以罗马尼亚问题为例，学习人工智能DFS/A*搜索算法

以罗马尼亚问题为例，学习人工智能的搜索算法

Romania problem

给出各个城市之间的距离及代价，包括A*算法需要的直线距离，求解从A到B点的最短路径。这里分别使用两种搜索算法求解–DFS和Astar。因为本问题的数据量较小，均使用邻接矩阵来表示图，程序均用C++实现。

无信息搜索算法：深度优先搜索

算法思路

使用图的深度优先算法求解，所以需要找出每一条路径，然后根据其总距离求出最短路径，图的深度优先搜索主要思路是，在图中从起点开始往任意有弧的节点扩展，直到没有出度（或在本题中找到目标节点）后保存为一条路径，把图中所有的点全部遍历一遍为止。

程序源码

#include 
#include 
#include 
#define VERTEX 13
using namespace std;//A must?

struct AdjacencyMatrix {
	string vertex[VERTEX];
	int edges[VERTEX][VERTEX];
	int isTraversed[VERTEX] = { 0 };
	vector<int> path;
	vector<pair<vector<int>, int>> pathcost;//?!
};
class Solution {
public:
	void createGraph(AdjacencyMatrix &graph) {
		//							  00  01  02  03  04  05  06  07  08  09  10  11  12
		int map[VERTEX][VERTEX] = { {  0, 75, -1,118, -1, -1, -1,140, -1, -1, -1, -1, -1 },//0
									{ -1,  0, 71, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },//1
									{ -1, -1,  0, -1, -1, -1, -1,151, -1, -1, -1, -1, -1 },//2
									{ -1, -1, -1,  0,111, -1, -1, -1, -1, -1, -1, -1, -1 },//3
									{ -1, -1, -1, -1,  0, 70, -1, -1, -1, -1, -1, -1, -1 },//4
									{ -1, -1, -1, -1, -1,  0, 75, -1, -1, -1, -1, -1, -1 },//5
									{ -1, -1, -1, -1, -1, -1,  0, -1, -1,120, -1, -1, -1 },//6
									{ -1, -1, -1, -1, -1, -1, -1,  0, 80, -1, 99, -1, -1 },//7
									{ -1, -1, -1, -1, -1, -1, -1, -1,  0,146, -1, 97, -1 },//8
									{ -1, -1, -1, -1, -1, -1, -1, -1, -1,  0, -1,138, -1 },//9
									{ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,  0, -1,211 },//10
									{ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,  0,101 },//11
									{ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,  0 } //12
		};//13 points
		for (int i = 0; i < VERTEX; i++)
			for (int j = 0; j < VERTEX; j++)
				if (map[i][j] != -1)
					map[j][i] = map[i][j];
		memcpy(graph.edges, map, sizeof(map));
		graph.vertex[0] = "Arad";
		graph.vertex[1] = "Zerind";
		graph.vertex[2] = "Oradea";
		graph.vertex[3] = "Timisoara";
		graph.vertex[4] = "Lugoj";
		graph.vertex[5] = "Mehadia";
		graph.vertex[6] = "Drobeta";
		graph.vertex[7] = "Sibiu";
		graph.vertex[8] = "Rimnicu-Vilcea";
		graph.vertex[9] = "Craiova";
		graph.vertex[10] = "Fagaras";
		graph.vertex[11] = "Pitesti";
		graph.vertex[12] = "Bucharest";
	}
	void outputGraph(AdjacencyMatrix &graph) {
		for (int i = 0; i < VERTEX; i++)
			for (int j = 0; j < VERTEX; j++)
				if (graph.edges[i][j] > 0)
					cout << graph.vertex[i] << " -> " << graph.vertex[j] << " : " << graph.edges[i][j] << endl;
	}
	int nextOutdegree(AdjacencyMatrix graph, int interrupt, int loc) {
		for (int index = interrupt + 1; index < VERTEX; index++) {
			if (graph.edges[loc][index] > 0 && graph.isTraversed[index] == 0)
				//cout << index << endl;
				return index;
		}
		return -1;
	}
	int computePathCost(AdjacencyMatrix graph) {
		int cost = 0, index = 0;
		for (; index < graph.path.size() - 1; index++)
			cost += graph.edges[graph.path[index]][graph.path[index + 1]];
		cost += graph.edges[graph.path[index]][12];
		return cost;
	}
	int minimumPathCost(vector<pair<vector<int>, int>> pathcost) {
		int minIndex = 0, index = 1;
		for (; index < pathcost.size(); index++)
			if (pathcost[minIndex].second >= pathcost[index].second)
				minIndex = index;
		return minIndex;
	}
	void depthFirstSearch(AdjacencyMatrix &graph, int start, int end) {
		int loc = 0, cost = 0;
		graph.isTraversed[start] = 1;
		graph.path.push_back(start);
		while ((loc = nextOutdegree(graph, loc, start)) != -1) {
			if (loc == end) {
				cout << "Path: ";
				for (int cur = 0; cur < graph.path.size(); cur++) {
					cout << graph.vertex[graph.path[cur]] << " ";
					cost = computePathCost(graph);
					graph.pathcost.push_back(pair<vector<int>, int>(graph.path, cost));
				}
				cout << graph.vertex[end] << ", Cost: " << cost << endl;//Here is an error.
				break;
			}
			depthFirstSearch(graph, loc, end);
			graph.path.pop_back();
			graph.isTraversed[loc] = 0;//Go back.
			//cout << "back path: " << loc << endl;
		}
	}
	/*void breadthFirstSearch(AdjacencyMatrix &graph) {
		//really?
	}*/
};
int main(int argc, char *argv[]) {
	int start = 0, end = 12, index = 0;
	AdjacencyMatrix graph;
	Solution sol;
	sol.createGraph(graph);
	sol.outputGraph(graph);
	sol.depthFirstSearch(graph, start, end);
	
