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.

/pdf/a-dynamic-survey-of-graph-labeling-198znmwxtq.pdf

A Dynamic Survey of Graph Labeling

The main way of analyzing biological sequences is by comparing and aligning them to each other. It remains difficult, however, to compare modern multi-billionbase DNA data sets. The difficulty is caused by the nonuniform (oligo)nucleotide composition of these sequences, rather than their size per se. To solve this problem, we modified the standard seed-and-extend approach (e.g., BLAST) to use adaptive seeds. Adaptive seeds are matches that are chosen based on their rareness, instead of using fixed-length matches. This method guarantees that the number of matches, and thus the running time, increases linearly, instead of quadratically, with sequence length. LAST, our open source implementation of adaptive seeds, enables fast and sensitive comparison of large sequences with arbitrarily nonuniform composition.

/pdf/adaptive-seeds-tame-genomic-sequence-comparison-58wsqwzwm9.pdf

Adaptive seeds tame genomic sequence comparison.

コンピュータ・サイエンス : ACM computing surveys

https://www.cs.usask.ca/~croy/papers/2009/RCK_SCP_Clones.pdf

Comparison and evaluation of code clone detection techniques and tools: A qualitative approach

$V$-order is a global order on strings related to Unique Maximal Factorization Families (UMFFs), which are themselves generalizations of Lyndon words. $V$-order has recently been proposed as an alternative to lexicographical order in the computation of suffix arrays and in the suffix-sorting induced by the Burrows-Wheeler transform. Efficient $V$-ordering of strings thus becomes a matter of considerable interest. In this paper we present new and surprising results on $V$-order in strings, then go on to explore the algorithmic consequences.

String Comparison in $V$-Order: New Lexicographic Properties & On-line Applications

An indeterminate string (or, more simply, just a string ) x=x[1..n] on an alphabet σ is a sequence of nonempty subsets of σ. We say that x[i1] and x[i2] match (written x[i1]≈x[i2]) if and only if x[i1]∩x[i2]≠θ. A feasible array is an array y=y[1..n] of integers such that y[1]=n and for every i∈2..n, y[i]∈0..n-i+1. A prefix table of a string x is an array π=π[1..n] of integers such that, for every i∈1..n, π[i]=j if and only if x[i..i+j-1] is the longest substring at position i of x that matches a prefix of x It is known from [6] that every feasible array is a prefix table of some indeterminate string. A prefix graph P=Py is a labelled simple graph whose structure is determined by a feasible array y In this paper we show, given a feasible array y , how to use Py to construct a lexicographically least indeterminate string on a minimum alphabet whose prefix table π=y.

Inferring an indeterminate string from a prefix graph

The number of segregating sites provides an indicator of the degree of DNA sequence variation that is present in a sample, and has been of great interest to the biological, pharmaceutical and medical professions. In this paper, we first provide linear- and expected-sublinear-time algorithms for finding all the segregating sites of a given set of DNA sequences. We also describe a data structure for tracking segregating sites in a set of sequences, such that every time the set is updated with the insertion of a new sequence or removal of an existing one, the segregating sites are updated accordingly without the need to re-scan the entire set of sequences.

/pdf/efficient-algorithms-for-counting-and-reporting-segregating-3e4i43dfr6.pdf

Efficient algorithms for counting and reporting segregating sites in genomic sequences

Stand-alone data manipulation programs

The study of strings is an important combinatorial field that precedes the digital computer. Strings can be very long, trillions of letters, so it is important to find compact representations. Here we first survey various forms of one potential compaction methodology, the cover of a given string x, initially proposed in a simple form in 1990, but increasingly of interest as more sophisticated variants have been discovered. We then consider covering by a seed; that is, a cover of a superstring of x. We conclude with many proposals for research directions that could make significant contributions to string processing in future.

William F. Smyth

Papers

String Comparison in $V$-Order: New Lexicographic Properties & On-line Applications

Inferring an indeterminate string from a prefix graph

Efficient algorithms for counting and reporting segregating sites in genomic sequences

Stand-alone data manipulation programs

String Covering: A Survey