Consider the following relaxation: for a given maze, remove some fraction p of the maze's obstacles at random. Any path through the original maze will be valid in this `thinned' maze, but there will be more feasible paths through the thinned maze and therefore it is `easier' to solve, potentially using something like A*-Manhattan. In particular, the distance from any point in the thinned maze to the goal is a lower bound on the distance from that point tothe goal in the original maze. As such, shortest path lengths in the thinned maze can be used as a heuristic in the original maze.
Try to determine whether or not this is a viable strategy. For an obstacle density of p = 0:3, write an A*-Manhattan solver and determine the largest size maze you can reliably, repeatedly solve in a reasonable amount of time. Then implement the -thinning strategy as an alternative solver, bootstrapped o the rst. Graph, as a function of , the fraction of nodes in the primary maze (non-thinned) expanded when solved by this thinning strategy (for each test value , generate and solve at least 100 mazes). Are there any where you see a savings over the straight A-Manhattan at this obstacle density? What additional time costs do you incur? Is the strategy viable?
Try to construct and analyze another relaxation-based strategy for generating a heuristic. Do you think this approach is viable? Why or why not. How does the representation of the maze factor in to this (sparse vs not sparse, etc)? Because certain strategies might have favorable or unfavorable regimes, could the strategy be variable based on how close you are to the goal (via Manhattan distance) or something similarly dynamic? How could the remainder of the maze be `approximated' in some way to produce a useful solution?
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