Finding Optimal Solution of 15 puzzle

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Finding Optimal Solution of 15 puzzle. B94902062 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

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### Finding Optimal Solution of 15 puzzle

B94902062 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

: fringe nodes

: expanded nodes h: heuristic f: evaluation function

Example

h=45, f=45

: fringe nodes

: expanded nodes h: heuristic f: evaluation function

Example

h=45, f=45

h=41, f=42

h=45,f=46

h=42, f=43

: fringe nodes

: expanded nodes h: heuristic f: evaluation function

Example

h=45, f=45

h=41, f=42

h=45,f=46

h=42, f=43

h=45, f=47

h=42, f=44

h=43, f=45

: fringe nodes

: expanded nodes h: heuristic f: evaluation function

Example

h=45, f=45

h=41, f=42

h=45,f=46

h=42, f=43

h=45, f=47

h=42, f=44

h=43, f=45

h=40,f=42

h=39, f=41

h=45,f=47

Pros and cons for A* search

• 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
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) 0.002 seconds

However, still too slow to solve 15 puzzle

More speed up technique
• 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
Pattern database
• Divide puzzle into 3 disjoint patterns
• Each pattern has 16!/10!≒5.7 * 106 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