12 research outputs found
Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs
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 , and the running time of the
best known deterministic algorithm is , where is the number of
vertices, is the number of vertices with at least one outgoing edge;
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.Comment: UCNC2019 Conference pape
Quantum-over-classical Advantage in Solving Multiplayer Games
We study the applicability of quantum algorithms in computational game theory
and generalize some results related to Subtraction games, which are sometimes
referred to as one-heap Nim games.
In quantum game theory, a subset of Subtraction games became the first
explicitly defined class of zero-sum combinatorial games with provable
separation between quantum and classical complexity of solving them. For a
narrower subset of Subtraction games, an exact quantum sublinear algorithm is
known that surpasses all deterministic algorithms for finding solutions with
probability .
Typically, both Nim and Subtraction games are defined for only two players.
We extend some known results to games for three or more players, while
maintaining the same classical and quantum complexities:
and respectively
Classical and Quantum Algorithms for Constructing Text from Dictionary Problem
We study algorithms for solving the problem of constructing a text (long
string) from a dictionary (sequence of small strings). The problem has an
application in bioinformatics and has a connection with the Sequence assembly
method for reconstructing a long DNA sequence from small fragments. The problem
is constructing a string of length from strings with
possible intersections. We provide a classical algorithm with running time
where is the sum of lengths
of . We provide a quantum algorithm with running time . Additionally, we show that the lower bound for the
classical algorithm is . Thus, our classical algorithm is optimal
up to a log factor, and our quantum algorithm shows speed-up comparing to any
classical algorithm in a case of non-constant length of strings in the
dictionary
Quantum Algorithms for the Most Frequently String Search, Intersection of Two String Sequences and Sorting of Strings Problems
We study algorithms for solving three problems on strings. The first one is
the Most Frequently String Search Problem. The problem is the following. Assume
that we have a sequence of strings of length . The problem is finding
the string that occurs in the sequence most often. We propose a quantum
algorithm that has a query complexity . This algorithm
shows speed-up comparing with the deterministic algorithm that requires
queries. The second one is searching intersection of two sequences
of strings. All strings have the same length . The size of the first set is
and the size of the second set is . We propose a quantum algorithm that
has a query complexity . This algorithm shows
speed-up comparing with the deterministic algorithm that requires
queries. The third problem is sorting of strings of length
. On the one hand, it is known that quantum algorithms cannot sort objects
asymptotically faster than classical ones. On the other hand, we focus on
sorting strings that are not arbitrary objects. We propose a quantum algorithm
that has a query complexity . This algorithm shows
speed-up comparing with the deterministic algorithm (radix sort) that requires
queries, where is a size of the alphabet.Comment: THe paper was presented on TPNC 201
Where Quantum Complexity Helps Classical Complexity
Scientists have demonstrated that quantum computing has presented novel
approaches to address computational challenges, each varying in complexity.
Adapting problem-solving strategies is crucial to harness the full potential of
quantum computing. Nonetheless, there are defined boundaries to the
capabilities of quantum computing. This paper concentrates on aggregating prior
research efforts dedicated to solving intricate classical computational
problems through quantum computing. The objective is to systematically compile
an exhaustive inventory of these solutions and categorize a collection of
demanding problems that await further exploration
Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs
© 2019, Springer Nature Switzerland AG. 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, and the running time of the best known deterministic algorithm is, where n is the number of vertices, 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 Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs
© 2019, Springer Nature Switzerland AG. 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, and the running time of the best known deterministic algorithm is, where n is the number of vertices, 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