目錄
- A* 算法簡介
- 關鍵代碼介紹
- 保存基本信息的地圖類
- 搜索到的節點類
- 算法主函數介紹
- 代碼的初始化
- 完整代碼
A* 算法簡介
A* 算法需要維護兩個數據結構：OPEN 集和 CLOSED 集。OPEN 集包含所有已搜索到的待檢測節點。初始狀態，OPEN集僅包含一個元素：開始節點。CLOSED集包含已檢測的節點。初始狀態，CLOSED集為空。每個節點還包含一個指向父節點的指針，以確定追蹤關系。
A* 算法會給每個搜索到的節點計算一個G+H 的和值F：
- F = G + H
- G：是從開始節點到當前節點的移動量。假設開始節點到相鄰節點的移動量為1，該值會隨著離開始點越來越遠而增大。
- H：是從當前節點到目標節點的移動量估算值。
- 如果允許向4鄰域的移動，使用曼哈頓距離。
- 如果允許向8鄰域的移動，使用對角線距離。
算法有一個主循環，重復下面步驟直到到達目標節點：
1 每次從OPEN集中取一個最優節點n（即F值最小的節點）來檢測。
2 將節點n從OPEN集中移除，然后添加到CLOSED集中。
3 如果n是目標節點，那么算法結束。
4 否則嘗試添加節點n的所有鄰節點n'。
- 鄰節點在CLOSED集中，表示它已被檢測過，則無需再添加。
- 鄰節點在OPEN集中：
- 如果重新計算的G值比鄰節點保存的G值更小，則需要更新這個鄰節點的G值和F值，以及父節點；
- 否則不做操作
- 否則將該鄰節點加入OPEN集，設置其父節點為n，并設置它的G值和F值。
有一點需要注意，如果開始節點到目標節點實際是不連通的，即無法從開始節點移動到目標節點，那算法在第1步判斷獲取到的節點n為空，就會退出
關鍵代碼介紹
保存基本信息的地圖類
地圖類用于隨機生成一個供尋路算法工作的基礎地圖信息
先創建一個map類， 初始化參數設置地圖的長度和寬度，并設置保存地圖信息的二維數據map的值為0， 值為0表示能移動到該節點。
class Map():
def __init__(self, width, height):
self.width = width
self.height = height
self.map = [[0 for x in range(self.width)] for y in range(self.height)]
在map類中添加一個創建不能通過節點的函數，節點值為1表示不能移動到該節點。
def createBlock(self, block_num):
for i in range(block_num):
x, y = (randint(0, self.width-1), randint(0, self.height-1))
self.map[y][x] = 1
在map類中添加一個顯示地圖的函數，可以看到，這邊只是簡單的打印出所有節點的值，值為0或1的意思上面已經說明，在后面顯示尋路算法結果時，會使用到值2，表示一條從開始節點到目標節點的路徑。
def showMap(self):
print("+" * (3 * self.width + 2))
for row in self.map:
s = '+'
for entry in row:
s += ' ' + str(entry) + ' '
s += '+'
print(s)
print("+" * (3 * self.width + 2))
添加一個隨機獲取可移動節點的函數
def generatePos(self, rangeX, rangeY):
x, y = (randint(rangeX[0], rangeX[1]), randint(rangeY[0], rangeY[1]))
while self.map[y][x] == 1:
x, y = (randint(rangeX[0], rangeX[1]), randint(rangeY[0], rangeY[1]))
return (x , y)
搜索到的節點類
每一個搜索到將到添加到OPEN集的節點，都會創建一個下面的節點類，保存有entry的位置信息（x，y），計算得到的G值和F值，和該節點的父節點（pre_entry）。
class SearchEntry():
def __init__(self, x, y, g_cost, f_cost=0, pre_entry=None):
self.x = x
self.y = y
# cost move form start entry to this entry
self.g_cost = g_cost
self.f_cost = f_cost
self.pre_entry = pre_entry
def getPos(self):
return (self.x, self.y)
算法主函數介紹
下面就是上面算法主循環介紹的代碼實現，OPEN集和CLOSED集的數據結構使用了字典，在一般情況下，查找，添加和刪除節點的時間復雜度為O(1), 遍歷的時間復雜度為O(n), n為字典中對象數目。
def AStarSearch(map, source, dest):
...
