A $t$-spanner of a graph is a subgraph that $t$-approximates pairwise distances. The greedy algorithm is one of the simplest and most well-studied algorithms for constructing a sparse spanner: it computes a $t$-spanner with $n^{1+O(1/t)}$ edges by repeatedly choosing any edge which does not close a cycle of chosen edges with $t+1$ or fewer edges. We demonstrate that the greedy algorithm computes a $t$-spanner with $n^{1 + O(1/t)}$ edges even when a matching of such edges are added in parallel. In particular, it suffices to repeatedly add any matching where each individual edge does not close a cycle with $t +1$ or fewer edges but where adding the entire matching might. Our analysis makes use of and illustrates the power of new advances in length-constrained expander decompositions.

翻译：图的一个$t$-稀疏子图是能够$t$-近似所有点对距离的子图。贪婪算法是构造稀疏子图最简单且研究最充分的算法之一：通过不断选择不会与已有边形成长度不超过$t+1$的环的任意边，从而得到具有$n^{1+O(1/t)}$条边的$t$-稀疏子图。我们证明，即使并行添加这样的边匹配，贪婪算法仍能计算出具有$n^{1 + O(1/t)}$条边的$t$-稀疏子图。特别地，只需反复添加任意匹配，其中每条独立边都不会与已有边形成长度不超过$t+1$的环，但添加整个匹配则可能生成这样的环。我们的分析利用并展示了长度约束扩展分解新进展的强大能力。