algorithm based on dynamic programming technique. Time complexity of this algorithm is O(m/sup 2/n/sup 2/). By using a property of the partial sums, we reduced the time complexity to O(mn(m+n)).

Let P be a property of graphs. An epsiv/-test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than epsiv/n/sup 2/ edges to make it satisfy P. The property P is called testable, if for every epsiv/ there exists an epsiv/-test for P whose total number of queries is independent of the size of the input graph. O. Goldreich et al. (1996) showed that certain graph properties admit an epsiv/-test. In this paper we make a first step towards a logical characterization of all testable graph properties, and show that properties describable by a very general type of coloring problem are testable. We use this theorem to prove that first order graph properties not containing a quantifier alternation of type /spl forall//spl exist/ are always testable, while we show that some properties containing this alternation are not. Our results are proven using a combinatorial lemma, a special case of which, that may be of independent interest, is the following. A graph H is called epsiv/-unavoidable in G if all graphs that differ from G in no more than epsiv/|G|/sup 2/ places contain an induced copy of H. A graph H is called delta/-abundant in G if G contains at least delta/|G|/sup |H|/ induced copies of H. If H is epsiv/-unavoidable in G then it is also delta/(/spl epsiv/, |H|)-abundant.

In this paper we develop a new approach for learning decision trees and multivariate polynomials via interpolation of multivariate polynomials. This new approach yields simple learning algorithms for multivariate polynomials and decision trees over finite fields under any constant bounded product distribution. The output hypothesis is a (single) multivariate polynomial that is an /spl epsiv/-approximation of the target under any constant bounded product distribution. The new approach demonstrates the learnability of many classes under any constant bounded product distribution and using membership queries, such as j-disjoint DNF and multivariate polynomial with bounded degree over any field. The technique shows how to interpolate multivariate polynomials with bounded term size from membership queries only. This in particular gives a learning algorithm for O(log n)-depth decision tree from membership queries only and a new learning algorithm of any multivariate polynomial over sufficiently large fields from membership queries only. We show that our results for learning from membership queries only are the best possible.

In this paper, we consider the problem of differential space-time (ST) block coding in the context of impulse-radio ultra-wideband (IR-UWB) communications with two transmit antennas and Pulse Position Modulation (PPM). We propose a novel family of codes where the information is encoded differentially through the number of transmitted pulses, the positions of these pulses and the locations of the non-idle symbol durations within each ST block. In order to maintain a low complexity of the UWB transmitter, unipolar transmissions are ensured by avoiding all forms of amplitude-domain encoding. The proposed scheme provides a full scalability capability where depending on the value taken by the scale parameter Θ, different levels of compromise between data rates and error rates (along with the decoding complexity) can be achieved. Finally, the proposed scheme lends itself to optimal detection with cross-correlation receivers that can be implemented in the analog domain in the absence of all forms of channel state information at the transmitter and receiver sides.

We show that for low-density parity-check (LDPC) codes whose Tanner graphs have sufficient expansion, the linear programming (LP) decoder of Feldman, Karger, and Wainwright can correct a constant fraction of errors. A random graph will have sufficient expansion with high probability, and recent work shows that such graphs can be constructed efficiently. A key element of our method is the use of a dual witness: a zero-valued dual solution to the decoding linear program whose existence proves decoding success. We show that as long as no more than a certain constant fraction of the bits are flipped by the channel, we can find a dual witness. This new method can be used for proving bounds on the performance of any LP decoder, even in a probabilistic setting. Our result implies that the word error rate of the LP decoder decreases exponentially in the code length under the binary-symmetric channel (BSC). This is the first such error bound for LDPC codes using an analysis based on "pseudocodewords." Recent work by Koetter and Vontobel shows that LP decoding and min-sum decoding of LDPC codes are closely related by the "graph cover" structure of their pseudocodewords; in their terminology, our result implies that that there exist families of LDPC codes where the minimum BSC pseudoweight grows linearly in the block length

Stochastic simulation of power systems with integrated intermittent renewable resources

Reduced-order autoregressive modeling for center-frequency estimation.

