Abstract
We use span programs to develop quantum algorithms for several graph problems. We give an algorithm that uses \(O(n \sqrt d)\) queries to the adjacency matrix of an n-vertex graph to decide if vertices s and t are connected, under the promise that they either are connected by a path of length at most d, or are disconnected. We also give O(n)-query algorithms that decide if a graph contains as a subgraph a path, a star with two subdivided legs, or a subdivided claw. These algorithms can be implemented time efficiently and in logarithmic space. One of the main techniques is to modify the natural st-connectivity span program to drop along the way “breadcrumbs,” which must be retrieved before the path from s is allowed to enter t.
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Belovs, A., Reichardt, B.W. (2012). Span Programs and Quantum Algorithms for st-Connectivity and Claw Detection. In: Epstein, L., Ferragina, P. (eds) Algorithms – ESA 2012. ESA 2012. Lecture Notes in Computer Science, vol 7501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33090-2_18
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DOI: https://doi.org/10.1007/978-3-642-33090-2_18
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