A*算法基本原理

Posted 2021-09-12 智低对商象

广义上而言，A（A-Star)算法就是一种动态规划算法，只不过A算法在每步迭代计算时作出了更为严格的限制。关于动态规划，可以参考我的另一篇博客：《动态规划及其在Apollo项目Planning模块的应用.

A*算法基于代价函数f(n)=g(n)+h(n)计算最短路径，其中f(n)是结点n的总代价函数，g(n)是从初始结点到结点n的移动代价，h(n)是从结点n到目标结点的启发代价。

如果不考虑具体实现代码，A*算法是相当简单的。有两个集合，OPEN集和CLOSED集。其中OPEN集保存待考察的结点。开始时，OPEN集只包含一个元素：初始结点。CLOSED集保存已考查过的结点。开始时，CLOSED集是空的。如果绘成图，OPEN集就是被访问区域的边境（frontier），而CLOSED集则是被访问区域的内（interior）。每个结点同时保存其父结点的指针，以便反向溯源。

在主循环中重复地从OPEN集中取出最好的结点n（f值最小的结点）并检查之。如果n是目标结点，则我们的任务完成了。否则，从OPEN集中删除结点n并将其加入CLOSED集。然后检查它的邻居n’。如果邻居n’在CLOSED集中，表明该邻居已被检查过，不必再次考虑（若你确实需要检查结点n’的g值是否更小，可进行相关检查，若其g值更小，则将该结点从CLOSED集中删除）；如果n’在OPEN集中，那么该结点今后肯定会被考察，现在不必考虑它。否则，把它加入OPEN集，把它的父结点设为n。到达n’的路径的代价g(n’)，设定为g(n) + movementcost(n, n’)。

下面是算法伪代码：

OPEN = priority queue containing START

CLOSED = empty set

while lowest rank in OPEN is not the GOAL:

current = remove lowest rank item from OPEN

add current to CLOSED

for neighbors of current:

cost = g(current) + movementcost(current, neighbor)

if neighbor in OPEN and cost less than g(neighbor):

  remove neighbor from OPEN, because new path is better

if neighbor in CLOSED and cost less than g(neighbor): **

  remove neighbor from CLOSED

if neighbor not in OPEN and neighbor not in CLOSED:

  set g(neighbor) to cost

  add neighbor to OPEN

  set priority queue rank to g(neighbor) + h(neighbor)

  set neighbor\'s parent to current 

reconstruct reverse path from goal to start by following parent pointers

(**) This should never happen if you have an admissible heuristic.
However in games we often have inadmissible heuristics.