Early computers replaced calculators and typewriters, and programmers focused on scientific computing (calculations involving numbers) and string processing (manipulating sequences of alphanumeric characters, or strings). Ironically, in modern applications, string processing is an integral part of scientific computing, as strings are an appropriate model of the natural world in a wide range of applications, notably computational biology and chemistry. Beyond scientific applications, strings are the lingua franca of modern computing, with billions of computers having immediate access to an almost unimaginable number of strings.

Decades of research have met the challenge of developing fundamental algorithms for string processing and mathematical models for strings and string processing that are suitable for scientific studies. Until now, much of this knowledge has been the province of specialists, requiring intimate familiarity with the research literature. The appearance of this new book is therefore a welcome development. It is a unique resource that provides a thorough coverage of the field and serves as a guide to the research literature. It is worthy of serious study by any scientist facing the daunting prospect of making sense of huge numbers of strings.

The development of an understanding of strings and string processing algorithms has paralleled the emergence of the field of analytic combinatorics, under the leadership of the late Philippe Flajolet, to whom this book is dedicated. Analytic combinatorics provides powerful tools that can synthesize and simplify classical derivations and new results in the analysis of strings and string processing algorithms. As disciples of Flajolet and leaders in the field nearly since its inception, Philippe Jacquet and Wojciech Szpankowski are well positioned to provide a cohesive modern treatment, and they have done a masterful job in this volume.

Repeated patterns and related phenomena in words are known to play a central role in many facets of computer science, telecommunications, coding, data compression, data mining, and molecular biology. One of the most fundamental questions arising in such studies is the frequency of pattern occurrences in a given string known as the text. Applications of these results include gene finding in biology, executing and analyzing tree-like protocols for multiaccess systems, discovering repeated strings in Lempel–Ziv schemes and other data compression algorithms, evaluating string complexity and its randomness, synchronization codes, user searching in wireless communications, and detecting the signatures of an attacker in intrusion detection.

The basic pattern matching problem is to find for a given (or random) pattern w or set of patterns W and a text X how many times W occurs in the text X and how long it takes for W to occur in X for the first time. There are many variations of this basic pattern matching setting which is known as exact string matching. In approximate string matching, better known as generalized string matching, certain words from W are expected to occur in the text while other words are forbidden and cannot appear in the text. In some applications, especially in constrained coding and neural data spikes, one puts restrictions on the text (e.g., only text without the patterns 000 and 0000 is permissible), leading to constrained string matching. Finally, in the most general case, patterns from the set W do not need to occur as strings (i.e., consecutively) but rather as subsequences; that leads to subsequence pattern matching, also known as hidden pattern matching.

These various pattern matching problems find a myriad of applications. Molecular biology provides an important source of applications of pattern matching, be it exact or approximate or subsequence pattern matching. There are examples in abundance: finding signals in DNA; finding split genes where exons are interrupted by introns; searching for starting and stopping signals in genes; finding tandem repeats in DNA.

In this chapter we consider generalized pattern matching, in which a set of patterns (rather than a single pattern) is given. We assume here that the pattern is a pair of sets of words (W0, W), where Wi consists of the sets Wi ⊂ Ami (i.e., all words in Wi have a fixed length mi). The set W0 is called the forbidden set. For W0 = ∅ one is interested in the number of pattern occurrences On(W), defined as the number of patterns from W occurring in a text generated by a (random) source. Another parameter of interest is the number of positions in where a pattern from W appears (clearly, several patterns may occur at the same positions but words from Wi must occur in different locations); this quantity we denote as Πn. If we define as the number of positions where a word from Wi occurs, then

Notice that at any given position of the text and for a given i only one word from Wi can occur.

For W0 ≠ ∅ one studies the number of occurrences On(W) under the condition that, that is, there is no occurrence of a pattern from W0 in the text. This could be called constrained pattern matching since one restricts the text to those strings that do not contain strings from W0. A simple version of constrained pattern matching was discussed in Chapter 3 (see also Exercises 3.3, 3.6, and 3.10).

In this chapter we first present an analysis of generalized pattern matching with W0 = ∅ and d = 1, which we call the reduced pattern set (i.e., no pattern is a substring of another pattern).

A taking-and-breaking game [Albert et al. 2007; Berlekamp et al. 2001] is an impartial combinatorial game, played with heaps of beans on a table. A move for either player consists of choosing a heap, removing a certain number of beans from the heap, and then possibly splitting the remainder into several heaps; the winner is the player making the last move. For example, both Grundy’s Game (choose a heap and split it into two unequal heaps) and Couples-Are-Forever (choose a heap with at least three beans and split it into two) are taking-and-breaking games with very simple rules, however neither has been solved.

We present an overview of the required theory of impartial games. The reader can consult the references above for a more in-depth grounding in the theory of, and for more details about, subtraction and octal games.

The numbers in parentheses are the old numbers used in each of the lists of unsolved problems given on pp. 183–189 of AMS Proc. Sympos. Appl. Math. 43 (1991), called PSAM 43 below; on pp. 475–491 of Games of No Chance, hereafter referred to as GONC; on pp. 457–473 of More Games of No Chance (MGONC); and on pp. 475–500 of Games of No Chance 3 (GONC3). Some numbers have little more than the statement of the problem if there is nothing new to be added. References [year] may be found in Fraenkel’s bibliography at the end of this volume. References [#] are at the end of this article. A useful reference for the rules and an introduction to many of the specific games mentioned below is M. Albert, R. J. Nowakowski and D. Wolfe, Lessons in Play: An Introduction to the Combinatorial Theory of Games, A. K. Peters, 2007 (LIP) or Berlekamp, Conway and Guy, Winning Ways for your Mathematical Plays, vol. 1–4, A. K. Peters, 2000–2004 (WW).

Subtraction games with finite subtraction sets are known to have periodic nim-sequences. Investigate the relationship between the subtraction set and the length and structure of the period. The same question can be asked about partizan subtraction games, in which each player is assigned an individual subtraction set. See Fraenkel and Kotzig [1987].