openlist = {}
closedlist = {}
location = SearchEntry(source[0], source[1], 0.0)
dest = SearchEntry(dest[0], dest[1], 0.0)
openlist[source] = location
while True:
location = getFastPosition(openlist)
if location is None:
# not found valid path
print("can't find valid path")
break;
if location.x == dest.x and location.y == dest.y:
break
closedlist[location.getPos()] = location
openlist.pop(location.getPos())
addAdjacentPositions(map, location, dest, openlist, closedlist)
#mark the found path at the map
while location is not None:
map.map[location.y][location.x] = 2
location = location.pre_entry
我們按照算法主循環的實現來一個個講解用到的函數。
下面函數就是從OPEN集中獲取一個F值最小的節點，如果OPEN集會空，則返回None。
# find a least cost position in openlist, return None if openlist is empty
def getFastPosition(openlist):
fast = None
for entry in openlist.values():
if fast is None:
fast = entry
elif fast.f_cost > entry.f_cost:
fast = entry
return fast
addAdjacentPositions 函數對應算法主函數循環介紹中的嘗試添加節點n的所有鄰節點n'。
# add available adjacent positions
def addAdjacentPositions(map, location, dest, openlist, closedlist):
poslist = getPositions(map, location)
for pos in poslist:
# if position is already in closedlist, do nothing
if isInList(closedlist, pos) is None:
findEntry = isInList(openlist, pos)
h_cost = calHeuristic(pos, dest)
g_cost = location.g_cost + getMoveCost(location, pos)
if findEntry is None :
# if position is not in openlist, add it to openlist
openlist[pos] = SearchEntry(pos[0], pos[1], g_cost, g_cost+h_cost, location)
elif findEntry.g_cost > g_cost:
# if position is in openlist and cost is larger than current one,
# then update cost and previous position
findEntry.g_cost = g_cost
findEntry.f_cost = g_cost + h_cost
findEntry.pre_entry = location
getPositions 函數獲取到所有能夠移動的節點，這里提供了2種移動的方式：
- 允許上，下，左，右 4鄰域的移動
- 允許上，下，左，右，左上，右上，左下，右下 8鄰域的移動
def getNewPosition(map, locatioin, offset):
x,y = (location.x + offset[0], location.y + offset[1])
if x 0 or x >= map.width or y 0 or y >= map.height or map.map[y][x] == 1:
return None
return (x, y)
def getPositions(map, location):
# use four ways or eight ways to move
offsets = [(-1,0), (0, -1), (1, 0), (0, 1)]
#offsets = [(-1,0), (0, -1), (1, 0), (0, 1), (-1,-1), (1, -1), (-1, 1), (1, 1)]
poslist = []
for offset in offsets:
pos = getNewPosition(map, location, offset)
if pos is not None:
poslist.append(pos)
return poslist
isInList 函數判斷節點是否在OPEN集 或CLOSED集中
# check if the position is in list
def isInList(list, pos):
if pos in list:
return list[pos]
return None
calHeuristic 函數簡單得使用了曼哈頓距離，這個后續可以進行優化。
getMoveCost 函數根據是否是斜向移動來計算消耗（斜向就是2的開根號，約等于1.4）
# imporve the heuristic distance more precisely in future
def calHeuristic(pos, dest):
return abs(dest.x - pos[0]) + abs(dest.y - pos[1])
def getMoveCost(location, pos):
if location.x != pos[0] and location.y != pos[1]:
return 1.4
else:
return 1
代碼的初始化
可以調整地圖的長度，寬度和不可移動節點的數目。
可以調整開始節點和目標節點的取值范圍。
WIDTH = 10
HEIGHT = 10
BLOCK_NUM = 15
map = Map(WIDTH, HEIGHT)
map.createBlock(BLOCK_NUM)
map.showMap()
source = map.generatePos((0,WIDTH//3),(0,HEIGHT//3))
dest = map.generatePos((WIDTH//2,WIDTH-1),(HEIGHT//2,HEIGHT-1))
print("source:", source)
print("dest:", dest)
AStarSearch(map, source, dest)
map.showMap()
執行的效果圖如下，第一個表示隨機生成的地圖，值為1的節點表示不能移動到該節點。
第二個圖中值為2的節點表示找到的路徑。
完整代碼
使用python3.7編譯
from random import randint
class SearchEntry():
def __init__(self, x, y, g_cost, f_cost=0, pre_entry=None):
self.x = x
self.y = y
# cost move form start entry to this entry
self.g_cost = g_cost
self.f_cost = f_cost
self.pre_entry = pre_entry
def getPos(self):
return (self.x, self.y)
class Map():
