Finding Optimal Solution of 15 puzzle B NTUCSIE Kai-Yung Chiang

15 puzzle introduction Initial states: a solvable puzzle, including numbered 1~15 tiles and a blank tile 0 Goal state: puzzle in which tiles 0 ~ 15 are in well permutation from row to column A (valid) move: swap blank tile with its neighboring tile The optimal solution: given a initial state, find the move sequence such that it is a solution taking the fewest moves to reach the goal state

Searching Algorithm A* search Each node has heuristic (the estimating distance from current state to goal state) Each node s evaluation function = heuristic + total moves from initial state A* always expands the fringe node whose evaluation function is minimum

Example h=45, f=45 : expanded nodes h: heuristic f: evaluation function: fringe nodes

Example h=45, f=45 h=41, f=42h=45, f=46h=42, f=43 : expanded nodes h: heuristic f: evaluation function: fringe nodes

Example h=45, f=45 h=41, f=42h=45, f=46h=42, f=43 h=45, f=47h=42, f=44h=43, f=45 : expanded nodes h: heuristic f: evaluation function: fringe nodes

Example h=45, f=45 h=41, f=42h=45, f=46h=42, f=43 h=45, f=47h=42, f=44h=43, f=45 h=45, f=47h=39, f=41h=40, f=42 : expanded nodes h: heuristic f: evaluation function: fringe nodes

Pros: Can find the optimal solution (or admissible) if heuristic function will never overestimate Eliminate number of expanded nodes Cons: Taking more time on expanding each nodes (compared with DFS or BFS): Find minimum nodes (in fringe) to expand Compute heuristic and evaluation function Check, eliminate and update repetitive states Maintain data structure Pros and cons for A* search

Speed up by good data structure Priority queue: Fibonacci heap Extract minimum and decrease key in (amortized) O(logn) Insert in O(1) Hash function: Used in checking repetitive states, eliminate number of searching nodes A* search with above data structure and Manhattan distance as heuristic function could solve 8 puzzle in (averaged) seconds However, still too slow to solve 15 puzzle

Still too many nodes need to be expanded (though eliminate repetitive states) If f(n) = 2h(n) + moves(n) … Could speed up much, but not admissible. Ideas: try to improve h(n) to be more close to truly needed optimal moves, while still maintaining admissible Disjoint pattern database heuristic instead of Manhattan distance New heuristic = i number of needed moves to well order tiles in pattern i More speed up technique

Pattern database Divide puzzle into 3 disjoint patterns Each pattern has 16!/10! 5.7 * 10 6 different combination orders of tiles, and each of them corresponds to a database entry Each entry records moves for aligning these pattern tiles to correct position

Building pattern database Constructed by bottom up approach Start with well ordered tiles BFS moving blank tile to new states, move is added only when tile in this pattern is swapped with blank tile Add new entry to pattern database if current state has not been traversed (using a bits of array to record this information)

Benchmark and outcome Randomly generate 100 solvable puzzles:

Benchmark and outcome Pattern database heuristic improves nearly 10 moves than Manhattan distance in averaged In most problem instances … Expanded nodes are less than 100,000 Could be solved in 30 seconds, and more than half could be solved in 10 seconds CPU% are less than 20%. It might suggest that lots of time are took in communication with database Pattern database heuristic improves a lot

Demo