In this paper, we describe a very simple (1+/spl epsi/)-approximation algorithm for the multicommodity flow problem. The algorithm runs in time that is polynomial in N (the number of nodes in the network) and /spl epsiv//sup -1/ (the closeness of the approximation to optimal). The algorithm is remarkable in that it is much simpler than all known polynomial time flow algorithms (including algorithms for the special case of one-commodity flow). In particular, the algorithm does not rely on augmenting paths, shortest paths, min-cost paths, or similar techniques to push flow through a network. In fact, no explicit attempt is ever made to push flow towards a sink during the algorithm. Because the algorithm is so simple, it can be applied to a variety of problems for which centralized decision making and flow planning is not possible. For example, the algorithm can be easily implemented with local control in a distributed network and it can be made tolerant to link failures. In addition, the algorithm appears to perform well in practice. Initial experiments using the DIMACS generator of test problems indicate that the algorithm performs as well as or better than previously known algorithms, at least for certain test problems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Current methods for optimization of low-density parity-check (LDPC) codes analyze the degree distribution pair asymptotically as block length approaches infinity. This effectively ignores the discrete nature of the space of valid degree distribution pairs for LDPC codes of finite block length. While large codes are likely to conform reasonably well to the infinite block length analysis, shorter codes have no such guarantee. We present and analyze an algorithm for completely enumerating the space of all valid degree distribution pairs for a given block length, code rate, maximum variable node degree, and maximum check node degree. We then demonstrate this algorithm on an example LDPC code of finite block length. Finally, we discuss how the result of this algorithm can be utilized by discrete optimization routines to form novel methods for the optimization of small block length LDPC codes.

It is well known that the performance of the wireless communication systems can be enhanced by using multiple transmit and receive antennas, which is generally referred to as the MIMO technique, and has been incorporated into the IEEE 802.11n standard, i.e. one of the Wi-Fi systems. The MIMO technique has also been adopted by the 3GPP Long Term Evolution (LTE) and the 3GPP2 Ultra Mobile Broadband (UMB) standards. The space-time coding is a promising way to realize the gain in the wireless communications system using MIMO. The space time block code, and the orthogonal space time block code in particular have been proposed in research and implemented in practical wireless communication systems due to their low computational complexity in maximum likelihood decoding implementation. However, the code rates of the orthogonal space time block codes are usually lower than $3/4$ when the number of transmit antennas is greater than two, which limit their throughput. To increase the code rate and the throughput of the orthogonal space time block code for more than two transmit antennas, a family of new space-time block codes (STBC) for 3 and 4 transmit antennas in a MIMO wireless communication system have been proposed in this Ph.D. research. These new codes are orthogonal and achieve full diversity, full rate using Triple QPSK'' modulation, which is proposed in this research. We investigated the performance of these STBCs when the channel state information (CSI) is perfectly known at the receiver. The CSI is not known to the transmitter in all of the studies in this dissertation. An analytical bound of the probability of bit error is developed and the Monte Carlo simulation is used when investigating these new STBC codes. In practical wireless communication system, the CSI is generally not available at the receiver and it is usually estimated by the receiver through the pilot signals or training signals known to the receiver. No matter what channel estimation method is used, the estimate of CSI will not match the CSI exactly and is noisy. The performance of these new STBCs using theTriple QPSK'' modulation when the estimate of the CSI is employed to detect the symbol is investigated in this dissertation. An upper bound of the bit error probability is developed when the noisy CSI is used to detect the symbols. The performance of the new STBCs using the Triple QPSK'' modulation is also evaluated using Monte Carlo simulation. When the CSI is not available to either the receiver or the transmitter, the differential encoding and decoding algorithm is applied to these new STBCs, which use theTriple QPSK'' modulation. The results showed that these orthogonal STBCs achieve full diversity full rate using ""Triple-QPSK"" modulation with differential detection. This provides a means of achieving space diversity and multiplexing gain without the CSI. A new differential encoding and decoding algorithm is proposed and analyzed for STBCs with 3 transmit antennas, whose transmission matrix is not square matrix, as part of this Ph.D. research. The STBCs using this algorithm achieve full diversity the MIMO system can possibly provide.