A binary string transmitted via a memoryless i.i.d. deletion channel is received as a subsequence of the original input. From this, one obtains a posterior distribution on the channel input, corresponding to a set of candidate supersequences weighted by the number of times the received subsequence can be embedded in them. In a previous work it is conjectured on the basis of experimental data that the entropy of the posterior is minimized and maximized by the constant and the alternating strings, respectively. In this work, we present an algorithm for counting the number of subsequence embeddings using a run-length encoding of strings. We then describe two different ways of clustering the space of supersequences and prove that their cardinality depends only on the length of the received subsequence and its Hamming weight, but not its exact form. Then, we consider supersequences that contain a single embedding of a fixed subsequence, referred to as singletons, and provide a closed form expression for enumerating them using the same run-length encoding. We prove an analogous result for the minimization and maximization of the number of singletons, by the alternating and the uniform strings, respectively. Next, we prove the original minimal entropy conjecture for the special cases of single and double deletions using similar clustering techniques and the same run-length encoding, which allow us to characterize the distribution of the number of subsequence embeddings in the space of compatible supersequences to demonstrate the effect of an entropy decreasing operation.
An important body of research in this area is dedicated to deriving tight bounds on the capacity of deletion channels @cite_12 @cite_7 @cite_20 @cite_11 and developing bounding techniques @cite_4 .
@cite_4 @cite_7 @cite_12 @cite_20 @cite_11
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We develop a systematic approach, based on convex programming and real analysis, for obtaining upper bounds on the capacity of the binary deletion channel and, more generally, channels with i.i.d. insertions and deletions. Other than the classical deletion channel, we give a special attention to the Poisson-repeat channel introduced by Mitzenmacher and Drinea (IEEE Transactions on Information Theory, 2006). Our framework can be applied to obtain capacity upper bounds for any repetition distribution (the deletion and Poisson-repeat channels corresponding to the special cases of Bernoulli and Poisson distributions). Our techniques essentially reduce the task of proving capacity upper bounds to maximizing a univariate, real-valued, and often concave function over a bounded interval. The corresponding univariate function is carefully designed according to the underlying distribution of repetitions and the choices vary depending on the desired strength of the upper bounds as well as the desired simplicity of the function (e.g., being only efficiently computable versus having an explicit closed-form expression in terms of elementary, or common special, functions). Among our results, we show that the capacity of the binary deletion channel with deletion probability d is at most (1−d) logϕ for d ≥ 1 2, and, assuming the capacity function is convex, is at most 1−d log(4 ϕ) for d Furthermore, we derive the first set of capacity upper bounds for the Poisson-repeat channel. Our results uncover further striking connections between this channel and the deletion channel, and suggest, somewhat counter-intuitively, that the Poisson-repeat channel is actually analytically simpler than the deletion channel and may be of key importance to a complete understanding of the deletion channel. Finally, we derive several novel upper bounds on the capacity of the deletion channel. All upper bounds are maximums of efficiently computable, and concave, univariate real functions over a bounded domain. In turn, we upper bound these functions in terms of explicit elementary and standard special functions, whose maximums can be found even more efficiently (and sometimes, analytically, for example for d=1 2). Along the way, we develop several new techniques of potentially independent interest. For example, we develop systematic techniques to study channels with mean constraints over the reals. Furthermore, we motivate the study of novel probability distributions over non-negative integers, as well as novel special functions which could be of interest to mathematical analysis. We develop several analytical lower bounds on the capacity of binary insertion and deletion channels by considering independent uniformly distributed (i.u.d.) inputs and computing lower bounds on the mutual information between the input and output sequences. For the deletion channel, we consider two different models: i.i.d. deletion-substitution channel and i.i.d. deletion channel with additive white Gaussian noise (AWGN). These two models are considered to incorporate effects of the channel noise along with the synchronization errors. For the insertion channel case, we consider Gallager's model in which the transmitted bits are replaced with two random bits and uniform over the four possibilities independently of any other insertion events. The general approach taken is similar in all cases, however the specific computations differ. Furthermore, the approach yields a useful lower bound on the capacity for a wide range of deletion probabilities of the deletion channels, while it provides a beneficial bound only for small insertion probabilities (less than 0.25) of the insertion model adopted. We emphasize the importance of these results by noting that: 1) our results are the first analytical bounds on the capacity of deletion-AWGN channels, 2) the results developed are the best available analytical lower bounds on the deletion-substitution case, 3) for the Gallager insertion channel model, the new lower bound improves the existing results for small insertion probabilities. We present two upper bounds on the capacity of the i.i.d. binary deletion channel, where each bit is independently deleted with a fixed probability d. The first can be numerically evaluated for any fixed d. The second provides an asymptotic upper bound as d goes to 1. These appear to be the first nontrivial upper bounds for this probabilistic deletion channel. In this paper, we consider the capacity C of the binary deletion channel for the limiting case where the deletion probability p goes to 0. It is known that for any p < 1 2, the capacity satisfies C ≥ 1−H(p), where H is the standard binary entropy. We show that this lower bound is essentially tight in the limit, by providing an upper bound C ≤ 1−(1−o(1))H(p), where the o(1) term is understood to be vanishing as p goes to 0. Our proof utilizes a natural counting argument that should prove helpful in analyzing related channels.