Abstract
In this paper, we present a quantum algorithm for dynamic programming approach for problems on directed acyclic graphs (DAGs). The running time of the algorithm is \(O(\sqrt{\hat{n}m}\log \hat{n})\), and the running time of the best known deterministic algorithm is \(O(n+m)\), where n is the number of vertices, \(\hat{n}\) is the number of vertices with at least one outgoing edge; m is the number of edges. We show that we can solve problems that use OR, AND, NAND, MAX and MIN functions as the main transition steps. The approach is useful for a couple of problems. One of them is computing a Boolean formula that is represented by Zhegalkin polynomial, a Boolean circuit with shared input and non-constant depth evaluating. Another two are the single source longest paths search for weighted DAGs and the diameter search problem for unweighted DAGs.
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Acknowledgements
This work was supported by Russian Science Foundation Grant 17-71-10152.
A part of work was done when K. Khadiev visited University of Latvia. We thank Andris Ambainis, Alexander Rivosh and Aliya Khadieva for help and useful discussions.
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Khadiev, K., Safina, L. (2019). Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs. In: McQuillan, I., Seki, S. (eds) Unconventional Computation and Natural Computation. UCNC 2019. Lecture Notes in Computer Science(), vol 11493. Springer, Cham. https://doi.org/10.1007/978-3-030-19311-9_13
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