We consider the problem of detecting a planted clique of size $k$ in a random graph on $n$ vertices. When the size of the clique exceeds $\Theta(\sqrt{n})$, polynomial-time algorithms for detection proliferate. We study faster—namely, sublinear time—algorithms in the high-signal regime when $k = \Theta(n^{1/2 + \delta})$, for some $\delta > 0$. To this end, we consider algorithms that non-adaptively query a subset $M$ of entries of the adjacency matrix and then compute a low-degree polynomial function of the revealed entries. We prove a computational phase transition for this class of \emph{non-adaptive low-degree algorithms}: under the scaling $\lvert M \rvert = \Theta(n^{\gamma})$, the clique can be detected when $\gamma > 3(1/2 - \delta)$ but not when $\gamma < 3(1/2 - \delta)$. As a result, the best known runtime for detecting a planted clique, $\widetilde{O}(n^{3(1/2-\delta)})$, cannot be improved without looking beyond the non-adaptive low-degree class. Our proof of the lower bound—based on bounding the conditional low-degree likelihood ratio—reveals further structure in non-adaptive detection of a planted clique. Using (a bound on) the conditional low-degree likelihood ratio as a potential function, we show that for \emph{every} non-adaptive query pattern, there is a highly structured query pattern of the same size that is at least as effective.

We study the planted clique problem in which a clique of size k is planted in an Erdős-Rényi graph G(n, 1/2), and one is interested in either detecting or recovering this planted clique. This problem is interesting because it is widely believed to show a statistical-computational gap at clique size k=sqrt{n}, and has emerged as the prototypical problem with such a gap from which average-case hardness of other statistical problems can be deduced. It also displays a tight computational connection between the detection and recovery variants, unlike other problems of a similar nature. This wide investigation into the computational complexity of the planted clique problem has, however, mostly focused on its time complexity. In this work, we ask- Do the statistical-computational phenomena that make the planted clique an interesting problem also hold when we use `space efficiency' as our notion of computational efficiency? It is relatively easy to show that a positive answer to this question depends on the existence of a O(log n) space algorithm that can recover planted cliques of size k = Omega(sqrt{n}). Our main result comes very close to designing such an algorithm. We show that for k=Omega(sqrt{n}), the recovery problem can be solved in O((log*{n}-log*{k/sqrt{n}}) log n) bits of space. 1. If k = omega(sqrt{n}log^{(l)}n) for any constant integer l > 0, the space usage is O(log n) bits. 2.If k = Theta(sqrt{n}), the space usage is O(log*{n} log n) bits. Our result suggests that there does exist an O(log n) space algorithm to recover cliques of size k = Omega(sqrt{n}), since we come very close to achieving such parameters. This provides evidence that the statistical-computational phenomena that (conjecturally) hold for planted clique time complexity also (conjecturally) hold for space complexity.

As magnetization and semiconductor based storage technologies approach their limits, bio-molecules, such as DNA, have been identified as promising media for future storage systems, due to their high storage density (petabytes/gram) and long-term durability (thousands of years). Furthermore, nanopore DNA sequencing enables high-throughput sequencing using devices as small as a USB thumb drive and thus is ideally suited for DNA storage applications. Due to the high insertion/deletion error rates associated with base-called nanopore reads, current approaches rely heavily on consensus among multiple reads and thus incur very high reading costs. We propose a novel approach which overcomes the high error rates in basecalled sequences by integrating a Viterbi error correction decoder with the basecaller, enabling the decoder to exploit the soft information available in the deep learning based basecaller pipeline. Using convolutional codes for error correction, we experimentally observed 3x lower reading costs than the state-of-the-art techniques at comparable writing costs.The code, data and Supplementary Material is available at https://github.com/shubhamchandak94/nanopore_dna_storage.

In distributed storage, erasure codes (like Reed-Solomon Codes) are often employed to provide reliability. In this setting, it is desirable to be able to repair one or more failed nodes while minimizing the repair bandwidth. In this paper, motivated by Reed-Solomon codes, we study the problem of repairing multiple failed nodes in a scalar MDS code. We extend the framework of (Guruswami and Wootters, 2017) to give a framework for constructing repair schemes for multiple failures in general scalar MDS codes in the centralized repair model. We then specialize our framework to Reed-Solomon codes, and also extend and improve upon recent results of (Dau et al., 2017).