Classical and quantum algorithms for constructing text from dictionary problem

We study algorithms for solving a problem of constructing text (a long string) from dictionary sequence small strings). The has an application in bioinformatics and connection with the assembly method reconstructing DNA fragments. Our is construction string t length n using strings $$s^1,\dots , s^m$$ possible overlapping. Firstly, we provide classical (randomized) algorithm running time $$O\left( n+L +m(\log n)^2\right) =\tilde{O}(n+L)$$ where L sum lengths ,s^m$$ . Secondly, quantum +\log n\cdot (\log m+\log \log n)\cdot \sqrt{m\cdot L}\right) =\tilde{O}\left( +\sqrt{m\cdot $$ Additionally, show that lower bound (randomized or deterministic) $$\varOmega (n+L)$$ Thus, our optimal up to log factor, shows speed-up when compared any case non-constant dictionary.