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We study quantum algorithms for several fundamental string problems, including Longest Common Substring, Lexicographically Minimal String Rotation, and Longest Square Substring. These problems have been widely studied in the stringology literature since the 1970s, and are known to be solvable by near-linear time classical algorithms. In this work, we give quantum algorithms for these problems with nearoptimal query complexities and time complexities. Specifically, we show that: Longest Common Substring can be solved by a quantum algorithm in Õ(n 2/3 ) time, improving upon the recent Õ(n 5/6

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A remarkable example of a nonempty closed convex set in the Euclidean plane for which the directional derivative of the metric projection mapping fails to exist was constructed by A. Shapiro. In this paper, we revisit and modify that construction to obtain a convex set with C 1,1 boundary which possesses the same property.Keywords Metric projection · Directional derivative A convex set with smooth boundaryDefine a strictly decreasing sequence of real numbers {α n } n∈N ⊂ (0, π/2] with α 0 = π/2, lim n→∞ α n = 0 and α n+1 < α n + α n+2 2 for all n ∈ N.(1.1) Now we identify R 2 equipped with the Euclidean norm · with C and let A n = e iα n . A beautiful and surprisingly simple example of a nonempty closed convex set for which the directional derivative of the metric projection mapping fails to exist was constructed by Shapiro in [13]. This set is essentially the convex hull J of the collection of points