Search Algorithm ML

Consider a fourth heuristic defined in terms of inversions: For a puzzle configuration, we say that a pair of tiles a and b are inverted if a < b but the position of b is before a in the left-toright, top-to-bottom ordering described through the goal state. For instance, in the configuration in Figure 2a, the pairs (5, 8), (7, 8), (6, 8), and (6, 7) are inverted. We define a new heuristic hfn inversions as the number of inversions in a configuration. So for the said configuration in Figure 2a, hfn inversions = 4. Which of the four heuristics are admissible? Suppose that for sliding a tile to the left we would change the cost from 1 to 0.5 and leave all the other moves the same cost. Does this affect the admissibility of the heuristics? Which of them are admissible now? For any which is not, why not? Now suppose we would change the cost for sliding a tile to the left to 2 and leave all the other moves the same cost. Does this now affect the admissibility of the four heuristics? Again, which of them are admissible? For any which is not, why not? In the former modification (sliding to the left costs 0.5), can you say for sure which heuristic will be the fastest (expand the least number of states) in finding a (not necessary optimal) solution? Explain. One can obtain another heuristic for the N-Puzzle by relaxing the problem as follows: let’s say that a tile can move from square A to square B if B is blank. The exact solution to this problem defines Gaschnig’s heuristic. Explain why Gaschnig’s heuristic is at least as accurate as hfn_misplaced . Show some cases where it is more accurate than both the hfn_misplaced and hfn_manhattan heuristics. Can you suggest a way to calculate Gaschnig’s heuristic efficiently?