A* 算法基本概念
参考
A*算法是一种寻路算法，常常被用在游戏智能ai的自动寻路过程等等，它较之于图论的最短路算法而言，更加适用于节点巨大的情况下，但是该算法是一种启发式搜索算法，并不能保证总是找到最优路径。
参考这个人基本概念讲解很清楚 来源
算法基本流程 A星算法伪码：
a、将开始点记录为当前点P
b、将当前点P放入封闭列表
c、搜寻点P所有邻近点，假如某邻近点既没有在开放列表或封闭列表里面，则计算出该邻近点的F值，并设父节点为P，然后将其放入开放列表
d、判断开放列表是否已经空了，如果没有说明在达到结束点前已经找完了所有可能的路径点，寻路失败，算法结束；否则继续。
e、从开放列表拿出一个F值最小的点，作为寻路路径的下一步。
f、判断该点是否为结束点，如果是，则寻路成功，算法结束；否则继续。
g、将该点设为当前点P，跳回步骤c
非常像我们的BFS 搜索过程，可以说就是
matlab 验证程序 参考来源 此博主的代码应该比较正宗 非常适合没有matlab 基础 而又C++ 基础的人 基本函数思想类似BFS 只是算法过程进行了可视化
一共分三个.m 文件
分别为 TestAstarandGij.m运行此代码就可以
%
% TestScript for Assignment 1
%
%% Define a small map
map = false(10);
% Add an obstacle
map (1:5, 6) = true;
start_coords = [6, 2];
dest_coords = [8, 10];
%%
close all;
%[route, numExpanded] = DijkstraGrid (map, start_coords, dest_coords);
% Uncomment following line to run Astar
[route, numExpanded] = AStarGrid (map, start_coords, dest_coords);
%HINT: With default start and destination coordinates defined above, numExpanded for Dijkstras should be 76, numExpanded for Astar should be 23.
A * 算法AStarGrid.m
function [route,numExpanded] = AStarGrid (input_map, start_coords, dest_coords)
% Run A* algorithm on a grid.
% Inputs :
% input_map : a logical array where the freespace cells are false or 0 and
% the obstacles are true or 1
% start_coords and dest_coords : Coordinates of the start and end cell
% respectively, the first entry is the row and the second the column.
% Output :
% route : An array containing the linear indices of the cells along the
% shortest route from start to dest or an empty array if there is no
% route. This is a single dimensional vector
% numExpanded: Remember to also return the total number of nodes
% expanded during your search. Do not count the goal node as an expanded node.
% set up color map for display用一个map矩阵来表示每个点的状态
% 1 - white - clear cell
% 2 - black - obstacle
% 3 - red = visited 相当于CLOSED列表的作用
% 4 - blue - on list 相当于OPEN列表的作用
% 5 - green - start
% 6 - yellow - destination
cmap = [1 1 1; ...
0 0 0; ...
1 0 0; ...
0 0 1; ...
0 1 0; ...
1 1 0; ...
0.5 0.5 0.5];
colormap(cmap);
% variable to control if the map is being visualized on every
% iteration
drawMapEveryTime = true;
[nrows, ncols] = size(input_map);
% map - a table that keeps track of the state of each grid cell用来上色的
map = zeros(nrows,ncols);
map(~input_map) = 1; % Mark free cells %普通格设置为1
map(input_map) = 2; % Mark obstacle cells %障碍物设置为2
% Generate linear indices of start and dest nodes将下标转换为线性的索引值
start_node = sub2ind(size(map), start_coords(1), start_coords(2));
dest_node = sub2ind(size(map), dest_coords(1), dest_coords(2));
map(start_node) = 5;
map(dest_node) = 6;
% meshgrid will `replicate grid vectors' nrows and ncols to produce
% a full grid
% type `help meshgrid' in the Matlab command prompt for more information
parent = zeros(nrows,ncols);%用来记录每个节点的父节点
%
% 返回两个矩阵 每个点的 x y 坐标 方便求曼哈顿距离
[X, Y] = meshgrid (1:ncols, 1:nrows);
xd = dest_coords(1); % 目标点的 x值
yd = dest_coords(2); % 目标点的 y值
% Evaluate Heuristic function, H, for each grid cell
% Manhattan distance用曼哈顿距离作为启发式函数
%提前计算好 每个点距离目标点的曼哈顿距离
H = abs(X - xd) + abs(Y - yd);
H = H';
% Initialize cost arrays
f = Inf(nrows,ncols);
g = Inf(nrows,ncols);
g(start_node) = 0;
f(start_node) = H(start_node);
% keep track of the number of nodes that are expanded
numExpanded = 0;
% Main Loop
while true
% Draw current map
map(start_node) = 5;
map(dest_node) = 6;
% make drawMapEveryTime = true if you want to see how the
% nodes are expanded on the grid.
