Planar network shortest paths and max flow solved faster in congested graphs
Exact Shortest Paths and Maximum Flow in Planar Networks in the CONGEST Model
Distributed, Parallel, and Cluster Computing
Summary
Finding shortest paths and maximum flow in networks can be slow, especially under strict communication limits like in the CONGEST model. The authors improved algorithms for exact shortest paths and max flow in directed planar networks, reducing the number of communication rounds close to the theoretical minimum based on network diameter. They use new approximate distance labels as a starting point and correct earlier methods to achieve faster exact solutions. Their results also apply to detecting negative cycles and minimum cuts in planar networks.
What this means in practice
- •For network schedulers: Implement faster routing protocols in planar network topologies by using the improved exact shortest path computations within communication limits.
- •For telecommunication operators: Optimize data flow and detect bottlenecks in planar network sections more quickly using the improved maximum flow algorithms under message size constraints.
Authors
Sebastian Senge
Abstract
In the CONGEST model, exact single-source shortest paths (SSSP) with nonnegative weights and exact maximum $st$-flow in directed planar networks have both been solved in $\tilde O(D^2)$ rounds, where $D$ is the hop diameter, while the lower bound is $Ω(D)$. Given the approximate distance labels of a recent preprint (arXiv:2606.31192), which we use as a black box, we solve exact directed SSSP in $\tilde O(D)$ rounds. It rests on a lemma that answers queries from a virtual source with the labels, and on a corrected discretization of the reduction of exact to approximate distances (arXiv:2210.16351). All further bounds of the form $\tilde O(D)$ and $D\cdot n^{o(1)}$ use these labels. Through the reduction of arXiv:2303.00811, SSSP with negative weights, or the detection of a negative cycle, takes $D\cdot n^{o(1)}$ rounds with high probability. We weight the face-disjoint graph of arXiv:2411.11718 so that its distances are dual distances, which turns distance problems in the dual into problems in a planar network of hop diameter $O(D)$. This gives exact maximum $st$-flow in directed planar networks in $\min\{\tilde O(D^2),D\cdot n^{o(1)}\}$ rounds with high probability, and in $\tilde O(D)$ rounds with $s,t$ on a common face. In undirected planar networks, the value of a minimum $st$-cut and a minimum cut take $\tilde O(D)$ rounds. For this we repair the auxiliary graphs of arXiv:2608.09500 at $s$ and $t$, and we show that its recursion, as we read it, can miss the minimum cut, also with exact shortest paths: on a network with $11$ vertices it reports $37$ for a minimum cut of $15$. Two more shortest-path computations per level make it exact. The virtual-source lemma, the weighting of the face-disjoint graph and the corrected discretization are machine-checked in Lean 4 with Mathlib; all other arguments are proved on paper only. Numerical checks are not used in any proof.