# Multicriteria Route Planning for In-Operation Mass Transit under Urban Data
## 2 Related Work  
In urban space, people tend to either travel by car or use public transport to move around and travel to their destinations [10]. Accordingly, most research focusing on route planning for mobility systems in urban space target the road network structure and public transit network structure. To be more precise, research on transportation route planning is not an uncharted area; where, in most cases, the original network is modeled as a directed graph for algorithms to retrieve a best route among vertices [11]. **General Approach for Routing in Transportation Network**. As one of the most fundamental solutions towards the shortest path planning, [PERSON]'s algorithm plays an important role in routing for transportation network. Accordingly, multiple techniques are proposed to enable [PERSON]'s algorithm to run in almost linear time or to use little memory [12; 13]. Furthermore, to facilitate practical applications in real-world scenarios consisting of millions of nodes with instant outputs required [14], methods that are able to speed up the searching process in a unimodal transportation network have flourished over the past decades. For instance, bidirectional search (graph search), goal direction [15; 16; 17], transportation hierarchy, separator-based methods [18], and reinforcement learning [19; 20] have been suggested [21; 22; 23; 24]. Among the abovementioned approaches, the A* algorithm [25] is able to locate the path with the least costs if the heuristic function is admissible; that is, A* can retrieve the optimal solution given a monotonic cost function in the problem space. However, with the nonmonotonicity in urban space, the efficacies for approaches with similar characteristics or prerequisites as A* turn out to be limited [26]. Nonetheless, by ensuring that the forward and backward potential functions are both consistent, general A* could be made bidirectional. Similarly, by modifying the stopping criteria for the spanning of A*, some propose to terminate the searching process after the algorithm finds the first node belonged to both closed sets [27; 28]. On the other hand, A* with Landmarks and Triangle inequality, or ALT, focuses on the triangle inequality and aims to obtain better bounds for A* [15]. Furthermore, by enabling much larger number of landmarks to be incorporated, the planning process can be made faster based on precomputation; however, this method requires exploring a previously explored node again during A* and faces the nonmonotonic as a result of a looser triangle inequality [29]. On the other hand, by reprojecting vertices into artificial coordinates with limited geoinformation, some propose to employ the Gauss-Seidel method to position the vertex in the weighted barycenter of its neighbors; next, gradient descend is adopted to disperse the barycentric layout into a grid-based graph. As it turns out, the nearest path for given queried locations can be derived from the graph [30]. Utilizing the concept of a geometric container, some propose to prune the searching at an early stage according to edges used by shortest paths towards certain vertices. Precisely, nodes that can be reached on a shortest path through certain edges are first labelled, and different geometric objects (e.g., bounding box, angular sector) are then adopted to contain and bind up those nodes that are to be reached so as to be used for a further pruning process [17]. Similar to some of the concepts of reprojection and pruning, some propose to adopt a separator to divide and modify the searching space. Such separator-based methods can be classified into different categories according to the features that are adopted as the separator (e.g., vertex, path, and arc) for the original searching space. First, for the vertex separator, a subset of vertices whose removal could decompose the original graph into cells is labelled, and shortcut edges are then added to this subset such that the distances between any pair of vertices are preserved.