if (drawMapEveryTime)
image(1.5, 1.5, map);
grid on;
axis image;
drawnow;
end
% Find the node with the minimum f value,其中的current是index值，需要转换
[min_f, current] = min(f(:));
if ((current == dest_node) || isinf(min_f))
break;
end;
% Update input_map
map(current) = 3;
f(current) = Inf; % remove this node from further consideration
numExpanded=numExpanded+1;
% Compute row, column coordinates of current node
% 计算每个格子的索引
[i, j] = ind2sub(size(f), current);
% *********************************************************************
% ALL YOUR CODE BETWEEN THESE LINES OF STARS
% Visit all of the neighbors around the current node and update the
% entries in the map, f, g and parent arrays
%
action=[-1 0; 1 0; 0 -1; 0 1];%上，下，左，右
for a=1:4
expand=[i,j]+action(a,:);
expand1=expand(1,1);
expand2=expand(1,2);
%不超出边界，不穿越障碍，不在CLOSED列表里，也不是起点，则进行扩展
% ~= 为不等于的意思
if ( expand1>=1 && expand1<=10 && expand2>=1 && expand2<=10 && map(expand1,expand2)~=2 && map(expand1,expand2)~=3 && map(expand1,expand2)~=5)
if ( g(expand1,expand2)> g(i,j)+1 )
g(expand1,expand2)= g(i,j)+1;
f(expand1,expand2)= g(expand1,expand2)+H(expand1,expand2);
parent(expand1,expand2)=current;
map(expand1,expand2)=4;
end
end
end
%*********************************************************************
end
%% Construct route from start to dest by following the parent links
% 判断目标点是不是为无穷大 如果是则证明没有找到目标点
if (isinf(f(dest_node)))
route = [];
else
disp('parent 矩阵');
disp(parent);
route = [dest_node];
while (parent(route(1)) ~= 0)
disp(route(1));
route = [parent(route(1)), route];
% 打印索引到命窗口
disp(route);
end
% Snippet of code used to visualize the map and the path
for k = 2:length(route) - 1
map(route(k)) = 7;
pause(0.1);
image(1.5, 1.5, map);
grid on;
axis image;
end
end
end
附带 dijksta验证程序
function [route,numExpanded] = DijkstraGrid (input_map, start_coords, dest_coords)
% Run Dijkstra's algorithm on a grid.
% Inputs :
% input_map : a logical array where the freespace cells are false or 0 and
% the obstacles are true or 1
% start_coords and dest_coords : Coordinates of the start and end cell
% respectively, the first entry is the row and the second the column.
% Output :
% route : An array containing the linear indices of the cells along the
% shortest route from start to dest or an empty array if there is no
% route. This is a single dimensional vector
% numExpanded: Remember to also return the total number of nodes
% expanded during your search. Do not count the goal node as an expanded node.
% set up color map for display
% 1 - white - clear cell
% 2 - black - obstacle
% 3 - red = visited
% 4 - blue - on list
% 5 - green - start
% 6 - yellow - destination
cmap = [1 1 1; ...
0 0 0; ...
1 0 0; ...
0 0 1; ...
0 1 0; ...
1 1 0; ...
0.5 0.5 0.5];
colormap(cmap);
% variable to control if the map is being visualized on every
% iteration
drawMapEveryTime = true;
[nrows, ncols] = size(input_map);
% map - a table that keeps track of the state of each grid cell
map = zeros(nrows,ncols);
map(~input_map) = 1; % Mark free cells
map(input_map) = 2; % Mark obstacle cells
% Generate linear indices of start and dest nodes
start_node = sub2ind(size(map), start_coords(1), start_coords(2));
dest_node = sub2ind(size(map), dest_coords(1), dest_coords(2));
map(start_node) = 5;
map(dest_node) = 6;
% Initialize distance array
distanceFromStart = Inf(nrows,ncols);
% For each grid cell this array holds the index of its parent
parent = zeros(nrows,ncols);
distanceFromStart(start_node) = 0;
% keep track of number of nodes expanded
numExpanded = 0;
% Main Loop
while true
% Draw current map
map(start_node) = 5;
map(dest_node) = 6;
% make drawMapEveryTime = true if you want to see how the
% nodes are expanded on the grid.
