There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.

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A Dynamic Survey of Graph Labeling

DNA sequencing with chain terminating inhibitors

The vast majority of coding variants are rare, and assessment of the contribution of rare variants to complex traits is hampered by low statistical power and limited functional data. Improved methods for predicting the pathogenicity of rare coding variants are needed to facilitate the discovery of disease variants from exome sequencing studies. We developed REVEL (rare exome variant ensemble learner), an ensemble method for predicting the pathogenicity of missense variants on the basis of individual tools: MutPred, FATHMM, VEST, PolyPhen, SIFT, PROVEAN, MutationAssessor, MutationTaster, LRT, GERP, SiPhy, phyloP, and phastCons. REVEL was trained with recently discovered pathogenic and rare neutral missense variants, excluding those previously used to train its constituent tools. When applied to two independent test sets, REVEL had the best overall performance (p −12 ) as compared to any individual tool and seven ensemble methods: MetaSVM, MetaLR, KGGSeq, Condel, CADD, DANN, and Eigen. Importantly, REVEL also had the best performance for distinguishing pathogenic from rare neutral variants with allele frequencies

REVEL: An Ensemble Method for Predicting the Pathogenicity of Rare Missense Variants

Standards and Guidelines for the Interpretation and Reporting of Sequence Variants in Cancer: A Joint Consensus Recommendation of the Association for Molecular Pathology, American Society of Clinical Oncology, and College of American Pathologists

Motifs are relatively short sequences that are biologically significant, and their discovery in molecular sequences is a well-researched subject A don’t care is a special letter that matches every letter in the alphabet Formally, a motif is a sequence of letters of the alphabet and don’t care letters A motif $\tilde{m}_{d,k}$ that occurs at least k times in a sequence is maximal if it cannot be extended (to the left or right) nor can it be specialised (that is, its $d' \le d$ don’t cares cannot be replaced with letters from the alphabet) without reducing its number of occurrences Here we present a new dynamic data structure, and the first on-line algorithm, to discover all maximal motifs in a sliding window of length $\ell $ on a sequence x of length n in $\mathcal {O}(nd\ell + d\lceil \frac{\ell }{w}\rceil \cdot \sum _{i = \ell }^{n-1} |{\textsc {diff}}_{i-1}^{i}|)$ time, where w is the size of the machine word and ${\textsc {diff}}_{i-1}^{i}$ is the symmetric difference of the sets of occurrences of maximal motifs at $x[i-\ell \mathinner {\,}i-1]$ and at $x[i-\ell +1 \mathinner {\,}i]$

Maximal Motif Discovery in a Sliding Window

For a partial word $w$ the longest common compatible prefix of two positions $i,j$, denoted $lccp(i,j)$, is the largest $k$ such that $w[i,i+k-1]\uparrow w[j,j+k-1]$, where $\uparrow$ is the compatibility relation of partial words (it is not an equivalence relation). The LCCP problem is to preprocess a partial word in such a way that any query $lccp(i,j)$ about this word can be answered in $O(1)$ time. It is a natural generalization of the longest common prefix (LCP) problem for regular words, for which an $O(n)$ preprocessing time and $O(1)$ query time solution exists. 
Recently an efficient algorithm for this problem has been given by F. Blanchet-Sadri and J. Lazarow (LATA 2013). The preprocessing time was $O(nh+n)$, where $h$ is the number of "holes" in $w$. The algorithm was designed for partial words over a constant alphabet and was quite involved. 
We present a simple solution to this problem with slightly better runtime that works for any linearly-sortable alphabet. Our preprocessing is in time $O(n\mu+n)$, where $\mu$ is the number of blocks of holes in $w$. Our algorithm uses ideas from alignment algorithms and dynamic programming.

A Note on the Longest Common Compatible Prefix Problem for Partial Words

Prague Stringology Conference 2008

The approximate string matching problem is to find all locations at which a query of length m matches a substring of a text of length n with k-or-fewer differences. Nowadays, with the advent of novel high throughput sequencing technologies, the approximate string matching algorithms are used to identify similarities, molecular functions and abnormalities in DNA sequences. We consider a generalization of this problem, the fixed-length approximate string matching problem: given a text t, a pattern ρ and an integer l, compute the optimal alignment of all substrings of ρ of length l and a substring of t. We present a practical parallel algorithm of comparable simplicity that requires only O( nm⌈l/w⌉ p ) time, where w is the word size of the machine (e.g. 32 or 64 in practice) and p the number of processors, by virtue of computing a bit representation of the relocatable dynamic programming matrix for the problem. Thus the algorithm’s performance is independent of k and the alphabet size |Σ|.

Costas S. Iliopoulos

Papers

Maximal Motif Discovery in a Sliding Window

A Note on the Longest Common Compatible Prefix Problem for Partial Words

Prague Stringology Conference 2008

A Parallel Algorithm for the Fixed-length Approximate String Matching Problem for High Throughput Sequencing Technologies

Problems in Computational Musicology