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

We introduce quasi suffix arrays as a generalization of suffix arrays for character strings. We show that a quasi suffix array encodes enough of the structure of the string to be a useful construct for many applications where the full power of suffix arrays is not necessary, notably in problems that do not require lexicographical order, for example, pattern-matching or calculation of repeating substrings. We are interested in quasi suffix arrays, for we believe that they can be calculated by simple, fast, and space efficient algorithms. As a first step towards this goal, we describe a family DIST of algorithms (inspired by the Crochemore's repetitions algorithm) that compute the quasi suffix array in the average-case in O(|x| log |x|) time, where x is the input string. Based on experiments conducted by one of us (Xiao), it appears that in practice our algorithms execute faster than all suffix tree and most suffix array construction algorithms. Though at this time we can only prove that the average-case complexity is O(|x| log |x|), tests carried out by one of us (Holub) strongly suggest that not only the worst-case complexity may be the same as the average-case complexity, but both may in fact be linear. Given the very recent results on computing suffix arrays in linear time by recursive algorithms, the only advantage quasi suffix arrays can have lies in the simplicity and space efficiency of DIST algorithms that do not use recursion.

Computing quasi suffix arrays

The k-coversproblem (kCP asks us to compute a minimum cardinality set of strings given length k>1 that covers a given string. It was shown in a recent paper, by reduction to 3 -SAT, that the k-covers problem is NP-complete. In this paper we introduce a new problem, that we call the Relaxed Vertex Cover Problem (RVCP), which we show is a special case of Set Cover (SCP). We show further the kCP is equivalent to RVCP restricted to certain classes GXk of graphs that represent all strings x. We discuss approximate solutions of kCP and we state a number of conjectures and open problems related to kCP and GXk.

The complexity of the minimum k-Cover Problem

Suppose a set W of strings contains exactly one rotation (cyclic shift) of every primitive string on some alphabet Σ. Then W is a circ-UMFF if and only if every word in Σ$^+$ has a unique maximal factorization over W. The classic circ-UMFF is the set of Lyndon words based on lexicographic ordering (1958). Duval (1983) designed a linear sequential Lyndon factorization algorithm; a corresponding PRAMparallel algorithmwas described by J. Daykin, Iliopoulos and Smyth (1994). Daykin and Daykin defined new circ-UMFFs based on various methods for totally ordering sets of strings (2003), and further described the structure of all circ-UMFFs (2008). Here we prove new combinatorial results for circ-UMFFs, and in particular for the case of Lyndon words. We introduce Acrobat and Flight Deck circ-UMFFs, and describe some of our results in terms of dictionaries. Applications of circ-UMFFs pertain to structured methods for concatenating and factoring strings over ordered alphabets, and those of Lyndon words are wide ranging and multidisciplinary.

/pdf/combinatorics-of-unique-maximal-factorization-families-umffs-4dra7cz0yl.pdf

Combinatorics of Unique Maximal Factorization Families (UMFFs)

The prefix table of a string x = x[1..n] is an array π = π[1..n] such that π[i] is the length of the longest substring beginning at i that equals a prefix of x. In this paper we describe and evaluate algorithms for prefix table construction, some previously proposed, others designed by us. We also describe and evaluate new linear-time algorithms for transformations between π and the border array.

Prefix Table Construction and Conversion

In this paper we extend previous work on Unique Maximal Factorization Families (UMFFs) and a total (but non-lexicographic) ordering of strings called V-order. We describe linear-time algorithms for string comparison and Lyndon factorization based on V-order. We propose extensions of these algorithms to other forms of order.

William F. Smyth

Papers

Computing quasi suffix arrays

The complexity of the minimum k-Cover Problem

Combinatorics of Unique Maximal Factorization Families (UMFFs)

Prefix Table Construction and Conversion

String comparison and Lyndon-like factorization using V-order in linear time