if (drawMapEveryTime)
image(1.5, 1.5, map);
grid on;
axis image;
drawnow;
end
% Find the node with the minimum distance
[min_dist, current] = min(distanceFromStart(:));
if ((current == dest_node) || isinf(min_dist))
break;
end;
% Update map
map(current) = 3; % mark current node as visited
numExpanded=numExpanded+1;
% Compute row, column coordinates of current node
[i, j] = ind2sub(size(distanceFromStart), current);
% *********************************************************************
% YOUR CODE BETWEEN THESE LINES OF STARS
% Visit each neighbor of the current node and update the map, distances
% and parent tables appropriately.
action=[-1 0; 1 0; 0 -1; 0 1];%上，下，左，右
for a=1:4
expand=[i,j]+action(a,:);
expand1=expand(1,1);
expand2=expand(1,2);
%不超出边界，不穿越障碍，不在CLOSED列表里，则进行扩展
if ( expand1>=1 && expand1<=10 && expand2>=1 && expand2<=10 && map(expand1,expand2)~=2 && map(expand1,expand2)~=3 && map(expand1,expand2)~=5 )
if ( distanceFromStart(expand1,expand2)> distanceFromStart(i,j)+1 )
distanceFromStart(expand1,expand2)= distanceFromStart(i,j)+1;
parent(expand1,expand2)=current;
map(expand1,expand2)=4;
end
end
end
distanceFromStart(current) = Inf; % remove this node from further consideration
%*********************************************************************
end
%% Construct route from start to dest by following the parent links
if (isinf(distanceFromStart(dest_node)))
route = [];
else
route = [dest_node];
while (parent(route(1)) ~= 0)
route = [parent(route(1)), route];
end
% Snippet of code used to visualize the map and the path
for k = 2:length(route) - 1
map(route(k)) = 7;
pause(0.1);
image(1.5, 1.5, map);
grid on;
axis image;
end
end
end
C++ 版本
代码参考 https://blog.youkuaiyun.com/u012234115/article/details/47152137
Astar.cpp
#include <math.h>
#include "Astar.h"
#include <iostream>
void Astar::InitAstar(std::vector<std::vector<int>>& _maze)
{
maze = _maze;
}
int Astar::calcG(Point *temp_start,Point *point)
{
int extraG = (abs(point->x-temp_start->x)+abs(point->y- point->y))==1?kCosntStraight:kConstSkew;
int parentG = point->parent == NULL ? 0 : point->parent->G;
return parentG + extraG;
}
int Astar::calcH(Point *point,Point *end)
{
// 利用欧式距离计算这个H
return sqrt((double)(end->x - point->x) * (double)(end->x - point->x) + (double)(end->y - point->y) * (double)(end->y - point->y));
}
int Astar::calcF(Point* point)
{
return point->G + point->H;
}
Point* Astar::getLeastFpoint()
{
if (!openList.empty())
{
auto resPoint = openList.front();
for (auto& point : openList)
{
// 找f值最小的结点
if (point->F < resPoint->F)
resPoint = point;
}
return resPoint;
}
return NULL;
}
Point* Astar::findPath(Point& startPoint, Point& endPoint, bool isIgnoreCorner)
{
openList.push_back(new Point(startPoint.x, startPoint.y));
while (!openList.empty())
{
// 找到值最小的点
auto curPoint = getLeastFpoint();
openList.remove(curPoint);
// 加入closelist
closeList.push_back(curPoint);
//1,找到当前周围八个格中可以通过的格子
auto surroundPoints = getSurroundPoints(curPoint, isIgnoreCorner);
for (auto& target : surroundPoints)
{
//2,对某一个格子，如果它不在开启列表中，加入到开启列表，设置当前格为其父节点，计算F G H
if (!isInList(openList, target))
{
target->parent = curPoint;
target->G = calcG(curPoint, target);
target->H = calcH(target, &endPoint);
target->F = calcF(target);
openList.push_back(target);
}
//3，对某一个格子，它在开启列表中，计算G值, 如果比原来的大, 就什么都不做, 否则设置它的父节点为当前点,并更新G和F
else
{
int tempG = calcG(curPoint, target);
if (tempG < target->G)
{
target->parent = curPoint;
target->G = tempG;
target->F = calcF(target);
}
}
