Finding the Minimum-Weight k-Path

Given a weighted $n$-vertex graph $G$ with integer edge-weights taken from a range $[-M,M]$, we show that the minimum-weight simple path visiting $k$ vertices can be found in time \tilde{O}(2^k \poly(k) M n^\omega) = O^*(2^k M). If the weights are reals in $[1,M]$, we provide a $(1+\varepsilon)$-approximation which has a running time of \tilde{O}(2^k \poly(k) n^\omega(\log\log M + 1/\varepsilon)). For the more general problem of $k$-tree, in which we wish to find a minimum-weight copy of a $k$-node tree $T$ in a given weighted graph $G$, under the same restrictions on edge weights respectively, we give an exact solution of running time \tilde{O}(2^k \poly(k) M n^3) and a $(1+\varepsilon)$-approximate solution of running time \tilde{O}(2^k \poly(k) n^3(\log\log M + 1/\varepsilon)). All of the above algorithms are randomized with a polynomially-small error probability.Comment: To appear at WADS 201

Hierarchical path-finding for Navigation Meshes (HNA*)

Path-finding can become an important bottleneck as both the size of the virtual environments and the number of agents navigating them increase. It is important to develop techniques that can be efficiently applied to any environment independently of its abstract representation. In this paper we present a hierarchical NavMesh representation to speed up path-finding. Hierarchical path-finding (HPA*) has been successfully applied to regular grids, but there is a need to extend the benefits of this method to polygonal navigation meshes. As opposed to regular grids, navigation meshes offer representations with higher accuracy regarding the underlying geometry, while containing a smaller number of cells. Therefore, we present a bottom-up method to create a hierarchical representation based on a multilevel k-way partitioning algorithm (MLkP), annotated with sub-paths that can be accessed online by our Hierarchical NavMesh Path-finding algorithm (HNA*). The algorithm benefits from searching in graphs with a much smaller number of cells, thus performing up to 7.7 times faster than traditional A¿ over the initial NavMesh. We present results of HNA* over a variety of scenarios and discuss the benefits of the algorithm together with areas for improvement.Peer ReviewedPostprint (author's final draft

Path planning algorithm for a car like robot based on Coronoi Diagram Method

The purpose of this study is to develop an efficient offline path planning algorithm that is capable of finding optimal collision-free paths from a starting point to a goal point. The algorithm is based on Voronoi diagram method for the environment representation combined with Dijkstra’s algorithm to find the shortest path. Since Voronoi diagram path exhibits sharp corners and redundant turns, path tracking was applied considering the robot’s kinematic constraints. The results has shown that the Voronoi diagram path planning method recorded fast computational time as it provides simpler, faster and efficient path finding. The final path, after considering robot’s kinematic constraints, provides shorter path length and smoother compared to the original one. The final path can be tuned to the desired path by tuning the parameter setting; velocity, v and minimum turning radius, Rmin. In comparison with the Cell Decomposition method, it shows that Voronoi diagram has a faster computation time. This leads to the reduced cost in terms of time. The findings of this research have shown that Voronoi Diagram and Dijkstra’s Algorithm are a good combination in the path planning problem in terms of finding a safe and shortest path