A* and Weighted A* Search
思路
啟發式搜索算法
要理解A*搜尋算法，還得從啟發式搜索算法開始談起。
所謂啟發式搜索，就在於當前搜索結點往下選擇下一步結點時，可以通過一個啟發函數(Heuristic Function)來進行選擇，選擇代價最少的結點作為下一步搜索結點而跳轉其上（遇到有一個以上代價最少的結點，不妨選距離當前搜索點最近一次展開的搜索點進行下一步搜索）。
DFS和BFS在展開子結點時均屬於盲目型搜索，也就是說，它不會選擇哪個結點在下一次搜索中更優而去跳轉到該結點進行下一步的搜索。在運氣不好的情形中，均需要試探完整個解集空間, 顯然，只能適用於問題規模不大的搜索問題中。
而與DFS,BFS不同的是，一個經過仔細設計的啟發函數，往往在很快的時間內就可得到一個搜索問題的最優解，對於NP問題，亦可在多項式時間內得到一個較優解。是的，關鍵就是如何設計這個啟發函數。
A*搜尋算法
A*搜尋算法，俗稱A星算法，作為啟發式搜索算法中的一種，這是一種在圖形平面上，有多個節點的路徑，求出最低通過成本的算法。常用於游戲中的NPC的移動計算，或線上游戲的BOT的移動計算上。該算法像Dijkstra算法一樣，可以找到一條最短路徑；也像BFS一樣，進行啟發式的搜索。
A*算法最為核心的部分，就在於它的一個估值函數的設計上：
\(f(n)=g(n)+h(n)\)
其中\(f(n)\)是每個可能試探點的估值，它有兩部分組成：
- 一部分，為g(n)，它表示從起始搜索點到當前點的代價（通常用某結點在搜索樹中的深度來表示）。
- 另一部分，即h(n)，它表示啟發式搜索中最為重要的一部分，即當前結點到目標結點的估值，
h(n)設計的好壞，直接影響着具有此種啟發式函數的啟發式算法的是否能稱為A*算法。
一種具有\(f(n)=g(n)+h(n)\)策略的啟發式算法能成為A*算法的充分條件是：
1、搜索樹上存在着從起始點到終了點的最優路徑。
2、問題域是有限的。
3、所有結點的子結點的搜索代價值\(>0\)。
4、\(h(n)=<h*(n)\) （\(h*(n)\)為實際問題的代價值）。
當此四個條件都滿足時，一個具有\(f(n)=g(n)+h(n)\)策略的啟發式算法能成為A*算法，並一定能找到最優解。
對於一個搜索問題，顯然，條件1,2,3都是很容易滿足的，而條件4： \(h(n)<=h*(n)\)是需要精心設計的，由於\(h*(n)\)顯然是無法知道的，所以，一個滿足條件4的啟發策略\(h(n)\)就來的難能可貴了。
不過，對於圖的最優路徑搜索和八數碼問題，有些相關策略h(n)不僅很好理解，而且已經在理論上證明是滿足條件4的，從而為這個算法的推廣起到了決定性的作用。
且\(h(n)\)距離\(h*(n)\)的呈度不能過大，否則\(h(n)\)就沒有過強的區分能力，算法效率並不會很高。對一個好的\(h(n)\)的評價是：\(h(n)\)在\(h*(n)\)的下界之下，並且盡量接近\(h*(n)\)。
示例程序
"""
A* grid planning
author: Atsushi Sakai(@Atsushi_twi)
Nikos Kanargias (nkana@tee.gr)
See Wikipedia article (https://en.wikipedia.org/wiki/A*_search_algorithm)
"""
import math
import matplotlib.pyplot as plt
show_animation = True
class AStarPlanner:
def __init__(self, ox, oy, reso, rr):
"""
Initialize grid map for a star planning
ox: x position list of Obstacles [m]
oy: y position list of Obstacles [m]
reso: grid resolution [m]
rr: robot radius[m]
"""
self.reso = reso
self.rr = rr
self.calc_obstacle_map(ox, oy)
self.motion = self.get_motion_model()
class Node:
def __init__(self, x, y, cost, pind):
self.x = x # index of grid
self.y = y # index of grid
self.cost = cost
self.pind = pind
def __str__(self):
return str(self.x) + "," + str(self.y) + "," + str(self.cost) + "," + str(self.pind)
def planning(self, sx, sy, gx, gy):
"""
A star path search
input:
sx: start x position [m]
sy: start y position [m]
gx: goal x position [m]
gy: goal y position [m]
output:
rx: x position list of the final path
ry: y position list of the final path
"""
nstart = self.Node(self.calc_xyindex(sx, self.minx),
self.calc_xyindex(sy, self.miny), 0.0, -1)
ngoal = self.Node(self.calc_xyindex(gx, self.minx),
self.calc_xyindex(gy, self.miny), 0.0, -1)
open_set, closed_set = dict(), dict()
open_set[self.calc_grid_index(nstart)] = nstart
while 1:
if len(open_set) == 0:
print("Open set is empty..")
