A* Search Algorithm

写在前面：第一次写博客，以后想把遇到的有意思的算法或者小知识记录下来。纯原创，部分图片来自课堂PPT，出自UCR，CS170课件。转载请声明出处。

算法简介

A* Search 是一种启发式搜索算法，该算法内置一个函数来表达当前状态和目标状态的差值。举个栗子，8-puzzle问题（一个游戏，目标是将对应数字的版块放到对应位置）。当前状态是下图

目标状态：

如果我们以两个状态有多少不符合启发函数，那么这里的启发函数的值是
1（PS:这里我们排除了空白格，图中当前态和目标态就一个‘8’不同）

接下来会详细介绍A算法并以8-puzzle游戏左右一个例子来说明该算法，在引入A search前，我会先简单介绍两个算法，Uniform Cost Search和Hill climbing Search。启发函数选择的是Manhattan距离，详解看下图。（h(n)就是启发函数）

先说说Hill climbing Search，该算法也是一种启发式搜索算法，思路很简单，直接使用Manhattan距离。在下面的例子中，这个算法运行的很好。（h(n)是启发函数）

这个算法运行的很快，可是会遇到问题。有时候该算法在一些状态会陷入找不到解的状态

另一种Uniform Cost Search算法，该算法更简单每次在拓展节点的时候，拓展最廉价的点（该点的耗费是到这个点的路径的累积消耗，拓展累积消耗最小的点）。该算法运行效率较低，但是可以保证每次找到一个解。

背景介绍完毕。接下来进入A*算法，这个算法结合了Uniform Cost Search和Hill climbing Search。将到达该节点的累积消耗g(n)和该节点到达目标节点的差h(n)作为启发函数

下图是一个A算法的例子：这是一个迷宫问题，初始状态是粉色的点，目标状态是蓝色的点。将粉点移动到蓝点即可。图中的树说明了用A树的搜索过程。H(n)依然使用Manhattan距离

