Solving 8-puzzle problem using A* algorithm
The 8-puzzle consists of an area divided into 3x3 (3 by 3) grid. Each grid with in the puzzle is known as tile and each tile contains a number ranged between 1 to 8, so that they can be uniquely identified. Tile adjacent to the empty grid can be moved to the empty space leaving its previous position empty until reaching the goal.
PROBLEM FORMULATION
- Goal: Goal State is initially given.
- States: Integer locations of tiles.
- Actions: Move the blank tile in left, up, down and right positions -Performance: Number of total moves in the solution
A* ALGORITHM
A* is an informed search algorithm used in path findings and graph traversals. It is a combination of uniform cost search and best first search, which avoids expanding expensive paths. A* star uses admissible heuristics which is optimal as it never over-estimates the path to goal. The evaluation function A* star uses for calculating distance is
- f(n) = g(n) + h(n)
- g(n) = cost so far to reach n
- h(n) = estimated cost from n to goal
- f(n) = estimated total cost of path through n to goal
Heuristic Functions
The heuristic function is a way to inform the search regarding the direction to a goal. It provides an information to estimate which neighboring node will lead to the goal. The two heuristic functions that we considered for solving 8-puzzle problem are
Misplaced Tile
The number of misplaced tiles calculated by comparing the current state and goal state.
Manhattan Distance
The distance between two tiles measured along the axes of right angles. It is the sum of absolute values of differences between goal state (i, j) coordinates and current state (l, m) coordinates respectively, i.e. |i - l|+ |j - m|