Online weighted matching problem is a fundamental problem in machine learning due to its numerous applications. Despite many efforts in this area, existing algorithms are either too slow or don't take $\mathrm{deadline}$ (the longest time a node can be matched) into account. In this paper, we introduce a market model with $\mathrm{deadline}$ first. Next, we present our two optimized algorithms (\textsc{FastGreedy} and \textsc{FastPostponedGreedy}) and offer theoretical proof of the time complexity and correctness of our algorithms. In \textsc{FastGreedy} algorithm, we have already known if a node is a buyer or a seller. But in \textsc{FastPostponedGreedy} algorithm, the status of each node is unknown at first. Then, we generalize a sketching matrix to run the original and our algorithms on both real data sets and synthetic data sets. Let $\epsilon \in (0,0.1)$ denote the relative error of the real weight of each edge. The competitive ratio of original \textsc{Greedy} and \textsc{PostponedGreedy} is $\frac{1}{2}$ and $\frac{1}{4}$ respectively. Based on these two original algorithms, we proposed \textsc{FastGreedy} and \textsc{FastPostponedGreedy} algorithms and the competitive ratio of them is $\frac{1 - \epsilon}{2}$ and $\frac{1 - \epsilon}{4}$ respectively. At the same time, our algorithms run faster than the original two algorithms. Given $n$ nodes in $\mathbb{R} ^ d$, we decrease the time complexity from $O(nd)$ to $\widetilde{O}(\epsilon^{-2} \cdot (n + d))$.

翻译：在线加权匹配问题是机器学习中的一个基础问题，具有众多应用。尽管该领域已有很多努力，但现有算法要么速度过慢，要么未考虑截止（节点可被匹配的最长时间）。本文首先引入一个带有截止的市场模型。接着，我们提出两种优化算法（FastGreedy 和 FastPostponedGreedy），并给出这些算法时间复杂度和正确性的理论证明。在 FastGreedy 算法中，我们事先已知节点是买方还是卖方；但在 FastPostponedGreedy 算法中，各节点的状态起初未知。然后，我们泛化一个素描矩阵，用于在真实数据集和合成数据集上运行原始算法及我们的算法。令 ε ∈ (0,0.1) 表示每条边真实权重的相对误差。原始 Greedy 和 PostponedGreedy 算法的竞争比分别为 1/2 和 1/4。基于这两种原始算法，我们提出了 FastGreedy 和 FastPostponedGreedy 算法，其竞争比分别为 (1-ε)/2 和 (1-ε)/4。同时，我们的算法运行速度比原始两种算法更快。给定 ℝ^d 中的 n 个节点，我们将时间复杂度从 O(nd) 降低到 Ṽ(ε^{-2}·(n + d))。