break
c_id = min(
open_set, key=lambda o: open_set[o].cost + self.calc_heuristic(ngoal, open_set[o]))
current = open_set[c_id]
# show graph
if show_animation: # pragma: no cover
plt.plot(self.calc_grid_position(current.x, self.minx),
self.calc_grid_position(current.y, self.miny), "xc")
# for stopping simulation with the esc key.
plt.gcf().canvas.mpl_connect('key_release_event',
lambda event: [exit(0) if event.key == 'escape' else None])
if len(closed_set.keys()) % 10 == 0:
plt.pause(0.001)
if current.x == ngoal.x and current.y == ngoal.y:
print("Find goal")
ngoal.pind = current.pind
ngoal.cost = current.cost
break
# Remove the item from the open set
del open_set[c_id]
# Add it to the closed set
closed_set[c_id] = current
# expand_grid search grid based on motion model
for i, _ in enumerate(self.motion):
node = self.Node(current.x + self.motion[i][0],
current.y + self.motion[i][1],
current.cost + self.motion[i][2], c_id)
n_id = self.calc_grid_index(node)
# If the node is not safe, do nothing
if not self.verify_node(node):
continue
if n_id in closed_set:
continue
if n_id not in open_set:
open_set[n_id] = node # discovered a new node
else:
if open_set[n_id].cost > node.cost:
# This path is the best until now. record it
open_set[n_id] = node
rx, ry = self.calc_final_path(ngoal, closed_set)
return rx, ry
def calc_final_path(self, ngoal, closedset):
# generate final course
rx, ry = [self.calc_grid_position(ngoal.x, self.minx)], [
self.calc_grid_position(ngoal.y, self.miny)]
pind = ngoal.pind
while pind != -1:
n = closedset[pind]
rx.append(self.calc_grid_position(n.x, self.minx))
ry.append(self.calc_grid_position(n.y, self.miny))
pind = n.pind
return rx, ry
@staticmethod
def calc_heuristic(n1, n2):
w = 1.0 # weight of heuristic
d = w * math.hypot(n1.x - n2.x, n1.y - n2.y)
return d
def calc_grid_position(self, index, minp):
"""
calc grid position
:param index:
:param minp:
:return:
"""
pos = index * self.reso + minp
return pos
def calc_xyindex(self, position, min_pos):
return round((position - min_pos) / self.reso)
def calc_grid_index(self, node):
return (node.y - self.miny) * self.xwidth + (node.x - self.minx)
def verify_node(self, node):
px = self.calc_grid_position(node.x, self.minx)
py = self.calc_grid_position(node.y, self.miny)
if px < self.minx:
return False
elif py < self.miny:
return False
elif px >= self.maxx:
return False
elif py >= self.maxy:
return False
# collision check
if self.obmap[node.x][node.y]:
return False
return True
def calc_obstacle_map(self, ox, oy):
self.minx = round(min(ox))
self.miny = round(min(oy))
self.maxx = round(max(ox))
self.maxy = round(max(oy))
print("minx:", self.minx)
print("miny:", self.miny)
print("maxx:", self.maxx)
print("maxy:", self.maxy)
self.xwidth = round((self.maxx - self.minx) / self.reso)
self.ywidth = round((self.maxy - self.miny) / self.reso)
print("xwidth:", self.xwidth)
print("ywidth:", self.ywidth)
# obstacle map generation
self.obmap = [[False for i in range(self.ywidth)]
for i in range(self.xwidth)]
for ix in range(self.xwidth):
x = self.calc_grid_position(ix, self.minx)
for iy in range(self.ywidth):
y = self.calc_grid_position(iy, self.miny)
for iox, ioy in zip(ox, oy):
d = math.hypot(iox - x, ioy - y)
if d <= self.rr:
self.obmap[ix][iy] = True
break
@staticmethod
def get_motion_model():
# dx, dy, cost
motion = [[1, 0, 1],
[0, 1, 1],
[-1, 0, 1],
[0, -1, 1],
[-1, -1, math.sqrt(2)],
[-1, 1, math.sqrt(2)],
[1, -1, math.sqrt(2)],
[1, 1, math.sqrt(2)]]
return motion
def main():
print(__file__ + " start!!")
