*2.1. Grid-Based Distance Maps*

In the context of robotics and game AI, the grid-based DM is a popular spatial representation applied in navigation and motion planning tasks. The principal component of the recent approaches for constructing or reconstructing grid-based DMs is the well-known Brushfire algorithm [20]. Intuitively, Brushfire launches wavefronts to propagate changes of maximal clearance (i.e., changes caused by insertion or deletion of obstacle cells), updating distance values from the source of the change, and terminates when the change does not affect any more cells. Brushfire represents the OPEN list as a priority queue to incrementally record the affected cells and propagate the wavefronts. The priority of an element in the OPEN queue is determined by its newly updated distance and all these elements are popped up in increasing priorities. Sequentially, new cells which are adjacent to the popped one are tested, among which, newly updated cells will again be inserted into the OPEN list so that the propagation continues.

Kalra et al. [6], in their fundamental work, proposed a dynamic version of Brushfire algorithm, Dynamic Brushfire, to incrementally update grid-based DMs by propagating two kinds of wavefronts named "lower" and "raise" which start at newly blocked or freed cells, respectively; therefore, the update can be constrained within local areas. However, the wavefronts launched by Dynamic Brushfire roughly accumulate 8-connected grid steps to approximate maximal clearance, which overestimates the true Euclidean distances and would possibly lead to either a collision risk or overly conservative movements. To this end, Scherer et al. [21] proposed a method to propagate obstacle locations rather than counts of the grid steps, which reduces the absolute overestimation error below an upper bound of 0.09 pixel units. In the method proposed by Cuisenaire and Macq [22], the shortest distance at which this propagation error can occur is 13 pixels, which yields a maximum relative error of 0.69%. Regarding propagating obstacle references, Lau et al. proposed an approach to provide the location of the closest obstacle rather than just the distance to it, which can be appealing for collision check tasks [23]. Moreover, Lau et al. extended their method to 3D by adding the possibility to limit the propagated distances to maintain online feasibility in large open spaces and outdoors as proposed by Scherer et al.

Although these dynamic algorithms are fast and efficient for dealing with local changes, they just indiscriminately expand all the adjacent cells surrounding a currently processed cell, which results in a lot of redundant cell visits and scales up the size of the OPEN list and restricts the overall efficiency of the algorithm. We introduce the Canonical Ordering strategy in our work to prune the search space.

#### *2.2. Canonical Ordering*

The idea of applying Canonical Ordering as a speedup technique for real-time pathfinding systems that operate on regular grids was proposed by Daniel Harabor [24] and N Pochter [25]. As mentioned in the literature, searching in grids often becomes overwhelmed by a high degree of path symmetry, which accounts for a major part of the computational costs. Two paths are viewed as symmetric if (1) they have the same start and goal cells; (2) they are of the same length; and (3) their respective sequences of moves (i.e., cardinal or diagonal moves) can be reordered into the other. With symmetries in the grids, a search task will explore multiple cells for multiple times from those symmetric paths and this severely undermines the efficiency.

To break such symmetries, an online algorithm called Jump Point Search (JPS) [26] was presented by Daniel Harabor et al. to apply Canonical Ordering to recursively prune redundant successors and selectively expand only certain cells, called jump points. Canonical Ordering is essentially a special case of partial orderings among all the symmetric paths and prefers the diagonal-first ones to other alternatives. We say that a path has the diagonal-first property if there is no straight-diagonal turning point can be mutated into a diagonal-straight one of the same length constrained by the obstacles. By its virtue, JPS visits much fewer cells than traditional searching strategies; therefore, it answers a path query averagely faster than A\* by an order of magnitude. After that, this algorithm's performance was further improved by a preprocessing based strategy and addition of Bounding Boxes, resulting in the algorithms JPS+ [26] and JPS+BB [27]. As an automatic move pruning technique for single-agent search [28], Canonical Ordering can not only be used in grids, but can also be built on general graphs and considerably reduce the number of cells generated by an A\* search [29].

The outstanding performance of the Canonical Ordering strategy in compressing search space for real-time pathfinding algorithms provides us with a novel method to guide the direction of the wavefronts which propagate the distance changes, making it possible to speed up the construction of grid-based DMs.
