发表机构
Peking University（北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种确定性归约，在递减有向图中维护近线性时间的近似单源距离估计，利用动态割平衡和势能函数实现几乎线性的最坏情况界。
AI 中文摘要
我们给出一种确定性归约，用于在在线递减有向图中维护同时的近似单源距离估计。对于[1,W]中的正整数权重，该算法维护一个显式的整数(1+ε)上估计数组，精确识别不可达顶点，并在常数时间内回答数值查询。设N=m+n，S=N+⌈1/ε⌉，并假设log W=polylog(S)。初始化和所有更新耗时O(Δ)+(N+Δ_eff+ℒ/ε)S^{o(1)}，其中Δ统计原始更新数，Δ_eff=O(m(1+log W)/ε)统计过滤后的更新数，ℒ衡量由可达子图的当前入度加权的有限距离增长。被移除的边和变得不可达的顶点不再产生后续增长费用。因此，最坏情况界为O(Δ)+(N/ε)S^{o(1)}，对于次多项式逆精度而言是近线性的。该归约使用了van den Brand等人（FOCS 2024）的动态最小比率割和精确可达性算法。一个度加权的截断对数势能将常数相对割平衡转化为每个目标处的精度；随后一个后继割恢复严格可行性。距离热启动和递减返回系数释放的能量产生改进的增长界。返回质量还放大了现有变化检测器的精度。公开估计可以是单调的，对于无权重有限增长度量ℒ≤ℒ，数组写入次数为O(n+𝒥/ε)。执行使用有理算术。一个单独的归约将最坏情况保证扩展到非负整数权重。该算法维护数值估计，不提供快速路径报告。
英文摘要
We give a deterministic reduction for maintaining simultaneous approximate single-source distance estimates in online decremental directed graphs. For positive integer weights in $[1,W]$, the algorithm maintains an explicit array of integer $(1+\varepsilon)$ upper estimates, identifies unreachable vertices exactly, and answers numerical queries in constant time. Let $N=m+n$, $S=N+\lceil1/\varepsilon\rceil$, and assume $\log W=\operatorname{polylog}(S)$. Initialization and all updates take $O(Δ)+(N+Δ_{\mathrm{eff}}+\mathcal{L}/\varepsilon)S^{o(1)}$ time, where $Δ$ counts raw updates, $Δ_{\mathrm{eff}}=O(m(1+\log W)/\varepsilon)$ counts filtered updates, and $\mathcal{L}$ measures finite distance growth weighted by the current indegrees of the reachable subgraph. Removed edges and vertices that become unreachable incur no subsequent growth charge. Consequently, the worst-case bound is $O(Δ)+(N/\varepsilon)S^{o(1)}$, which is almost linear for subpolynomial inverse accuracy. The reduction uses the dynamic minimum-ratio cut and exact reachability algorithms of van den Brand et al. (FOCS 2024). A degree-weighted clipped logarithmic potential turns constant relative cut balance into accuracy at every target; a successor cut then restores strict feasibility. A distance warm start and the energy released by decreasing return coefficients yield the refined growth bound. Return mass also amplifies the accuracy of the existing change detector. The public estimates can be monotone, with $O(n+\mathcal{J}/\varepsilon)$ array writes for an unweighted finite-growth measure $\mathcal{J}\le\mathcal{L}$. Execution uses rational arithmetic. A separate reduction extends the worst-case guarantee to nonnegative integer weights. The algorithm maintains numerical estimates and does not provide fast path reporting.
Comments20 pages, no figures