# start and goal position
sx = 10.0 # [m]
sy = 10.0 # [m]
gx = 50.0 # [m]
gy = 50.0 # [m]
grid_size = 2.0 # [m]
robot_radius = 1.0 # [m]
# set obstable positions
ox, oy = [], []
for i in range(-10, 60):
ox.append(i)
oy.append(-10.0)
for i in range(-10, 60):
ox.append(60.0)
oy.append(i)
for i in range(-10, 61):
ox.append(i)
oy.append(60.0)
for i in range(-10, 61):
ox.append(-10.0)
oy.append(i)
for i in range(-10, 40):
ox.append(20.0)
oy.append(i)
for i in range(0, 40):
ox.append(40.0)
oy.append(60.0 - i)
if show_animation: # pragma: no cover
plt.plot(ox, oy, ".k")
plt.plot(sx, sy, "og")
plt.plot(gx, gy, "xb")
plt.grid(True)
plt.axis("equal")
a_star = AStarPlanner(ox, oy, grid_size, robot_radius)
rx, ry = a_star.planning(sx, sy, gx, gy)
if show_animation: # pragma: no cover
plt.plot(rx, ry, "-r")
plt.pause(0.001)
plt.show()
if __name__ == '__main__':
main()
核心代碼
def planning(self, sx, sy, gx, gy):
"""
A star path search
input:
sx: start x position [m]
sy: start y position [m]
gx: goal x position [m]
gy: goal y position [m]
output:
rx: x position list of the final path
ry: y position list of the final path
"""
nstart = self.Node(self.calc_xyindex(sx, self.minx),
self.calc_xyindex(sy, self.miny), 0.0, -1)
ngoal = self.Node(self.calc_xyindex(gx, self.minx),
self.calc_xyindex(gy, self.miny), 0.0, -1)
open_set, closed_set = dict(), dict()
open_set[self.calc_grid_index(nstart)] = nstart
while 1:
if len(open_set) == 0:
print("Open set is empty..")
break
c_id = min(
open_set, key=lambda o: open_set[o].cost + self.calc_heuristic(ngoal, open_set[o]))
current = open_set[c_id]
# show graph
if show_animation: # pragma: no cover
plt.plot(self.calc_grid_position(current.x, self.minx),
self.calc_grid_position(current.y, self.miny), "xc")
# for stopping simulation with the esc key.
plt.gcf().canvas.mpl_connect('key_release_event',
lambda event: [exit(0) if event.key == 'escape' else None])
if len(closed_set.keys()) % 10 == 0:
plt.pause(0.001)
if current.x == ngoal.x and current.y == ngoal.y:
print("Find goal")
ngoal.pind = current.pind
ngoal.cost = current.cost
break
# Remove the item from the open set
del open_set[c_id]
# Add it to the closed set
closed_set[c_id] = current
# expand_grid search grid based on motion model
for i, _ in enumerate(self.motion):
node = self.Node(current.x + self.motion[i][0],
current.y + self.motion[i][1],
current.cost + self.motion[i][2], c_id)
n_id = self.calc_grid_index(node)
# If the node is not safe, do nothing
if not self.verify_node(node):
continue
if n_id in closed_set:
continue
if n_id not in open_set:
open_set[n_id] = node # discovered a new node
else:
if open_set[n_id].cost > node.cost:
# This path is the best until now. record it
open_set[n_id] = node
rx, ry = self.calc_final_path(ngoal, closed_set)
return rx, ry
其中，
c_id = min(open_set, key=lambda o: open_set[o].cost + self.calc_heuristic(ngoal, open_set[o])) current = open_set[c_id]
把min函數的比較函數換成open_set[o].cost + self.calc_heuristic(ngoal, open_set[o])，也即\(f(n)=g(n)+h(n)\)，通過這兩行，找到在open_set中cost與heuristic值最小的節點。比如，在常見的機器人搜索路徑問題中，cost指的是從起點到該節點走過的距離，heuristic是指從該節點到達終點的距離。那么很好理解，每一步都能選取到一個最佳路徑點。
參考
- P. E. Hart, N. J. Nilsson, and B. Raphael. A formal basis for the heuristic determination of minimum cost paths in graphs. IEEE Trans. Syst. Sci. and Cybernetics, SSC-4(2):100-107, 1968
- https://blog.csdn.net/v_JULY_v/article/details/6093380