Algorithmic Combinatorics on Partial Words: 1st Edition (Hardback) book cover

Algorithmic Combinatorics on Partial Words

1st Edition

By Francine Blanchet-Sadri

Chapman and Hall/CRC

392 pages | 69 B/W Illus.

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Hardback: 9781420060928
pub: 2007-11-19
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pub: 2007-11-19
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The discrete mathematics and theoretical computer science communities have recently witnessed explosive growth in the area of algorithmic combinatorics on words. The next generation of research on combinatorics of partial words promises to have a substantial impact on molecular biology, nanotechnology, data communication, and DNA computing. Delving into this emerging research area, Algorithmic Combinatorics on Partial Words presents a mathematical treatment of combinatorics on partial words designed around algorithms and explores up-and-coming techniques for solving partial word problems as well as the future direction of research.

This five-part book begins with a section on basics that covers terminology, the compatibility of partial words, and combinatorial properties of words. The book then focuses on three important concepts of periodicity on partial words: period, weak period, and local period. The next part describes a linear time algorithm to test primitivity on partial words and extends the results on unbordered words to unbordered partial words while the following section introduces some important properties of pcodes, details a variety of ways of defining and analyzing pcodes, and shows that the pcode property is decidable using two different techniques. In the final part, the author solves various equations on partial words, presents binary and ternary correlations, and covers unavoidable sets of partial words.

Setting the tone for future research in this field, this book lucidly develops the central ideas and results of combinatorics on partial words.

Table of Contents



Preliminaries on Partial Words

Alphabets, letters, and words

Partial functions and partial words


Factorizations of partial words

Recursion and induction on partial words

Containment and compatibility

Combinatorial Properties of Partial Words




Fine and Wilf’s Theorem

The case of zero or one hole

The case of two or three holes

Special partial words

Graphs associated with partial words

The main result

Related results

Critical Factorization Theorem


The zero-hole case

The main result: First version

The main result: Second version


Guibas and Odlyzko’s Theorem

The zero-hole case

The main result

The algorithm


Primitive Partial Words

Testing primitivity on partial words

Counting primitive partial words

Exact periods

First counting method

Second counting method

Existence of primitive partial words

Unbordered Partial Words

Concatenations of prefixes

More results on concatenations of prefixes

Critical factorizations



Pcodes of Partial Words

Binary relations


Pcodes and monoids

Prefix and suffix orderings

Border ordering

Commutative ordering

Circular pcodes

Deciding the Pcode Property

First algorithm

Second algorithm


Equations on Partial Words

The equation xmyn

The equation x2ymz

The equation xmynzp

Correlations of Partial Words

Binary and ternary correlations

Characterizations of correlations

Distributive lattices

Unavoidable Sets of Partial Words

Unavoidable sets

Classifying unavoidable sets of size two

The case where k = 1 and l = 1

The case where k = 1 and l = 2

Larger values of k and l

Solutions to Selected Exercises



Numerous Exercises as well as Website and Bibliographic Notes appear at the end of each chapter.

About the Series

Discrete Mathematics and Its Applications

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
COMPUTERS / Operating Systems / General
COMPUTERS / Programming / Algorithms
MATHEMATICS / Combinatorics