$\text{String-matching}$

$\text{Overview}$

$\text{String-matching Problem}$

String matching is the problem of finding all occurrences of a string pattern $P$ of length $m$ in a text $T$ of lenght $n$, where $m\le n$. The characters in the pattern and the text come from a finite set $\mathrm{\Sigma }$ called the $\text{alphabet}$.

We say that pattern $P$ occurs with shift $s$ in text $T$ if $0\le s\le n-m$ and $T[s:s+m-1]=P[0:m-1]$. If $P$ occurs with shift $s$ in $T$, then $s$ is called a $\text{valid shift}$ for $P$ in $T$, otherwise $s$ is called an $\text{invalid shift}$ for $P$ in $T$.

In other words, the string-matching problem is to find all valid shifts for $P$ in $T$, i.e. the shift set of $P$ in $T$. Using the terminology below, the string-matching problem is to $\text{find all shifts }s$ $\text{such that }P⊐T_{s+m}$, where $0\le s\le n-m$ and $T_{s+m}$ is the prefix of $T$ of length $s+m$.

$\text{Terminology and properties}$

• ${\mathrm{\Sigma }}^{\ast }$: the set of all finite-length strings over $\mathrm{\Sigma }$
• $ϵ$: the empty string, i.e. the string of length $0$
• $|x|$: the length of string $x$
• $xy$: the concatenation of strings $x$ and $y$, with $|xy|=|x|+|y|$
• $T_q$: the q-character prefix of string $T$, i.e. $T_q=T[0:q-1]$
• $x⊏y$: string $x$ is a $\text{prefix}$ of string $y$, i.e. $y=xz$ for some string $z\in {\mathrm{\Sigma }}^{\ast }$
• $x⊐y$: string $x$ is a $\text{suffix}$ of string $y$, i.e. $y=zx$ for some string $z\in {\mathrm{\Sigma }}^{\ast }$
• $x⊏y\Rightarrow |x|\le |y|$
• $x⊐y\Rightarrow |x|\le |y|$
• $x⊏y\wedge |x|<|y|$: $x$ is a proper prefix of $y$
• $x⊐y\wedge |x|<|y|$: $x$ is a proper suffix of $y$
• $x\in {\mathrm{\Sigma }}^{\ast }\Rightarrow ϵ⊏x$
• $x\in {\mathrm{\Sigma }}^{\ast }\Rightarrow ϵ⊐x$
• $\mathrm{\forall }a\in \mathrm{\Sigma }:x⊐y\iff xa⊐ya$
• $\mathrm{\forall }z\in {\mathrm{\Sigma }}^{\ast }:x⊐y\wedge y⊐z\Rightarrow x⊐z$
• $\mathrm{\forall }z\in {\mathrm{\Sigma }}^{\ast }:x⊏y\wedge y⊏z\Rightarrow x⊏z$