Quantum Pattern Matching in Generalised Degenerate Strings

Abstract

A degenerate string is a sequence of sets of characters. A generalized degenerate (GD) string extends this notion to the sequence of sets of strings, where strings of the same set are of equal length. Finding an exact match for a pattern string inside a GD string can be done in $O(mn+N)$ time (Ascone et al., WABI 2024), where $m$ is the pattern length, $n$ is the number of strings and $N$ the total length of strings constituting the GD string. We modify this algorithm to work under a quantum model of computation, achieving running time $\tilde{O}(\sqrt{mnN})$.

Figures (2)

• Figure 1: A GD string $T[1..5]$ with $T[1]=\{\mathtt{ACG,TAA,CGT,GTA}\}$, $T[2]=\{\mathtt{GATC,CGGT}\}$, $T[3]=\{\mathtt{AC,GT,CA}\}$, $T[4]=\{\mathtt{TAAGT,ATGCA}\}$, and $T[5]=\{\mathtt{ACG,TTA}\}$. Underlined characters illustrate a match for pattern $\mathtt{GTGTTAA}$.
• Figure 2: Abstract representation of different threads trying to match pattern $P$ in GD string $T$ starting from different position. Each dash symbol represents a single character, thus $|P|$ has $m=5$ characters and $T$ has $N=52$. Each thread $t_h$ tries to match $P$ column by column, with a shift of $h-1$ positions w.r.t. thread $t_1$, namely $t_1$ is shifted by $0$ positions and $t_5$ is shifted by $2$ positions. The characters highlighted in green show that thread $t_2$ finds a match at position $h + r\cdot m=2+1\cdot 5=7$. The grayed-out characters represents comparisons that will be tested but that cannot become full matches. Variable $i$ counts the iterations of the main for-loop.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2: Brassard2002Grover96
• Lemma 3
• proof
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['thm:main-quantum-algo']}
• Lemma 5
• proof