Quantum Query Complexity of Some Graph Problems

• Published in: SIAM journal on computing Vol. 35; no. 6; pp. 1310 - 1328
• Main Authors: Dürr, Christoph, Heiligman, Mark, HOyer, Peter, Mhalla, Mehdi
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2006
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Arrays; Combinatorics; Combinatorics. Ordered structures; Computer Science; Computer science; control theory; systems; Connectivity; Exact sciences and technology; Graph theory; Graphs; Information retrieval. Graph; Mathematics; Queries; R&D; Research & development; Sciences and techniques of general use; Theoretical computing; 68W20; Lower bound; Quantum algorithm; Error estimation; 05C85; minimum spanning tree; Shortest path; Adjacency matrix; Graph theory; Complexity; Upper bound; 81-04; connectivity; Graph connectivity; Spanning tree; 68R10; single source shortest paths
• ISSN: 0097-5397; 1095-7111
• DOI: 10.1137/050644719

Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example, we show that the query complexity of Minimum Spanning Tree is in $\Theta(n^{3/2})$ in the matrix model and in $\Theta(\sqrt{nm})$ in the array model, while the complexity of Connectivity is also in $\Theta(n^{3/2})$ in the matrix model but in $\Theta(n)$ in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.