We consider the minimum spanning tree problem with predictions, using the weight-arrival model, i.e., the graph is given, together with predictions for the weights of all edges. Then the actual weights arrive one at a time and an irrevocable decision must be made regarding whether or not the edge should be included into the spanning tree. In order to assess the quality of our algorithms, we define an appropriate error measure and analyze the performance of the algorithms as a function of the error. We prove that, according to competitive analysis, the simplest algorithm, Follow-the-Predictions, is optimal. However, intuitively, one should be able to do better, and we present a greedy variant of Follow-the-Predictions. In analyzing that algorithm, we believe we present the first random order analysis of a non-trivial online algorithm with predictions, by which we obtain an algorithmic separation. This may be useful for distinguishing between algorithms for other problems when Follow-the-Predictions is optimal according to competitive analysis.

翻译：我们考虑带预测的最小生成树问题，采用权重到达模型（weight-arrival model），即给定图以及所有边权重的预测值。随后实际权重逐一到达，且必须就边是否应纳入生成树做出不可撤销的决策。为评估算法质量，我们定义了适当的误差度量，并分析算法性能随误差变化的函数。根据竞争比分析，我们证明最简单的算法"跟随预测"(Follow-the-Predictions)是最优的。然而直觉上应有改进空间，为此提出"跟随预测"的贪心变体。在分析该算法时，我们首次对带预测的非平凡在线算法进行随机顺序分析，由此获得算法分离性结果。当竞争比分析显示"跟随预测"为最优时，该方法或可用于区分其他问题的算法优劣。