A probabilistic version of Dijkstra's algorithm does not exist in general; however, for some problems, it can be made to work. The algorithm in Figure 10.8 is adapted to the probabilistic case by using
The probabilistic version of Dijkstra's algorithm can be successfully applied if there exists a plan, , for which from any there is probability one that . What does this condition mean? From any , all possible next states that have nonzero probability of occurring must have a lower cost value. If all edge costs are positive, this means that all paths in the multi-stage forward projection will make monotonic progress toward the goal. In the deterministic case, this always occurs if is always positive. If nonmonotonic paths are possible, then Dijkstra's algorithm breaks down because the region in which cost-to-go values change is no longer contained within a propagating band, which arises in Dijkstra's algorithm for deterministic problems.
Steven M LaValle 2020-08-14