Serial and parallel algorithms for order-preserving pattern matching based on the duel-and-sweep paradigm

Abstract

Given a text and a pattern over an alphabet, the classic exact matching problem searches for all occurrences of the pattern in the text. Unlike exact matching, order-preserving pattern matching (OPPM) considers the relative order of elements, rather than their exact values. In this paper, we propose efficient algorithms for the OPPM problem using the “duel-and-sweep” paradigm. For a pattern of length m and a text of length n, our serial algorithm runs in $O(n + m\log m)$ time, and our parallel algorithm runs in $O(\log ^2 m)$ time and $O(n \log ^2 m)$ work with $O(\log m)$ time and $O(m \log m)$ work pattern preprocessing on the Priority Concurrent Read Concurrent Write Parallel Random-Access Machines (P-CRCW PRAM).

Appendix A. Comparison with the parallel SCER matching algorithms

Our proposed parallel algorithm is largely based on the general matching algorithm for arbitrary SCERs proposed by Jargalsaikhan et al. [19, 20]. An equivalence relation $\cong$ over $\Sigma ^*$ is called an SCER just in the case where $X \cong Y$ implies $|X|=|Y|$ and $X[i \mathbin {:} j] \cong Y[i \mathbin {:} j]$ for any $1 \le i \le j \le |X|$. Clearly, the OPM relation $\approx$ is an SCER. The algorithm in [19] uses an encoding that satisfies the following conditions.

Definition 34

($\cong$-encoding, [19, Definition 3]) Let $\Sigma$ and $\Delta$ be (possibly infinite) alphabets. We say a function $f:\Sigma ^* \rightarrow \Delta ^*$ is an $\cong$-encoding if

• (1) $f(X) = f(Y)$ iff $X \cong Y$,

• (2) $f(X[1:i]) = f(X)[1:i]$ for any $i \le |X|$, and

• (3) $f(X)[i] = f(Y)[i]$ implies $f(X[j+1:k])[i-j] = f(Y[j+1:k])[i-j]$ for any $j < i \le k$.

If $X \not \cong Y$, one can find a witness position i such that $f(X)[i] \ne f(Y)[i]$. The conditions (2) and (3) imply that when a position witnesses mismatch between substrings of X and Y, then that position witnesses the mismatch between the whole strings X and Y as well. This is an important property to “transfer” witnesses on an offset for other offsets in their algorithm. Accordingly, the efficiency of their algorithm depends on the efficiency of the calculation of the encoding of a string as well as that of recalculating the encoding of a substring from the encoding of the whole string.

Indeed, every SCER $\cong$ admits an encoding satisfying the above: let $\Delta$ be the power set of $\Sigma ^*$, and $f(\varepsilon )=\varepsilon$ and $f(Xc) = f(X) \cdot [Xc]_{\cong }$ for $c \in \Sigma$ where $[Y]_{\cong }$ is the equivalence class of $Y \in \Sigma ^*$ under $\cong$ (or, $\Delta$ can be any set of symbols that can represent those equivalence classes). This construction guarantees that their algorithm works for every SCER in theory, but computing this encoding is apparently expensive. Actually, many SCERs, like exact match, parameterized match, and Cartesian match, admit computationally cheaper encodings satisfying Definition 3435. However, we do not yet know if there is such a reasonable encoding for OPM.

Concerning the OPM relation $\approx$, recall that $X \approx Y$ if and only if $Lmax _{X} = Lmax _{Y}$ and $Lmin _{X} = Lmin _{Y}$. The encoding $Lmix _X$ of X defined as $Lmix _X[i]= \langle Lmin _{X}[i], Lmax _{X}[i] \rangle$ fulfills (1)–(2) of Definition 3435, but not (3). For example, consider the mismatch between $X=(4,6,1,5)$ and $Y=(4,6,9,5)$. We have

$$\begin{aligned} Lmix _X&= (\langle 0,0 \rangle ,\ \langle 0,1 \rangle ,\ \langle 1,0 \rangle ,\ \langle 2,1 \rangle ) \,,\\ Lmix _Y&= (\langle 0,0 \rangle ,\ \langle 0,1 \rangle ,\ \langle 0,1 \rangle ,\ \langle 2,1 \rangle ) \,. \end{aligned}$$

The mismatch is witnessed at position 3, but not at the last position, with this encoding. On the other hand, the mismatch of their suffixes $X'=(1,5) \not \approx Y'=(9,5)$ is witnessed at the last position.

$$\begin{aligned} Lmix _{X'}&= (\langle 0,0 \rangle ,\ \langle 0,1 \rangle ) \,,\\ Lmix _{Y'}&= (\langle 0,0 \rangle ,\ \langle 1,0 \rangle ) \,. \end{aligned}$$

Therefore, the algorithm proposed in [19] does not work with this encoding $Lmix$. Instead, we have designed an algorithm that uses two positions as a mismatch witness, where we do not compare the encoded characters. This modification exempts us from the encoding costs. Furthermore, the OPM relation $\approx$ is closed under reversal, i.e., $X \approx Y$ iff $\textrm{rev}(X) \approx \textrm{rev}(Y)$, which is not guaranteed in general SCERs. Thanks to this property, our proposed pattern preprocessing (Algorithm 9 more specifically) runs faster than the one for general SCERs in [19].