We study the maximum $s,t$-flow oracle problem on planar directed graphs where the goal is to design a data structure answering max $s,t$-flow value (or equivalently, min $s,t$-cut value) queries for arbitrary source-target pairs $(s,t)$. For the case of polynomially bounded integer edge capacities, we describe an exact max $s,t$-flow oracle with truly subquadratic space and preprocessing, and sublinear query time. Moreover, if $(1-\epsilon)$-approximate answers are acceptable, we obtain a static oracle with near-linear preprocessing and $\tilde{O}(n^{3/4})$ query time and a dynamic oracle supporting edge capacity updates and queries in $\tilde{O}(n^{6/7})$ worst-case time. To the best of our knowledge, for directed planar graphs, no (approximate) max $s,t$-flow oracles have been described even in the unweighted case, and only trivial tradeoffs involving either no preprocessing or precomputing all the $n^2$ possible answers have been known. One key technical tool we develop on the way is a sublinear (in the number of edges) algorithm for finding a negative cycle in so-called dense distance graphs. By plugging it in earlier frameworks, we obtain improved bounds for other fundamental problems on planar digraphs. In particular, we show: (1) a deterministic $O(n\log(nC))$ time algorithm for negatively-weighted SSSP in planar digraphs with integer edge weights at least $-C$. This improves upon the previously known bounds in the important case of weights polynomial in $n$, and (2) an improved $O(n\log{n})$ bound on finding a perfect matching in a bipartite planar graph.

翻译：我们研究平面有向图上的最大 $s,t$-流黑盒问题，目标是为任意源-目标对 $(s,t)$ 设计一种支持最大 $s,t$-流值（或等价地，最小 $s,t$-割值）查询的数据结构。针对边容量为多项式有界整数的情况，我们描述了一个精确的最大 $s,t$-流黑盒，其空间和预处理复杂度为真正次二次，查询时间为次线性。此外，如果允许 $(1-\epsilon)$-近似答案，我们可得到一个静态黑盒（预处理时间近线性，查询时间为 $\tilde{O}(n^{3/4})$）和一个动态黑盒（支持边容量更新和查询，最坏情况时间为 $\tilde{O}(n^{6/7})$）。据我们所知，对于平面有向图，即使未加权情况也尚未有（近似）最大 $s,t$-流黑盒的描述，仅已知涉及无预处理或预计算所有 $n^2$ 种可能答案的平凡权衡。我们在此过程中开发的一项关键技术工具是在所谓稠密距离图中寻找负环的算法（其复杂度与边数呈次线性关系）。通过将其嵌入先前的框架，我们改进了平面有向图其他基本问题的界。具体而言，我们证明：（1）对于边权重为整数且下界为 $-C$ 的负权平面有向图，存在确定性 $O(n\log(nC))$ 时间的单源最短路径算法，这改进了权重为 $n$ 的多项式时的重要情况下的先前已知界；（2）在二分平面图中寻找完美匹配的改进 $O(n\log{n})$ 界。