We consider global problems, i.e. problems that take at least diameter time, even when the bandwidth is not restricted. We show that all problems considered admit efficient solutions in low-treewidth graphs. By ``efficient'' we mean that the running time has polynomial dependence on the treewidth, a linear dependence on the diameter (which is unavoidable), and only a polylogarithmic dependence on $n$, the number of nodes in the graph. We present the algorithms solving the following problems in the CONGEST model which all attain $\tilde{O(\tau^{O(1)}D)}$-round complexity (where $\tau$ and $D$ denote the treewidth and diameter of the graph, respectively): (1) Exact single-source shortest paths (actually, the more general problem of computing a distance labeling scheme) for weighted and directed graphs, (2) exact bipartite unweighted maximum matching, and (3) the weighted girth for both directed and undirected graphs. We derive all of our results using a single unified framework, which consists of two novel technical ingredients, The first is a fully polynomial-time distributed tree decomposition algorithm, which outputs a decomposition of width $O(\tau^2\log n)$ in $\tilde{O}(\tau^{O(1)}D)$ rounds (where $n$ is the number of nodes in the graph). The second ingredient, and the technical highlight of this paper, is the novel concept of a \emph{stateful walk constraint}, which naturally defines a set of feasible walks in the input graph based on their local properties (e.g., augmenting paths). Given a stateful walk constraint, the constrained version of the shortest paths problem (or distance labeling) requires the algorithm to output the shortest \emph{constrained} walk (or its distance) for a given source and sink vertices. We show that this problem can be efficiently solved in the CONGEST model by reducing it to an \emph{unconstrained} version of the problem.
翻译:我们考虑的是全球性问题,即至少需要直径时间的问题,即使带宽不受限制。我们显示所有问题都被认为承认低树枝图形中的有效解决方案。“效率'”是指运行时间对树枝图有多元依赖性,对直径有线性依赖(这是不可避免的),只有多式依赖美元,图中节点的数量。我们展示的是解决CONEST模型中以下问题的算法,所有这些模型都达到$tilde{O(tau}O)O(美元)O(美元)O(美元)O(美元)O(美元)D)}美元(美元))) 美元周期复杂性(美元) 美元(美元) 数字(美元) 运行时间(美元) 运行时间(美元) 运行时间(美元) 数字(美元) 数字(美元) 数字(美元) 数字(美元) 数字(美元) 数字(美元) 数字(美元) 数字(我们用一个单一的统一框架来计算我们所有的结果。