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$t$-Spanner是其一个子图，能够$t$-近似任意点对之间的距离。贪心算法是构造稀疏Spanner最简单且研究最充分的算法之一：通过反复选择任意一条不会与已选边形成长度不超过$t+1$的环的边，构建一个边数为$n^{1+O(1/t)}$的$t$-Spanner。我们证明，即使在并行添加此类边的匹配时，贪心算法仍能计算出一个边数为$t^3\cdot \log^3 n \cdot n^{1 + O(1/t)}$的$t$-Spanner。具体而言，只需反复添加任意匹配，其中每条单独的边不会与已选边形成长度不超过$t+1$的环，但添加整个匹配可能导致这种环的形成。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。