An auto-adaptive convex map generating path-finding algorithm: Genetic Convex A^*

Abstract

Path-finding is a fundamental problem in many applications, such as robot control, global positioning system and computer games. Since A^* is time-consuming when applied to large maps, some abstraction methods have been proposed. Abstractions can greatly speedup on-line path-finding by combing the abstract and the original maps. However, most of these methods do not consider obstacle distributions, which may result in unnecessary storage and non-optimal paths in certain open areas. In this paper, a new abstract graph-based path-finding method named Genetic Convex A^* is proposed. An important convex map concept which guides the partition of the original map is defined. It is proven that the path length between any two nodes within a convex map is equal to their Manhattan distance. Based on the convex map, a fitness function is defined to improve the extraction of key nodes; and genetic algorithm is employed to optimize the abstraction. Finally, the on-line refinement is accelerated by Convex A^*, which is a fast alternative to A^* on convex maps. Experimental results demonstrated that the proposed abstraction generated by Genetic Convex A^*guarantees the optimality of the path whilst searches less nodes during the on-line processing.

Appendices

Appendix 1

Property 2

If t(i, j) is a free tile in a convex map M , and t(i′, j′) is a free tile adjacent to M , then the length of the shortest path between them is |i − i′| + |j − j′| or |i − i′| + |j − j′| + 2.

Proof

There are four conditions that t(i′, j′) is adjacent to M:

• 1. If E(t(i′, j′)) ∈ M, then the shortest path between t(i, j) and t(i′, j′) is |i − i′| + |j − (j′ + 1)| + 1. And if j > j′, |i − i′| + |j − (j′ + 1)| + 1 = 0, Otherwise, |i − i′| + |j − (j′ + 1)| + 1 = 2.

• 2. If W(t(i′, j′)) ∈ M, then the shortest path between t(i, j) and t(i′, j′) is |i − i′| + |j − (j′ − 1)| + 1. And if j < j′, |i − i′| + |j − (j′ − 1)| + 1 = 0, Otherwise, |i − i′| + |j − (j′ − 1)| + 1 = 2.

• 3. If S(t(i′, j′)) ∈ M, then the shortest path between t(i, j) and t(i′, j′) is |i − (i′ + 1)| + |j − j′| + 1. And if i > i′, |i − (i′ + 1)| + |j − j′| + 1 = 0, Otherwise, |i − (i′ + 1)| + |j − j′| + 1 = 2.

• 4. If N(t(i′, j′)) ∈ M, then the shortest path between t(i, j) and t(i′, j′) is |i − (j′ − 1)| + |j − j′| + 1. And if i < i′, |i − (i′ − 1)| + |j − j′| + 1 = 0, Otherwise, |i − (i′ − 1)| + |j − j′| + 1 = 2.

The proof of Property 2 indicates that when inserting a tile into the abstract graph, and if the inserted tile is from a convex map, then the shortest path length between the tile and its neighbors could be found by calculation rather than by searching.

Property 3

if t(i, j) and t(i′, j′) are two free tiles adjacent to a convex map M , then the shortest path length between them is |i − i′| + |j − j′| or |i − i′| + |j − j′| + 2 or |i − i′| + |j − j′| + 4.

The proof of Property 3 is similar to the proof of Property 2. To save space, we do not describe it in detail here. It should be noted that in the condition where t(i, j) and t(i′, j′) are directly adjacent to each other, their shortest path length is |i − i′| + |j − j′| = 1.

Property 3 suggests that if two free tiles are adjacent to the same convex map, then the shortest path length could also be calculated from their axis information. Therefore, the weights of edges in the abstract graph are also found by simple calculation instead of searching.

Appendix 2

If the two tiles are both on the same shortcut, then they could be on a straight path, so the refinement algorithm should search the tiles on the shortcut and the tiles within the adjacent convex map. Otherwise, the refined path has to go through the convex map. Table 5 shows the refinement algorithm of G-CA^*.

Table 5 Refinement algorithm for G-CA^*