Constant Time Algorithm for Computing a Collision-Free Path on R-Mesh with Path Quality Analysis

The reconfigurable mesh (R-Mesh) was shown to be a very powerful model capable of extremely fast solutions to many problems. R-Mesh has a wide range of applications such as arithmetic problems, image processing and robotics. The 2D R-Mesh was shown to be able to solve the path planning problem very fast. In this paper, we propose an algorithm to compute a collision-free path, P, between a source and a destination in an environment with the existence of obstacles. Independent of the number of obstacles, k, the proposed algorithm runs in constant time and requires O(log(2)N) preprocessing time where N is the size of the R-Mesh. This is in contrast to the previous work that requires O(k) time with the same pre-processing time. We then consider the quality of the generated path. We present a constant-time modification to enhance the length of the path and analyze the generated path P in terms of the number of bends in P. We derive the number of bends in P for any set of obstacles. We also derive a necessary condition for the minimum number of bends in the path P, i.e., a lower bound on the number of bends. We finally identify a class of obstacles for which the above necessary condition is sufficient as well (tight bound).