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

An O(log n) time CRCW PRAM algorithm for computing the least lexicographic rotation of a circular string (of length n) over a fixed alphabet is presented here. The logarithmic running time is achieved by using O(nlog n)processors and its space complexity is linear. A second algorithm for unbounded alphabets requires O(log n log log n) units of time, also using O(nlog n) processors.

Optimal algorithms for computing the canonical form of a circular string

The Sum Number of Complete Bipartite Graphs

A family of sparse graphs of large sum number

Unique substrings appear scattered in the stringology literature and have important applications in bioinformatics. In this paper we initiate a study of minimum unique substrings in a given string; that is, substrings that occur exactly once while all their substrings are repeats. We discover a strong duality between minimum unique substrings and maximum repeats which, in particular, allows fast computation of one from the other. We give several optimal algorithms, some of which are very simple and efficient. Their combinatorial properties are investigated and a number of open problems are proposed.

/pdf/minimum-unique-substrings-and-maximum-repeats-4e5ntnuntq.pdf

Minimum Unique Substrings and Maximum Repeats

Given a string x = x[1..n] on an alphabet of size alpha, and a threshold pmin 1 greater than or equal to, we first describe a new algorithm PSY1 that, based on suffix array construction, computes all the complete nonextendible repeats in x of length p greater than or equal to pmin. PSY1 executes in theta(n) time independent of alphabet size and is an order of magnitude faster than the two other algorithms previously proposed for this problem. Second, we describe a new fast algorithm PSY2 for computing all complete supernonextendible repeats in x that also executes in theta(n) time independent of alphabet size, thus asymptotically faster than methods previously proposed. Both algorithms require 9n bytes of storage, including preprocessing (with a minor caveat for PSY1). We conclude with a brief discussion of applications to bioinformatics and data compression.

/pdf/fast-optimal-algorithms-for-computing-all-the-repeats-in-a-52sit3f66q.pdf

William F. Smyth

Papers

Optimal algorithms for computing the canonical form of a circular string

The Sum Number of Complete Bipartite Graphs

A family of sparse graphs of large sum number

Minimum Unique Substrings and Maximum Repeats

Fast Optimal Algorithms for Computing All the Repeats in a String