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Quantum Physics

arXiv:1410.6220 (quant-ph)
[Submitted on 23 Oct 2014]

Title:Quantum algorithms for shortest paths problems in structured instances

Authors:Aran Nayebi, Virginia Vassilevska Williams
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Abstract:We consider the quantum time complexity of the all pairs shortest paths (APSP) problem and some of its variants. The trivial classical algorithm for APSP and most all pairs path problems runs in $O(n^3)$ time, while the trivial algorithm in the quantum setting runs in $\tilde{O}(n^{2.5})$ time, using Grover search. A major open problem in classical algorithms is to obtain a truly subcubic time algorithm for APSP, i.e. an algorithm running in $O(n^{3-\varepsilon})$ time for constant $\varepsilon>0$. To approach this problem, many truly subcubic time classical algorithms have been devised for APSP and its variants for structured inputs. Some examples of such problems are APSP in geometrically weighted graphs, graphs with small integer edge weights or a small number of weights incident to each vertex, and the all pairs earliest arrivals problem. In this paper we revisit these problems in the quantum setting and obtain the first nontrivial (i.e. $O(n^{2.5-\varepsilon})$ time) quantum algorithms for the problems.
Subjects: Quantum Physics (quant-ph); Data Structures and Algorithms (cs.DS)
Cite as: arXiv:1410.6220 [quant-ph]
  (or arXiv:1410.6220v1 [quant-ph] for this version)
  https://doi.org/10.48550/arXiv.1410.6220

Submission history

From: Aran Nayebi [view email]
[v1] Thu, 23 Oct 2014 01:09:06 UTC (16 KB)
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