Auctions are widely used in exchanges to match buy and sell requests. Once the buyers and sellers place their requests, the exchange determines how these requests are to be matched. The two most popular objectives used while determining the matching are maximizing volume at a uniform price and maximizing volume with dynamic pricing. In this work, we study the algorithmic complexity of the problems arising from these matching tasks. We present a linear time algorithm for uniform price matching which is an improvement over the previous algorithms that take $O(n\log n)$ time to match $n$ requests. For dynamic price matching, we establish a lower bound of $\Omega(n \log n)$ on the running time, thereby proving that the currently known best algorithm is time-optimal.

翻译：拍卖在交易所中被广泛用于匹配买卖请求。一旦买卖双方提交请求，交易所将决定这些请求如何配对。在确定匹配方案时，最常见的两个目标是在统一价格下最大化成交量，以及在动态定价下最大化成交量。本研究探讨了这些匹配任务中所涉及问题的算法复杂度。我们提出了一种用于统一价格匹配的线性时间算法，相比此前处理n个请求需要O(n log n)时间的算法有所改进。对于动态价格匹配，我们证明了运行时间的下界为Ω(n log n)，从而证实当前已知的最优算法在时间上已达到最优。