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

The notion of the cover is a generalization of a period of a string, and there are linear time algorithms for finding the shortest cover. The seed is a more complicated generalization of periodicity, it is a cover of a superstring of a given string, and the shortest seed problem is of much higher algorithmic difficulty. The problem is not well understood, no linear time algorithm is known. In the paper we give linear time algorithms for some of its versions --- computing shortest left-seed array, longest left-seed array and checking for seeds of a given length. The algorithm for the last problem is used to compute the seed array of a string (i.e., the shortest seeds for all the prefixes of the string) in $O(n^2)$ time. We describe also a simpler alternative algorithm computing efficiently the shortest seeds. As a by-product we obtain an $O(n\log{(n/m)})$ time algorithm checking if the shortest seed has length at least $m$ and finding the corresponding seed. We also correct some important details missing in the previously known shortest-seed algorithm (Iliopoulos et al., 1996).

Efficient Seeds Computation Revisited

The approximate string matching problem is to find all locations which a pattern of length m matches a substring of a text of length n with at most k differences. The program agrep is a simple and practical bit-vector algorithm for this problem. In this paper we consider the following incremental version of the problem: given an appropriate encoding of a comparison between A and bB, can one compute the answer for A and B, and the answer for A and Bc with equal efficiency, where b and c are additional symbols? Here we present an elegant and veryeasy to implement bit-vector algorithm for answering these questions that requires only O(n⌈m/w⌉) time, where n is the length of A, m is the length of B and w is the number of bits in a machine word. We also present an O(nm⌈h/w⌉) algorithm for the fixed-length approximate string matching problem: given a text t, a pattern p and an integer h, compute the optimal alignment of all substrings of p of length h and a substring of t.

The Max-Shift Algorithm for Approximate String Matching

Closed factorization

We study the complexity of the longest common subsequence (LCS) problem from a new perspective. By an indeterminate string (istring, in short) we mean a sequence e X = e X[1] e X[2] . . . e X[n], where e X[i] ⊆ Σ for each i, and Σ is a given alphabet of potentially large size. A subsequence of e X is any usual string over Σ which is an element of the finite (but usually of exponential size) language e X[i1] e X[i2] . . . e X[ip], where 1 ≤ i1 < i2 < i3 . . . < ip ≤ n, p ≥ 0. Similarly, we define a supersequence of x. Our first version of the LCS problem is Problem ILCS: for given i-strings e X and e Y , find their longest common subsequence. From the complexity point of view, new parameters of the input correspond to |Σ| and maximum size ` of the subsets in e X and e Y . There is also a third parameter R, which gives a measure of similarity between e X and e Y . The smaller the R, the lesser is the time for solving Problem ILCS. Our second version of the LCS problem is Problem CILCS (constrained ILCS): for given i-strings e X and e Y and a plain string Z, find the longest common subsequence of e X and e Y which is, at the same time, a supersequence of Z. In this paper, we present several efficient algorithms to solve both ILCS and CILCS problems. The efficiency in our algorithms are obtained in particular by using an efficient data structure for special type of range maxima queries and fast multiplication of boolean matrices. ? Part of this research work was carried out when Costas Iliopoulos and M. Sohel Rahman were visiting Institute of Informatics, Warsaw University. ?? Supported by EPSRC and Royal Society grants. ? ? ? Supported by the Commonwealth Scholarship Commission in the UK under the Commonwealth Scholarship and Fellowship Plan (CSFP). † On Leave from Department of CSE, BUET, Dhaka-1000, Bangladesh. ‡ Supported by the grant of the Polish Ministery of Science and Higher Education N

Costas S. Iliopoulos

Papers

Efficient Seeds Computation Revisited

The Max-Shift Algorithm for Approximate String Matching

Closed factorization

Algorithms for Two Versions of LCS Problem for Indeterminate Strings

String matching with Gaps for Musical Melodic Recognition