We prove that any $n$-node graph $G$ with diameter $D$ admits shortcuts with congestion $O(delta D log n)$ and dilation $O(delta D)$, where $delta$ is the maximum edge-density of any minor of $G$. Our proof is simple, elementary, and constructive - featuring a $tilde{Theta}(delta D)$-round distributed construction algorithm. Our results are tight up to $tilde{O}(1)$ factors and generalize, simplify, unify, and strengthen several prior results. For example, for graphs excluding a fixed minor, i.e., graphs with constant $delta$, only a $tilde{O}(D^2)$ bound was known based on a very technical proof that relies on the Robertson-Seymour Graph Structure Theorem. A direct consequence of our result is that many graph families, including any minor-excluded ones, have near-optimal $tilde{Theta}(D)$-round distributed algorithms for many fundamental communication primitives and optimization problems including minimum spanning tree, minimum cut, and shortest-path approximations.
تحميل البحث