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 $t^3\cdot \log^3 n \cdot n^{1 + O(1/t)}$ 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$t$-稀疏子图是原图的一个子图，它能够$t$-近似所有点对之间的距离。贪婪算法是构造稀疏子图最简单且最被深入研究的算法之一：它通过反复选择任意一条不会与已选边构成包含$t+1$条或更少边环路的边，从而计算出具有$n^{1+O(1/t)}$条边的$t$-稀疏子图。我们证明，即使并行添加这样的匹配边，贪婪算法仍能计算出具有$t^3\cdot \log^3 n \cdot n^{1 + O(1/t)}$条边的$t$-稀疏子图。特别地，只需反复添加任意一个匹配，其中每条单独边不会与已选边构成包含$t+1$条或更少边的环路，但整体添加该匹配却可能形成这样的环路。我们的分析利用并展示了长度约束扩展分解领域最新进展的威力。