We study the query complexity of determining if a graph is connected with global queries. The first model we look at is matrix-vector multiplication queries to the adjacency matrix. Here, for an $n$-vertex graph with adjacency matrix $A$, one can query a vector $x \in \{0,1\}^n$ and receive the answer $Ax$. We give a randomized algorithm that can output a spanning forest of a weighted graph with constant probability after $O(\log^4(n))$ matrix-vector multiplication queries to the adjacency matrix. This complements a result of Sun et al.\ (ICALP 2019) that gives a randomized algorithm that can output a spanning forest of a graph after $O(\log^4(n))$ matrix-vector multiplication queries to the signed vertex-edge incidence matrix of the graph. As an application, we show that a quantum algorithm can output a spanning forest of an unweighted graph after $O(\log^5(n))$ cut queries, improving and simplifying a result of Lee, Santha, and Zhang (SODA 2021), which gave the bound $O(\log^8(n))$. In the second part of the paper, we turn to showing lower bounds on the linear query complexity of determining if a graph is connected. If $w$ is the weight vector of a graph (viewed as an $\binom{n}{2}$ dimensional vector), in a linear query one can query any vector $z \in \mathbb{R}^{n \choose 2}$ and receive the answer $\langle z, w\rangle$. We show that a zero-error randomized algorithm must make $\Omega(n)$ linear queries in expectation to solve connectivity. As far as we are aware, this is the first lower bound of any kind on the unrestricted linear query complexity of connectivity. We show this lower bound by looking at the linear query \emph{certificate complexity} of connectivity, and characterize this certificate complexity in a linear algebraic fashion.

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关于全局查询连通性的查询复杂度

我们研究了确定图是否与全局查询相关的查询复杂性。我们看到的第一个模型是对邻接矩阵的矩阵向量乘法查询。这里，对于具有邻接矩阵 $A$ 的 $n$-顶点图，可以查询向量 $x \in \{0,1\}^n$ 并得到答案 $Ax$。我们给出了一个随机算法，该算法可以在对邻接矩阵进行 $O(\log^4(n))$ 矩阵向量乘法查询后以恒定概率输出加权图的生成森林。这补充了 Sun 等人的结果。\ (ICALP 2019) 给出了一种随机算法，该算法可以在对有符号顶点进行 $O(\log^4(n))$ 矩阵向量乘法查询后输出图的生成森林-图的边关联矩阵。作为应用程序，我们表明，量子算法可以在 $O(\log^5(n))$ 切割查询后输出未加权图的生成森林，改进和简化了 Lee、Santha 和 Zhang（SODA 2021）的结果，它给出了边界 $O(\log^8(n))$。在论文的第二部分，我们开始展示确定图是否连通的线性查询复杂度的下限。如果 $w$ 是图的权重向量（视为 $\binom{n}{2}$ 维向量），则在线性查询中可以查询任何向量 $z \in \mathbb{R}^{n \choose 2}$ 并得到答案 $\langle z, w\rangle$。我们表明零错误随机算法必须进行 $\Omega(n)$ 线性查询以解决连通性问题。据我们所知，这是连接的无限制线性查询复杂度的第一个下界。