Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs

• Published in: Unconventional Computation and Natural Computation Vol. 11493; pp. 150 - 163
• Main Authors: Khadiev, Kamil, Safina, Liliya
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2019; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: AND-OR-NOT formula; Boolean formula; Boolean formula evaluation; Classical vs. quantum; Computational complexity; DAG; DNF; Dynamic programming; Graph; NAND; Quantum algorithm; Quantum computation; Quantum models; Query model; Zhegalkin polynomial
• ISBN: 9783030193102; 3030193101
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-030-19311-9_13

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\sqrt{\hat{n}m}\log \hat{n})$$\end{document}, and the running time of the best known deterministic algorithm is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n+m)$$\end{document}, where n is the number of vertices, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{n}$$\end{document} 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.