Point* resPoint = isInList(openList, &endPoint);
{
if (resPoint)
return resPoint; //返回列表里的节点指针，不要用原来传入的endpoint指针，因为发生了深拷贝
}
}
}
return NULL;
}
std::vector<Point*> Astar::getSurroundPoints(const Point* point, bool isIgnoreCorner) const
{
std::vector<Point*> surroundPoints;
for (int x = point->x - 1; x <= point->x + 1; x++)
{
for (int y = point->y - 1; y <= point->y + 1; y++)
{
if (isCanreach(point, new Point(x, y), isIgnoreCorner))
{
surroundPoints.push_back(new Point(x,y));
}
}
}
return surroundPoints;
}
//判断某点是否可以用于下一步判断
bool Astar::isCanreach(const Point* point, const Point* target, bool isIgnoreCorner) const
{
if (target->x<0 || target->x>maze.size() - 1
|| target->y<0 || target->y>maze[0].size() - 1
|| maze[target->x][target->y] == 1 // 1 代表是障碍物
|| target->x == point->x && target->y == point->y
|| isInList(closeList, target)
)
return false;
else {
if (abs(point->y - target->y) + abs(point->x-point->x) == 1)
return true;
else
{
//斜对角线
if (maze[point->x][point->y] == 0 && maze[target->x][target->y] == 0)
return true;
else
return isIgnoreCorner;
}
}
}
//判断开启 / 关闭列表中是否包含某点
Point* Astar::isInList(const std::list<Point*>& list, const Point* point) const
{
for (auto p : list) {
if (p->x == point->x && p->y == point->y)
return p;
}
return NULL;
}
std::list<Point*> Astar::GetPath(Point& startPoint, Point& endPoint, bool isIgnoreCorner)
{
Point* result = findPath(startPoint, endPoint, isIgnoreCorner);
std::list<Point*> path;
while (result)
{
path.push_back(result);
result = result->parent;
}
openList.clear();
closeList.clear();
return path;
}
Astar.h
#pragma once
#include<vector>
#include<list>
// 直接移动一格消耗
const int kCosntStraight = 10;
// 斜移移动消耗
const int kConstSkew = 14;
struct Point
{
// 点坐标
int x, y;
int F, G, H; //F=g+h
Point* parent;
Point(int _x, int _y)
// 变量初始化
:x(_x), y(_y), F(0), G(0), H(0),parent(NULL)
{
}
};
class Astar
{
public:
void InitAstar(std::vector<std::vector<int>>& _maze);
std::list<Point*> GetPath(Point& startPoint, Point& endPoint, bool isIgnoreCorner);
private:
Point* findPath(Point& startPoint, Point& endPoint, bool isIgnoreCorner);
std::vector<Point*> getSurroundPoints(const Point* point, bool isIgnoreCorner) const;
bool isCanreach(const Point* point, const Point* target, bool isIgnoreCorner)const; //判断某点是否可以用于下一步判断
Point* isInList(const std::list<Point*>& list, const Point* point) const ; //判断开启/关闭列表中是否包含某点
Point* getLeastFpoint(); //从开启列表中返回F值最小的节点
//计算FGH值
int calcG(Point* temp_start, Point* point);
int calcH(Point* point, Point* end);
int calcF(Point* point);
private:
std::vector<std::vector<int>> maze;
std::list<Point*> openList; //开启列表
std::list<Point*> closeList; //关闭列表
};
testmain.c
#include <iostream>
#include "Astar.h"
/******************************************
*日期：2020/06/2
*来源https://blog.youkuaiyun.com/u012234115/article/details/47152137
*作者：
*
******************************************/
using namespace std;
int main()
{
//初始化地图，用二维矩阵代表地图，1表示障碍物，0表示可通
vector<vector<int>> maze = {
{1,1,1,1,1,1,1,1,1,1,1,1},
{1,0,0,1,1,0,1,0,0,0,0,1},
{1,0,0,1,1,0,0,0,0,0,0,1},
{1,0,0,0,0,0,1,0,0,1,1,1},
{1,1,1,0,0,0,0,0,1,1,0,1},
{1,1,0,1,0,0,0,0,0,0,0,1},
{1,0,1,0,0,0,0,1,0,0,0,1},
{1,1,1,1,1,1,1,1,1,1,1,1}
};
Astar astar;
astar.InitAstar(maze);
//设置起点和终点
Point start(1, 1);
Point end(6, 10);
// A*算法寻路
list<Point*> path = astar.GetPath(start,end,false);
// 反转起点到终点打印
path.reverse();
// 打印
for (auto& p : path)
{
cout << "(" << p->x << "," << p->y << ")" << endl;
}
return 0;
}
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