Quantum Algorithm for Lexicographically Minimal String Rotation
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Lexicographically minimal string rotation (LMSR) is a problem to find the minimal one among all rotations of a string in the lexicographical order, which is widely used in equality checking of graphs, polygons, automata and chemical structures. In this paper, we propose an O(n^3/4) quantum query algorithm for LMSR. In particular, the algorithm has average-case query complexity O(√(n)log n), which is shown to be asymptotically optimal up to a polylogarithmic factor, compared with its Ω(√(n/log n)) lower bound. Furthermore, we claim that our quantum algorithm outperforms any (classical) randomized algorithms in both worst-case and average-case query complexities by showing that every (classical) randomized algorithm for LMSR has worst-case query complexity Ω(n) and average-case query complexity Ω(n/log n). Our quantum algorithm for LMSR is developed in a framework of nested quantum algorithms, based on two new results: (i) an O(√(n)) (optimal) quantum minimum finding on bounded-error quantum oracles; and (ii) its O(√(n log(1/ε))) (optimal) error reduction. As a byproduct, we obtain some better upper bounds of independent interest: (i) O(√(N)) (optimal) for constant-depth MIN-MAX trees on N variables; and (ii) O(√(n log m)) for pattern matching which removes polylog(n) factors.
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