def __init__(self, width, height):
self.width = width
self.height = height
self.map = [[0 for x in range(self.width)] for y in range(self.height)]
def createBlock(self, block_num):
for i in range(block_num):
x, y = (randint(0, self.width-1), randint(0, self.height-1))
self.map[y][x] = 1
def generatePos(self, rangeX, rangeY):
x, y = (randint(rangeX[0], rangeX[1]), randint(rangeY[0], rangeY[1]))
while self.map[y][x] == 1:
x, y = (randint(rangeX[0], rangeX[1]), randint(rangeY[0], rangeY[1]))
return (x , y)
def showMap(self):
print("+" * (3 * self.width + 2))
for row in self.map:
s = '+'
for entry in row:
s += ' ' + str(entry) + ' '
s += '+'
print(s)
print("+" * (3 * self.width + 2))
def AStarSearch(map, source, dest):
def getNewPosition(map, locatioin, offset):
x,y = (location.x + offset[0], location.y + offset[1])
if x 0 or x >= map.width or y 0 or y >= map.height or map.map[y][x] == 1:
return None
return (x, y)
def getPositions(map, location):
# use four ways or eight ways to move
offsets = [(-1,0), (0, -1), (1, 0), (0, 1)]
#offsets = [(-1,0), (0, -1), (1, 0), (0, 1), (-1,-1), (1, -1), (-1, 1), (1, 1)]
poslist = []
for offset in offsets:
pos = getNewPosition(map, location, offset)
if pos is not None:
poslist.append(pos)
return poslist
# imporve the heuristic distance more precisely in future
def calHeuristic(pos, dest):
return abs(dest.x - pos[0]) + abs(dest.y - pos[1])
def getMoveCost(location, pos):
if location.x != pos[0] and location.y != pos[1]:
return 1.4
else:
return 1
# check if the position is in list
def isInList(list, pos):
if pos in list:
return list[pos]
return None
# add available adjacent positions
def addAdjacentPositions(map, location, dest, openlist, closedlist):
poslist = getPositions(map, location)
for pos in poslist:
# if position is already in closedlist, do nothing
if isInList(closedlist, pos) is None:
findEntry = isInList(openlist, pos)
h_cost = calHeuristic(pos, dest)
g_cost = location.g_cost + getMoveCost(location, pos)
if findEntry is None :
# if position is not in openlist, add it to openlist
openlist[pos] = SearchEntry(pos[0], pos[1], g_cost, g_cost+h_cost, location)
elif findEntry.g_cost > g_cost:
# if position is in openlist and cost is larger than current one,
# then update cost and previous position
findEntry.g_cost = g_cost
findEntry.f_cost = g_cost + h_cost
findEntry.pre_entry = location
# find a least cost position in openlist, return None if openlist is empty
def getFastPosition(openlist):
fast = None
for entry in openlist.values():
if fast is None:
fast = entry
elif fast.f_cost > entry.f_cost:
fast = entry
return fast
openlist = {}
closedlist = {}
location = SearchEntry(source[0], source[1], 0.0)
dest = SearchEntry(dest[0], dest[1], 0.0)
openlist[source] = location
while True:
location = getFastPosition(openlist)
if location is None:
# not found valid path
print("can't find valid path")
break;
if location.x == dest.x and location.y == dest.y:
break
closedlist[location.getPos()] = location
openlist.pop(location.getPos())
addAdjacentPositions(map, location, dest, openlist, closedlist)
#mark the found path at the map
while location is not None:
map.map[location.y][location.x] = 2
location = location.pre_entry
WIDTH = 10
HEIGHT = 10
BLOCK_NUM = 15
map = Map(WIDTH, HEIGHT)
map.createBlock(BLOCK_NUM)
map.showMap()
source = map.generatePos((0,WIDTH//3),(0,HEIGHT//3))
dest = map.generatePos((WIDTH//2,WIDTH-1),(HEIGHT//2,HEIGHT-1))
print("source:", source)
print("dest:", dest)
AStarSearch(map, source, dest)
map.showMap()
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