这完全是一个度的问题：使用度信息来优化多路连接
It's all a matter of degree: Using degree information to optimize multiway joins
摘要：
我们使用度信息优化关系表上的多路对等连接。我们给出了一个新的界限，使用度信息更紧密地约束查询的最大输出大小。在真实数据上，我们对社交网络中三角形数量的限制可以比现有的最坏情况限制严格$95$倍。我们表明，只使用一个常数量的程度信息，我们能够获得连接算法的运行时间，有一个较小的指数比现有的算法-{ \em的任何数据库实例}。我们还表明，这种程度的信息可以在接近线性的时间，这产生渐近更快的算法在串行设置和较低的通信算法在MapReduce设置。在串行设置中，连接处理的数据复杂度可以表示为关于输入大小$\IN$和输出大小$\OUT$的函数$O(\IN^x + \OUT)$，其中$x$取决于查询。分数超树宽给出了$x$的上界。我们感兴趣的情况下，我们可以得到的算法，其中$x$是严格小于分数超树宽。如果$x < 2$，我们说连接可以在次二次时间内处理。基于用于在二次时间内处理循环连接的AYZ算法，对于我们称之为$1$ -串-并行图的限制性连接类别，我们获得了用于识别次二次可解性的完整决策过程（服从需要二次时间的$3$ -SUM问题）。我们的$3$ -SUM为基础的二次下界是紧的，使其成为唯一已知的紧界连接，不需要任何假设的矩阵乘法指数$ω$。我们还给出了一个MapReduce算法，满足我们改进的通信边界和处理本质上是最佳的并行性。
英文摘要：
We optimize multiway equijoins on relational tables using degree information. We give a new bound that uses degree information to more tightly bound the maximum output size of a query. On real data, our bound on the number of triangles in a social network can be up to $95$ times tighter than existing worst case bounds. We show that using only a constant amount of degree information, we are able to obtain join algorithms with a running time that has a smaller exponent than existing algorithms--{\em for any database instance}. We also show that this degree information can be obtained in nearly linear time, which yields asymptotically faster algorithms in the serial setting and lower communication algorithms in the MapReduce setting. In the serial setting, the data complexity of join processing can be expressed as a function $O(\IN^x + \OUT)$ in terms of input size $\IN$ and output size $\OUT$ in which $x$ depends on the query. An upper bound for $x$ is given by fractional hypertreewidth. We are interested in situations in which we can get algorithms for which $x$ is strictly smaller than the fractional hypertreewidth. We say that a join can be processed in subquadratic time if $x < 2$. Building on the AYZ algorithm for processing cycle joins in quadratic time, for a restricted class of joins which we call $1$-series-parallel graphs, we obtain a complete decision procedure for identifying subquadratic solvability (subject to the $3$-SUM problem requiring quadratic time). Our $3$-SUM based quadratic lower bound is tight, making it the only known tight bound for joins that does not require any assumption about the matrix multiplication exponent $ω$. We also give a MapReduce algorithm that meets our improved communication bound and handles essentially optimal parallelism.
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