We consider the problem of communicating over a channel that breaks the message block into fragments of random lengths, shuffles them out of order, and deletes a random fraction of the fragments. Such a channel is motivated by applications in molecular data storage and forensics, and we refer to it as the torn-paper channel. We characterize the capacity of this channel under arbitrary i.i.d. fragment length distributions and deletion probabilities. Precisely, we show that the capacity is given by a closed-form expression that can be interpreted as F−A, where F is the coverage fraction, i.e., the fraction of the input codeword that is covered by output fragments, and A is an alignment cost incurred due to the lack of ordering in the output fragments. We then consider a noisy version of the problem, where the fragments are corrupted by binary symmetric noise. We derive upper and lower bounds to the capacity, both of which can be seen as F−A expressions. These bounds match for specific choices of fragment length distributions, and they are approximately tight in cases where there are not too many short fragments.

MOTIVATION: Pairwise sequence alignment is a heavy computational burden, particularly in the context of third-generation sequencing technologies. This issue is commonly addressed by approximately estimating sequence similarities using a hash-based method such as MinHash. In MinHash, all k-mers in a read are hashed and the minimum hash value, the min-hash, is stored. Pairwise similarities can then be estimated by counting the number of min-hash matches between a pair of reads, across many distinct hash functions. The choice of the parameter k controls an important tradeoff in the task of identifying alignments: larger k-values give greater confidence in the identification of alignments (high precision) but can lead to many missing alignments (low recall), particularly in the presence of significant noise. RESULTS: In this work, we introduce LexicHash, a new similarity estimation method that is effectively independent of the choice of k and attains the high precision of large-k and the high sensitivity of small-k MinHash. LexicHash is a variant of MinHash with a carefully designed hash function. When estimating the similarity between two reads, instead of simply checking whether min-hashes match (as in standard MinHash), one checks how "lexicographically similar" the LexicHash min-hashes are. In our experiments on 40 PacBio datasets, the area under the precision-recall curves obtained by LexicHash had an average improvement of 20.9% over MinHash. Additionally, the LexicHash framework lends itself naturally to an efficient search of the largest alignments, yielding an O(n) time algorithm, and circumventing the seemingly fundamental O(n2) scaling associated with pairwise similarity search. AVAILABILITY AND IMPLEMENTATION: LexicHash is available on GitHub at https://github.com/gcgreenberg/LexicHash.

Most DNA sequencing technologies are based on the shotgun paradigm: many short reads are obtained from random unknown locations in the DNA sequence. A fundamental question, studied in [1], is what read length and coverage depth (i.e., the total number of reads) are needed to guarantee reliable sequence reconstruction. Motivated by DNA-based storage, we study the coded version of this problem; i.e., the scenario in which the DNA molecule being sequenced is a codeword from a predefined codebook. Our main result is an exact characterization of the capacity of the resulting shotgun sequencing channel as a function of the read length and coverage depth. In particular, our results imply that while in the uncoded case, O(n) reads of length greater than 2logn are needed for reliable reconstruction of a length-n binary sequence, in the coded case, only O(n/log n) reads of length greater than log n are needed for the capacity to be arbitrarily close to 1.

The capacity region of several multiuser models in information theory can be enlarged by utilizing feedback of the received symbols. This is in contradiction to the discrete memoryless case, where feedback is known not to change the capacity. In this paper, we consider two broadcast models with noisy feedback from both the receivers. The models are derived from a standard memoryless scalar GBC, where two intermediate passive nodes are assumed to be observing the transmissions via separate noisy links corrupted by independent AWGN. In our first model, the scalar output from each intermediate node is passed through two additional independent AWGN links, called feedback and forward links. The output of the feedback link is observed by the transmitter as feedback, whereas only the forward link is observed by the corresponding decoder. We derive conditions that are both necessary and sufficient for feedback to enlarge the capacity region. In the second model, the two outputs of a standard GBC are observed by the respective decoders, but the transmitter observes the sum of the symbols at the receivers using causal feedback. We show that such a feedback has no effect on the capacity region.

We study the problem of transmitting a message over a channel that randomly breaks the message block into small fragments, deletes a subset of them, and shuffles the remaining fragments. We characterize the capacity of the binary torn-paper channel under arbitrary fragment length distribution and fragment deletion probabilities. We show that, for a message with block length$n$, discarding fragments shorter than$\log(n)$does not affect the achievable rates, and that the capacity is given by a simple closed-form expression that can be understood as “coverage minus reordering-cost”.

It is well known that the capacity region of an average transmit power constrained Gaussian Broadcast Channel (GBC) with independent noise realizations at the receivers is enlarged by the presence of causal noiseless feedback. When the noise variances at the receivers are identical, even passive feedback via independent memoryless Gaussian links can lead to a capacity region enlargement. The last fact remains true even when the feedback noise variance is very high, and available only from one of the receivers. While such capacity enlargements are feasible for several other feedback models in the Gaussian BC setting, it is also known that feedback does not change the capacity region for physically degraded broadcast channels. In this paper, we consider a two user GBC with independent noise realizations at the receivers, where the feedback links from the receivers are corrupted by independent additive Gaussian noise processes. We investigate the set of four noise variances, two forward and two feedback, for which no capacity enlargement is possible. A sharp characterization of this region is derived, i.e., any quadruple outside the presented region will lead to a capacity enlargement, whereas quadruples inside will leave the capacity region unchanged. Our results lead to the conclusion that when the forward noise variances are different, too noisy a feedback from one of the receivers alone is not always beneficial for enlarging the capacity region, be it from the stronger user or the weaker one, in sharp contrast to the case of equal forward noise variances.

Feedback is known to enlarge the capacity region of a Gaussian Broadcast Channel (GBC) with independent noise realizations at the receivers, and an average power constraint at the transmitter. The capacity enlargement may occur even when there is noisy feedback from only one of the two receivers. However, recent results show the existence of a feedback noise threshold, beyond which one-sided feedback from only the stronger receiver is futile in enlarging the capacity region. The current paper presents a tight characterization of the feedback noise threshold, which separates the regimes where feedback from only the stronger receiver enlarges the capacity or leaves it unchanged. The scheme used to prove this result also leads to some interesting observations on noisy feedback from only the weak receiver.