三角形查找的扩展学习图
Extended Learning Graphs for Triangle Finding
摘要：
本文提出了一种新的三角形查找量子算法，改进了现有算法在稠密和稀疏情况下的量子查询复杂度.对于$n$顶点上的稠密图，我们得到了$O(n^{5/4})$的查询复杂度，而没有Le Gall算法中的任何额外的对数因子.对于具有$m\geq n^{5/4}$边的稀疏图，我们得到了一个查询复杂度为$O(n^{11/12}m^{1/6}\sqrt{\log n})$，优于Le Gall和Nakajima [ISAAC'15]在$m \geq n^{3/2}$ .我们还得到了一个算法的查询复杂度${O}(n^{5/6}(m\log n)^{1/6}+d_2\sqrt{n})$，其中$d_2$是度分布的方差。我们的算法的设计和分析，在一个新的模型的学习图，我们称之为扩展学习图。此外，我们提出了一个框架，以便轻松地组合和分析它们。因此，我们得到了比Le Gall { \it et al}基于MNRS量子行走框架[SICOMP'11]的先前算法简单得多的算法和分析。
英文摘要：
We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and spare instances.For dense graphs on $n$ vertices, we get a query complexity of $O(n^{5/4})$ without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS'14]. For sparse graphs with $m\geq n^{5/4}$ edges, we get a query complexity of $O(n^{11/12}m^{1/6}\sqrt{\log n})$, which is better than the one obtained by Le Gall and Nakajima [ISAAC'15] when $m \geq n^{3/2}$. We also obtain an algorithm with query complexity ${O}(n^{5/6}(m\log n)^{1/6}+d_2\sqrt{n})$ where $d_2$ is the variance of the degree distribution. Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall {\it et al} based on the MNRS quantum walk framework [SICOMP'11].
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