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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

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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

• 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