We present streaming algorithms for the graph $k$-matching problem in both the insert-only and dynamic models. Our algorithms, with space complexity matching the best upper bounds, have optimal or near-optimal update time, significantly improving on previous results. More specifically, for the insert-only streaming model, we present a one-pass algorithm with optimal space complexity $O(k^2)$ and optimal update time $O(1)$, that with high probability computes a maximum weighted $k$-matching of a given weighted graph. The update time of our algorithm significantly improves the previous upper bound of $O(\log k)$, which was derived only for $k$-matching on unweighted graphs. For the dynamic streaming model, we present a one-pass algorithm that with high probability computes a maximum weighted $k$-matching in $O(Wk^2 \cdot \mbox{polylog}(n)$ space and with $O(\mbox{polylog}(n))$ update time, where $W$ is the number of distinct edge weights. Again the update time of our algorithm improves the previous upper bound of $O(k^2 \cdot \mbox{polylog}(n))$. This algorithm, when applied to unweighted graphs, gives a streaming algorithm on the dynamic model whose space and update time complexities are both near-optimal. Our results also imply a streaming approximation algorithm for maximum weighted $k$-matching whose space complexity matches the best known upper bound with a significantly improved update time.

翻译：我们提出了针对图上k-匹配问题的流式算法，适用于仅插入模型和动态模型。我们的算法在空间复杂度上匹配最佳上界，并具有最优或近最优的更新时间，显著改进了先前结果。具体而言，对于仅插入流式模型，我们提出了一种单遍算法，其空间复杂度最优为$O(k^2)$，更新时间最优为$O(1)$，并以高概率计算给定加权图的最大加权k-匹配。我们算法的更新时间显著改进了先前$O(\log k)$的上界，而该上界仅针对无权图上的k-匹配导出。对于动态流式模型，我们提出了一种单遍算法，以高概率在$O(Wk^2 \cdot \mbox{polylog}(n)$空间和$O(\mbox{polylog}(n))$更新时间内计算最大加权k-匹配，其中$W$是不同边权重的数量。同样，我们算法的更新时间改进了先前$O(k^2 \cdot \mbox{polylog}(n))$的上界。该算法应用于无权图时，在动态模型上提供了空间和更新时间复杂度均近最优的流式算法。我们的结果还蕴涵了一种最大加权k-匹配的流式近似算法，其空间复杂度匹配已知最佳上界，且更新时间显著提升。