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

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( √n m logn), and the running time of the best known deterministic algorithm is O(n + m), where n is the number of vertices,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.Quantum computing [32,9] is one of the hot topics in computer science of last decades. There are many problems where quantum algorithms outperform the best known classical algorithms [18,28]. superior of quantum over classical was shown for different computing models like query model, streaming processing models, communication models and others [6,5,4,3,2,1,30,29,27,31].In this paper, we present the quantum algorithm for the class of problems on directed acyclic graphs (DAGs) that uses a dynamic programming approach. The dynamic programming approach is one of the most useful ways to solve problems in computer science [17]. The main idea of the method is to solve a problem using pre-computed solutions of the same problem, but with smaller parameters. Examples of such problems for DAGs that are considered in this paper are the single source longest path search problem for weighted DAGs and the diameter search problem for unweighted DAGs.

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