Ridesharing has become a promising travel mode recently due to the economic and social benefits. As an essential operator, "insertion operator" has been extensively studied over static road networks. When a new request appears, the insertion operator is used to find the optimal positions of a worker's current route to insert the origin and destination of this request and minimize the travel time of this worker. Previous works study how to conduct the insertion operation efficiently in static road networks, however, in reality, the route planning should be addressed by considering the dynamic traffic scenario (i.e., a time-dependent road network). Unfortunately, existing solutions to the insertion operator become in efficient under this setting. Thus, this paper studies the insertion operator over time-dependent road networks. Specially, to reduce the high time complexity $O(n^3)$ of existing solution, we calculate the compound travel time functions along the route to speed up the calculation of the travel time between vertex pairs belonging to the route, as a result time complexity of an insertion can be reduced to $O(n^2)$. Finally, we further improve the method to a linear-time insertion algorithm by showing that it only needs $O(1)$ time to find the best position of current route to insert the origin when linearly enumerating each possible position for the new request's destination. Evaluations on two real-world and large-scale datasets show that our methods can accelerate the existing insertion algorithm by up to 25 times.

翻译：拼车因其经济和社会效益，近年来已成为一种有前景的出行模式。作为关键算子，"插入算子"在静态路网中已被广泛研究。当新请求出现时，插入算子用于寻找当前工作路径的最优位置，以插入该请求的起点和终点，并最小化工作者的行程时间。以往研究关注如何在静态路网中高效执行插入操作，然而在实际中，路径规划需考虑动态交通场景（即时间依赖路网）。遗憾的是，现有插入算子解决方案在此设置下效率低下。因此，本文研究时间依赖路网上的插入算子。具体而言，为降低现有方案 $O(n^3)$ 的高时间复杂度，我们沿路径计算复合行程时间函数，以加速路径上顶点对间行程时间的计算，从而将插入时间复杂度降至 $O(n^2)$。最后，我们进一步将该方法改进为线性时间插入算法，证明当线性枚举新请求终点的每个可能位置时，仅需 $O(1)$ 时间即可找到当前路径插入起点的最佳位置。在两个大规模真实数据集上的评估表明，我们的方法可将现有插入算法加速高达25倍。