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電子情報通信学会総合大会講演論文集

A-1-20 On the Quantum Query Complexity of All-Pairs Shortest Paths

Fuwei Cai, Satoshi Tayu, Shuichi Ueno

Published Mar 7, 2007• 1 citation
OpenAlex

Abstract

We show lower bounds for the quantum query complex-ity of the all-pairs shortest paths problem (APSP) for non-negatively weighted directed graphs (digraphs), both in the adjacency matrix model and in an adja-

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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 Θ(n3/2)\Theta(n^{3/2})Θ(n3/2) in the matrix model and in Θ(nm)\Theta(\sqrt{nm})Θ(nm​) in the array model, while the complexity of Connectivity is also in Θ(n3/2)\Theta(n^{3/2})Θ(n3/2) in the matrix model but in Θ(n)\Theta(n)Θ(n) in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis we study the power of quantum query algorithms and learning graphs; the latter essentially being very specialized quantum query algorithms themselves. We almost exclusively focus on proving lower bounds for these computational models. First, we study lower bounds on learning graph complexity. We consider two types of learning graphs: adaptive and, more restricted, non-adaptive learning graphs. We express both adaptive and non-adaptive learning graph complexities of Boolean-valued functions (i.e., decision problems) as semidefinite minimization problems, and derive their dual problems. For various functions, we construct feasible solutions to these dual problems, thereby obtaining lower bounds on the learn-ing graph complexity of the functions. Most notably, we prove an almost optimal Ω(n9/7/ log n)
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(n3)O(n^3)O(n3) time, while the trivial algorithm in the quantum setting runs in O~(n2.5)\tilde{O}(n^{2.5})O~(n2.5) 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(n3−ε)O(n^{3-\varepsilon})O(n3−ε) time for constant ε>0\varepsilon>0ε>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(n2.5−ε)O(n^{2.5-\varepsilon})O(n2.5−ε) time) quantum algorithms for the problems.