Jacqueline W. Daykin
Royal Holloway, University of London
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Featured researches published by Jacqueline W. Daykin.
Journal of Discrete Algorithms | 2003
David E. Daykin; Jacqueline W. Daykin
We say a family W of strings is an UMFF if every string has a unique maximal factorization over W. Then W is an UMFF iff xy, yz ∈ W and y non-empty imply xyz ∈ W. Let L-order denote lexicographic order. Danh and Daykin discovered V-order, B-order and T-order. Let R be L, V, B or T. Then we call r an R-word if it is strictly first in R-order among the cyclic permutations of r. The set of R-words form an UMFF. We show a large class of B-like UMFF. The well-known Lyndon factorization of Chen, Fox and Lyndon is the L case, and it motivated our work.
International Journal of Foundations of Computer Science | 2008
David E. Daykin; Jacqueline W. Daykin
We say a family of strings over an alphabet is an UMFF if every string has a unique maximal factorization over . Foundational work by Chen, Fox and Lyndon established properties of the Lyndon circ-UMFF, which is based on lexicographic ordering. Commencing with the circ-UMFF related to V-order, we then proved analogous factorization families for a further 32 Block-like binary orders. Here we distinguish between UMFFs and circ-UMFFs, and then study the structural properties of circ-UMFFs. These properties give rise to the complete construction of any circ-UMFF. We prove that any circ-UMFF is a totally ordered set and a factorization over it must be monotonic. We define atom words and initiate a study of u, v-atoms. Applications of circ-UMFFs arise in string algorithmics.
Theoretical Computer Science | 2013
David E. Daykin; Jacqueline W. Daykin; William F. Smyth
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 present new combinatorial results for V-order, in particular concatenation under V-order. We propose linear-time RAM algorithms for string comparison in V-order and for Lyndon-like factorization of a string into V-words. This asymptotic efficiency thus matches that of the corresponding algorithms for lexicographical order. Finally, we introduce Hybrid Lyndon words as a generalization of standard Lyndon words, and hence propose extensions of factorization algorithms to other forms of order.
Fundamenta Informaticae | 2009
David E. Daykin; Jacqueline W. Daykin; William F. Smyth
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
combinatorial pattern matching | 2011
David E. Daykin; Jacqueline W. Daykin; William F. Smyth
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Siam Journal on Algebraic and Discrete Methods | 1985
David E. Daykin; Jacqueline W. Daykin
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.
Theoretical Computer Science | 2014
Jacqueline W. Daykin; William F. Smyth
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.
Information Theory, Combinatorics, and Search Theory | 2013
David E. Daykin; Jacqueline W. Daykin; Costas S. Iliopoulos; William F. Smyth
We study order preserving maps from a finite poset to the integers. When these maps are bijective they are called linear extensions. For both kinds we give many elementary properties and inequalities. A positive correlation inequality was proved by Graham, Yao and Yao. Then contributions were made by Graham, Kleitman, Shearer, Shepp and others. We obtain the corresponding negative correlation inequalities. Most authors have used the FKG inequality; we use an inequality of Daykin instead. Graham made a conjecture concerning range posets so we characterise these, and prove various cases of the conjecture. Finally we give necessary and sufficient conditions for a map defined on a subposet to extend to the whole poset. The results have applications in computer science.
Mathematics in Computer Science | 2015
Jacqueline W. Daykin; Costas S. Iliopoulos; Mirka Miller; Oudone Phanalasy
In this paper we introduce the V-transform (V-BWT), a variant of the classic Burrows–Wheeler Transform. The original BWT uses lexicographic order, whereas we apply a distinct total ordering of strings called V-order. V -order string comparison and Lyndon-like factorization of a string x=x[1..n]x=x[1..n] into V-words have recently been shown to be linear in their use of time and space (Daykin et al., 2011) [18]. Here we apply these subcomputations, along with Θ(n)Θ(n) suffix-sorting (Ko and Aluru, 2003) [26], to implement linear V-sorting of all the rotations of a string. When it is known that the input string x is a V-word, we compute the V -transform in Θ(n)Θ(n) time and space, and also outline an efficient algorithm for inverting the V-transform and recovering x. We further outline a bijective algorithm in the case that x is arbitrary. We propose future research into other variants of transforms using lex-extension orderings (Daykin et al., 2013) [19]. Motivation for this work arises in possible applications to data compression.
Fundamenta Informaticae | 2015
Ali Alatabbi; Jacqueline W. Daykin; Mohammad Sohel Rahman; William F. Smyth
In this paper we describe algorithms for factoring words over sets of strings known as circ-UMFFs, generalizations of the well-known Lyndon words based on lexorder, whose properties were first studied in 1958 by Chen, Fox and Lyndon. In 1983 Duval designed an elegant linear-time sequential (RAM) Lyndon factorization algorithm; a corresponding parallel (PRAM) algorithm was described in 1994 by Daykin, Iliopoulos and Smyth. In 2003 Daykin and Daykin introduced various circ-UMFFs, including one based on V-words and V-ordering; in 2011 linear string comparison and sequential factorization algorithms based on V-order were given by Daykin, Daykin and Smyth. Here we first describe generic RAM and PRAM algorithms for factoring a word over any circ-UMFF; then we show how to customize these generic algorithms to yield optimal parallel Lyndon-like V-word factorization.