This paper demonstrates theoretically and empirically that a greedy algorithm called orthogonal matching pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous results, which require O(m2) measurements. The new results for OMP are comparable with recent results for another approach called basis pursuit (BP). In some settings, the OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal recovery problems.

/pdf/signal-recovery-from-random-measurements-via-orthogonal-5adhd8dwmq.pdf

Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit

This report demonstrates theoretically and empirically that a greedy algorithm called
Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension
d given O(mln d) random linear measurements of that signal. This is a massive improvement
over previous results, which require O(m2) measurements. The new results for OMP are comparable
with recent results for another approach called Basis Pursuit (BP). In some settings, the
OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal
recovery problems.

/pdf/signal-recovery-from-random-measurements-via-orthogonal-2uks7d8txo.pdf

Signal Recovery from Random Measurements Via Orthogonal Matching Pursuit: The Gaussian Case

We study the problem of searching on data that is encrypted using a public key system. Consider user Bob who sends email to user Alice encrypted under Alice’s public key. An email gateway wants to test whether the email contains the keyword “urgent” so that it could route the email accordingly. Alice, on the other hand does not wish to give the gateway the ability to decrypt all her messages. We define and construct a mechanism that enables Alice to provide a key to the gateway that enables the gateway to test whether the word “urgent” is a keyword in the email without learning anything else about the email. We refer to this mechanism as Public Key Encryption with keyword Search. As another example, consider a mail server that stores various messages publicly encrypted for Alice by others. Using our mechanism Alice can send the mail server a key that will enable the server to identify all messages containing some specific keyword, but learn nothing else. We define the concept of public key encryption with keyword search and give several constructions.

https://link.springer.com/content/pdf/10.1007%2F978-3-540-24676-3_30.pdf

Public Key Encryption with Keyword Search

Motivation: Next-generation sequencing captures sequence differences in reads relative to a reference genome or transcriptome, including splicing events and complex variants involving multiple mismatches and long indels. We present computational methods for fast detection of complex variants and splicing in short reads, based on a successively constrained search process of merging and filtering position lists from a genomic index. Our methods are implemented in GSNAP (Genomic Short-read Nucleotide Alignment Program), which can align both single-and paired-end reads as short as 14 nt and of arbitrarily long length. It can detect short-and long-distance splicing, including interchromosomal splicing, in individual reads, using probabilistic models or a database of known splice sites. Our program also permits SNP-tolerant alignment to a reference space of all possible combinations of major and minor alleles, and can align reads from bisulfite-treated DNA for the study of methylation state.

Results: In comparison testing, GSNAP has speeds comparable to existing programs, especially in reads of ≥70 nt and is fastest in detecting complex variants with four or more mismatches or insertions of 1–9 nt and deletions of 1–30 nt. Although SNP tolerance does not increase alignment yield substantially, it affects alignment results in 7–8% of transcriptional reads, typically by revealing alternate genomic mappings for a read. Simulations of bisulfite-converted DNA show a decrease in identifying genomic positions uniquely in 6% of 36 nt reads and 3% of 70 nt reads.

Availability: Source code in C and utility programs in Perl are freely available for download as part of the GMAP package at http://share.gene.com/gmap.

Contact: [email protected]

https://www.bioinf.jku.at/teaching/ss2010/spkb_chelbat/papers/ToReport/WU_NGS2.pdf

Fast and SNP-tolerant detection of complex variants and splicing in short reads

The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity. Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems. As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem. Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject. Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.

Digraphs Theory Algorithms And Applications

Group testing was first proposed for blood tests, but soon found its way to many industrial applications. Combinatorial group testing studies the combinatorial aspect of the problem and is particularly related to many topics in combinatorics, computer science and operations research. Recently, the idea of combinatorial group testing has been applied to experimental designs, coding, multiaccess computer communication, clone library screening and other fields. This book is the first attempt to cover the theory and applications of combinatorial group testing in one place.

Combinatorial Group Testing and Its Applications

d-Disjunct Matrices and d-Separable Matrices Geometric and Algebraic Designs Random Designs Subset-Containment Designs Array and Transversal Designs The Complex Model The Inhibitor Model Contig Sequencing Gene Detection.

Pooling Designs And Nonadaptive Group Testing: Important Tools For Dna Sequencing

Suppose a population containing items classified either as defectives or as non-defectives. This article presents a new group testing method to detect all defectives in the population. The proposed method applies either when the actual number of defectives (or an upper bound) is known or when its probability distribution is known. The method is surprisingly simple yet compares favorably with other existing methods by the number of tests required to detect all defectives.

A Method for Detecting all Defective Members in a Population by Group Testing

In this paper we present a new algorithm for merging two linearly ordered sets which requires substantially fewer comparisons than the commonly used tape merge or binary insertion algorithms. Bounds on the difference between the number of comparisons required by this algorithm and the information theory lower bounds are derived. Results from a computer implementation of the new algorithm are given and compared with a similar implementation of the tape merge algorithm.

A Simple Algorithm for Merging Two Disjoint Linearly Ordered Sets

This is the first book to cover comprehensively the mathematical theory of nonblocking switching networks since Benes' book published 30 years ago. Not only is the material on the classical theory of nonblocking and rearrangeable networks updated, but the modern topics on multicast and multirate switching are also surveyed. The author had spent more than 25 years working on switching networks at Bell Laboratories before he started teaching the course at Chiao-Tung University. He has published about 40 papers and obtained a dozen patents on multistage interconnection networks.

Frank K. Hwang

Papers

Combinatorial Group Testing and Its Applications

Pooling Designs And Nonadaptive Group Testing: Important Tools For Dna Sequencing

A Method for Detecting all Defective Members in a Population by Group Testing

A Simple Algorithm for Merging Two Disjoint Linearly Ordered Sets

The Mathematical Theory of Nonblocking Switching Networks