The \emph{Single-Source Personalized PageRank} (SSPPR) query is central to graph OLAP, measuring the probability $π(s,t)$ that an $α$-decay random walk from node $s$ terminates at $t$. Despite decades of research, a significant gap remains between upper and lower bounds for its computational complexity. Existing upper bounds are $O\left(\min\left(\frac{\log(1/ε)}{ε^2}, \frac{\sqrt{m \log n}}ε, m \log \frac{1}ε\right)\right)$ for SSPPR-A and $O\left(\min\left(\frac{\log(1/n)}δ, \sqrt{m \log(n/δ)}, m \log \left(\frac{\log(n)}{mδ}\right)\right)\right)$ for SSPPR-R, with trivial lower bounds of $Ω(\min(n,1/ε))$ and $Ω(\min(n,1/δ))$. This work narrows or closes this gap. We improve the upper bounds for SSPPR-A and SSPPR-R to $O\left(\frac{1}{ε^2}\right)$ and $O\left(\min\left(\frac{\log(1/δ)}δ, m + n \log(n) \log \left(\frac{\log(n)}{mδ}\right)\right)\right)$, respectively, offering improvements by factors of $\log(1/ε)$ and $\log\left(\frac{\log(n)}{mδ}\right)$. On the lower bound side, we establish stronger results: $Ω(\min(m, 1/ε^2))$ for SSPPR-A and $Ω(\min(m, \frac{\log(1/δ)}δ))$ for SSPPR-R, strengthening theoretical foundations. Our upper and lower bounds for SSPPR-R coincide for graphs with $m \in Ω(n \log^2 n)$ and any threshold $δ, 1/δ\in O(\text{poly}(n))$, achieving theoretical optimality in most graph regimes. The SSPPR-A query attains partial optimality for large error thresholds, matching our new lower bound. This is the first optimal result for SSPPR queries. Our techniques generalize to the Single-Target Personalized PageRank (STPPR) query, improving its lower bound from $Ω(\min(n, 1/δ))$ to $Ω(\min(m, \frac{n}δ \log n))$, matching the upper bound and revealing its optimality.

翻译：\emph{单源个性化PageRank}（SSPPR）查询是图OLAP的核心操作，用于度量从节点$s$出发的$\alpha$-衰减随机游走终止于节点$t$的概率$\pi(s,t)$。尽管经过数十年研究，其计算复杂性的上下界之间仍存在显著差距。现有SSPPR-A的上界为$O\left(\min\left(\frac{\log(1/\varepsilon)}{\varepsilon^2}, \frac{\sqrt{m \log n}}{\varepsilon}, m \log \frac{1}{\varepsilon}\right)\right)$，SSPPR-R的上界为$O\left(\min\left(\frac{\log(1/n)}{\delta}, \sqrt{m \log(n/\delta)}, m \log \left(\frac{\log(n)}{m\delta}\right)\right)\right)$，而平凡下界为$\Omega(\min(n,1/\varepsilon))$和$\Omega(\min(n,1/\delta))$。本文缩小或消除了这一差距。我们将SSPPR-A和SSPPR-R的上界分别改进为$O\left(\frac{1}{\varepsilon^2}\right)$和$O\left(\min\left(\frac{\log(1/\delta)}{\delta}, m + n \log(n) \log \left(\frac{\log(n)}{m\delta}\right)\right)\right)$，改进因子分别为$\log(1/\varepsilon)$和$\log\left(\frac{\log(n)}{m\delta}\right)$。在下界方面，我们建立了更强的结论：SSPPR-A的下界为$\Omega(\min(m, 1/\varepsilon^2))$，SSPPR-R的下界为$\Omega(\min(m, \frac{\log(1/\delta)}{\delta}))$，强化了理论基础。对于满足$m \in \Omega(n \log^2 n)$且任意阈值$\delta$（满足$1/\delta\in O(\text{poly}(n))$）的图，SSPPR-R的上下界重合，在大多数图场景下达到理论最优性。SSPPR-A查询在大误差阈值下达到部分最优性，与我们的新下界匹配。这是SSPPR查询首次获得最优结果。我们的技术可推广至单目标个性化PageRank（STPPR）查询，将其下界从$\Omega(\min(n, 1/\delta))$改进为$\Omega(\min(m, \frac{n}{\delta} \log n))$，与上界匹配并揭示其最优性。