说到这，A算法的基本概念已经介绍完毕。一下附上A算法的实现代码（Python）。以解决8-puzzle问题作为背景。

实现代码

import numpy as np
import Queue
import copy
DEFAULT_PUZZLE = np.array([1,2,3,4,8,0,7,6,5]).reshape((3,3))
GOAL_STATE = np.array([1,2,3,4,5,6,7,8,0]).reshape((3,3))
STATE_TABLE = dict() # store the state already checkclass Node():''':description: define the Node class'''def __init__(self,puzzle,depth,total_cost,blank_index_i,blank_index_j):self.puzzle = puzzleself.depth = depthself.total_cost = total_costself.blank_index_i = blank_index_iself.blank_index_j = blank_index_jdef get_depth(self):return self.depthdef get_puzzle(self):return self.puzzledef get_total_cost(self):return self.total_costdef get_blank_index_i(self):return self.blank_index_idef get_blank_index_j(self):return self.blank_index_jdef __lt__(self, other):return self.total_cost < other.total_costdef Init_input_puzzle():''':description:Initializing the input puzzle matrix, and choose the algorithm:input: None:return:puzzle: the puzzle need to solvedkey:  the choice of algorithmblank_index_i : the blank block indexblank_index_j : the blank block index'''print "Welcome to Geyeshui's 8-Puzzle solver"key = input("Inpute 1 to use the defualt puzzle, or 2 to enter your own puzzle: \n")if key == 1:puzzle = DEFAULT_PUZZLEelse:print "Enter your puzzle use a zero to represent the blank"puzzle_input = []for i in range(3):row = raw_input("Enter the "+str(i+1)+"th row, use SPACE bewtween numbers: \n")puzzle_input = puzzle_input + [eval(i) for i in row.split()]puzzle = np.array(puzzle_input).reshape((3,3))print "Enter your choice of algorithm"print "1.Uniform cost search"print "2.A* with the Misplaced title heuristic"key = input("3.A* with the Manhattan distance heuristic \n")# find the blank indexfor i in range(3):for j in range(3):if puzzle[i][j]==0:blank_index_i = iblank_index_j = jreturn puzzle, key, blank_index_i, blank_index_jdef Get_heuristic_cost(key,puzzle):''':description:according the algorithm you choice return the corresponding h(n) value:inpute:key : the algorithm index numberpuzzle : the puzzle needed to estimate the h(n):return:h_n : the h(n) value'''h_n = 0if key == 1:h_n=0elif key ==2:for i in range(3):  # calculate the misplace number, excluding the blank.for j in range(3):if puzzle[(i,j)] != GOAL_STATE[(i,j)]:if i==2 and j==2:continueelse:h_n = h_n+1else:for i in range(3): # calculate the manhattan distancefor j in range(3):num = puzzle[(i,j)]if num==0:continueelse:index_num_i = (num-1)/3index_num_j = (num-1)%3h_n = h_n + (abs(i-index_num_i)+abs(j-index_num_j))return h_ndef Is_goal_state(puzzle):''':description: return Ture if the puzzle is the goal state, False otherwise:input:puzzle: the puzzle needs to be check:return:Ture: if puzzle is the goal stateFalse: if it is not'''if sum(sum(puzzle==GOAL_STATE))==9:return Trueelse:return Falsedef Move_up(puzzle,index_i,index_j):''':description:move up the blank block if it can.:parampuzzle: the puzzle which needs the operation:return:puzzle: puzzle after move upTrue: if it can move upFalse: if it is illegal to move up'''if index_i>0:puzzle[index_i-1][index_j],puzzle[index_i][index_j] \= puzzle[index_i][index_j],puzzle[index_i-1][index_j]if STATE_TABLE.get(str(puzzle),0) == 1:return None,Falseelse:STATE_TABLE[str(puzzle)] = 1return puzzle,Trueelse:return None,Falsedef Move_down(puzzle,index_i,index_j):''':description:move down the blank block if it can.:parampuzzle: the puzzle which needs the operation:return:puzzle: puzzle after move downTrue: if it can move downFalse: if it is illegal to move down'''if index_i<2:puzzle[index_i+1][index_j],puzzle[index_i][index_j] \= puzzle[index_i][index_j],puzzle[index_i+1][index_j]if STATE_TABLE.get(str(puzzle),0) == 1:return None,Falseelse:STATE_TABLE[str(puzzle)] = 1return puzzle,Trueelse:return None,Falsedef Move_left(puzzle,index_i,index_j):''':description:move left the blank block if it can.:parampuzzle: the puzzle which needs the operation:return:puzzle: puzzle after move leftTrue: if it can move leftFalse: if it is illegal to move left'''if index_j>0:puzzle[index_i][index_j-1],puzzle[index_i][index_j] \= puzzle[index_i][index_j],puzzle[index_i][index_j-1]if STATE_TABLE.get(str(puzzle),0) == 1:return None,Falseelse:STATE_TABLE[str(puzzle)] = 1return puzzle,Trueelse:return None,Falsedef Move_right(puzzle,index_i,index_j):''':description:move right the blank block if it can.:parampuzzle: the puzzle which needs the operation:return:puzzle: puzzle after move rightTrue: if it can move rightFalse: if it is illegal to move right'''if index_j<2:puzzle[index_i][index_j+1],puzzle[index_i][index_j] \= puzzle[index_i][index_j],puzzle[index_i][index_j+1]if STATE_TABLE.get(str(puzzle),0) == 1:return None,Falseelse:STATE_TABLE[str(puzzle)] = 1return puzzle,Trueelse:return None,Falseif __name__ == '__main__':ans = None# key is the choice index of algorithmpuzzle, key, blank_index_i, blank_index_j= Init_input_puzzle()STATE_TABLE[str(puzzle)] = 1global_step = 0 # store the how many iteration we runsize_of_pq = 0 # store the max size of priority_queuepq = Queue.PriorityQueue()pq.put(Node(puzzle,0,Get_heuristic_cost(key,puzzle),blank_index_i,blank_index_j))while not pq.empty():size_of_pq = max(size_of_pq,pq.qsize())node = pq.get()global_step = global_step + 1print node.get_puzzle()if Is_goal_state(node.get_puzzle()):ans = nodebreakelse:blank_index_i = node.get_blank_index_i()blank_index_j = node.get_blank_index_j()up_puzzle, up_flag = Move_up(copy.deepcopy(node.get_puzzle()),blank_index_i,blank_index_j)down_puzzle, down_flag = Move_down(copy.deepcopy(node.get_puzzle()),blank_index_i,blank_index_j)right_puzzle, right_flag = Move_right(copy.deepcopy(node.get_puzzle()),blank_index_i,blank_index_j)left_puzzle, left_flag = Move_left(copy.deepcopy(node.get_puzzle()),blank_index_i,blank_index_j)if up_flag==True:pq.put(Node(up_puzzle,node.get_depth()+1,node.get_depth()+1+Get_heuristic_cost(key,up_puzzle),blank_index_i-1,blank_index_j))if down_flag==True:pq.put(Node(down_puzzle,node.get_depth()+1,node.get_depth()+1+Get_heuristic_cost(key,down_puzzle),blank_index_i+1,blank_index_j))if right_flag==True:pq.put(Node(right_puzzle,node.get_depth()+1,node.get_depth()+1+Get_heuristic_cost(key,right_puzzle),blank_index_i,blank_index_j+1))if left_flag==True:pq.put(Node(left_puzzle,node.get_depth()+1,node.get_depth()+1+Get_heuristic_cost(key,left_puzzle),blank_index_i,blank_index_j-1))print ans.get_puzzle(),ans.get_depth(),global_step,size_of_pq

以上就是A* Search Algorithm, 有什么说的不对的地方欢迎指正