# Speed Optimization for Incremental Updating of Grid-Based Distance Maps
## 2 Related Work
### Subgoal Graphs  
A subgoal graph is a kind of sparse spatial representation which can be precomputed by abstracting the skeletons of the underlying grids into undirected graphs. The algorithm which accounts for computing subgoal graphs was proposed by [PERSON] et al. [30]. It firstly introduces the basic version of the subgoal graph, called Simple Subgoal Graphs (SSGs). On grid maps, SSGs are constructed by placing subgoals at the convex corners of obstacles and connecting direct-h-reachable subgoals in the preprocessing stage. Two subgoals are mutually direct-h-reachable if all the optimal paths between them are valid and traverse no other subgoals.  
SSGs reduce the search space and accelerate the process of finding shortest paths by abstracting the key points of the grid maps and reducing all symmetric paths between subgoals into only one edge. It can be proved that, for any given start and goal cells which are reachable, there is a shortest path that can be divided into segments between subgoals. Each segment connects direct-h-reachable subgoals. Therefore, when finding shortest paths on SSGs, one first connects the start and goal vertices to their direct-h-reachable subgoals in SSGs, and then searches the modified graphs via A* to get the shortest high-level path, which consists of a sequence of subgoals between the start and goal vertices. By refining the shortest high-level path, one can get the shortest path between start and goal on the grid map.  
Moreover, a Two-level Subgoal Graph (TSG) is constructed from a SSG by partitioning the subgoals into global and local ones and only the global ones belong to the TSG. When removing the local subgoals, one has to add some extra edges so that the shortest paths between global subgoals remain the same. Finding the shortest path on a TSG is similar to finding a shortest path on a SSG. One first connects the start and goal vertices to their direct-h-reachable subgoals, among which the local ones should be temporarily connected to the TSG. Then, the search and refining processes are executed. It should be noted that the refining process of TSGs is slower than SSGs because TSGs connect h-reachable subgoals instead of direct-h-reachable subgoals. But, TSGs still find optimal paths faster due to the smaller search space. To reduce the search space even further, [PERSON] et al. generalized the idea of partitioning [31; 32], created a hierarchy among the vertices, and repeatedly divided the highest-level subgoals into global and local subgoals and increased the level of global subgoals by one. The resulting graphs are called N-Level Subgoal Graphs.  
Subgoal Graphs can be used not only to find grid paths, but also to find any-angle paths which are shorter and more realistic [33; 34; 35]. [PERSON] and [PERSON] [36] exploited the similarities between Subgoal Graphs and visibility graphs and used Subgoal Graphs to quickly find any-angle paths with some small modifications. Their algorithm is up to two orders of magnitude faster than Theta*, a well-known any-angle path planning algorithm.  
All the above algorithms only take regular grids as input and build subgoal graphs which do not take physical radius into considerations. Therefore, the paths planned from the resulting subgoal graphs would possibly be unable to meet the collision-free requirements in practical application. In this paper, we attempt to modify the existing algorithm so that it can make use of the maximal clearance values provided by the grid-based DMs and to build new type of subgoal graph that can efficiently search collision-free paths.