	index = sol.minimumPathCost(graph.pathcost);
	cout << endl << "So, mincost path: ";
	for (int cur = 0; cur < graph.pathcost[index].first.size(); cur++)
		cout << graph.vertex[graph.pathcost[index].first[cur]] << " -> ";
	cout << graph.vertex[end] << endl;//Here is an error.

	vector<int>().swap(graph.path);//Clear and release memory.
	system("pause");
	return 0;
}

最后输出的So, mincost path: Arad -> Sibiu -> Rimnicu-Vilcea -> Pitesti -> Bucharest即为结果。

代码分析

流程图描述

感觉图和代码都还算清晰，就不用自然语言描述了。

有信息搜索算法：A*搜索

算法思路

部分A*算法的思路来自https://www.gamedev.net/articles/programming/artificial-intelligence/a-pathfinding-for-beginners-r2003/，要维护两个表，分别是open list和close list，由公式求出f最小的点存入close list，根据这个节点扩展出所有邻接节点存入open list，并存入计算出的f值，具体过程如下：

1.把起点加入 open list 。
2.重复如下过程：
	a.遍历 open list ，查找 F 值最小的节点，把它作为当前要处理的节点。
	b.把这个节点移到 close list 。
	c.对当前方格的 8 个相邻方格的每一个方格？
		Ⅰ.如果它是不可抵达的或者它在 close list 中，忽略它。否则，做如下操作。
		Ⅱ.如果它不在 open list 中，把它加入 open list ，并且把当前方格设置为它的父亲，记录该方格的 F ， G 和 H 值。
		Ⅲ.如果它已经在 open list 中，检查这条路径 ( 即经由当前方格到达它那里 ) 是否更好，用 G 值作参考。更小的 G 值表示这是更好的路径。如果是这样，把它的父亲设置为当前方格，并重新计算它的 G 和 F 值。如果 open list 是按 F 值排序的话，改变后可能需要重新排序。
	d.停止，当
		Ⅰ.把终点加入到了 open list 中，此时路径已经找到了，或者
		Ⅱ.查找终点失败，并且 open list 是空的，此时没有路径。
3.保存路径。从终点开始，每个方格沿着父节点移动直至起点，这就是所求路径。

程序源码

#include 
#include 
#include 
#define VERTEX 13
using namespace std;//A must?

//https://www.gamedev.net/articles/programming/artificial-intelligence/a-pathfinding-for-beginners-r2003/
struct AstarNode {
	int id;
	AstarNode *parent;
	AstarNode() :
		id(-1), parent(NULL) {}
	AstarNode(int id) :
		id(id), parent(NULL) {}
	AstarNode(int id, AstarNode *parent) :
		id(id), parent(parent) {}
};
struct AdjacencyMatrix {
	string vertex[VERTEX];
	int edges[VERTEX][VERTEX];
	int direct[VERTEX];

	AstarNode openlist[VERTEX];
	AstarNode closelist[VERTEX];
	int fvalue[VERTEX];

	int isTraversed[VERTEX] = { 0 };
	vector<int> path;
};
class Solution {
public:
	void createGraph(AdjacencyMatrix &graph) {
		//							  00  01  02  03  04  05  06  07  08  09  10  11  12
		int map[VERTEX][VERTEX] = { {  0, 75, -1,118, -1, -1, -1,140, -1, -1, -1, -1, -1 },//0
									{ -1,  0, 71, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },//1
									{ -1, -1,  0, -1, -1, -1, -1,151, -1, -1, -1, -1, -1 },//2
									{ -1, -1, -1,  0,111, -1, -1, -1, -1, -1, -1, -1, -1 },//3
									{ -1, -1, -1, -1,  0, 70, -1, -1, -1, -1, -1, -1, -1 },//4
									{ -1, -1, -1, -1, -1,  0, 75, -1, -1, -1, -1, -1, -1 },//5
									{ -1, -1, -1, -1, -1, -1,  0, -1, -1,120, -1, -1, -1 },//6
									{ -1, -1, -1, -1, -1, -1, -1,  0, 80, -1, 99, -1, -1 },//7
									{ -1, -1, -1, -1, -1, -1, -1, -1,  0,146, -1, 97, -1 },//8
									{ -1, -1, -1, -1, -1, -1, -1, -1, -1,  0, -1,138, -1 },//9
									{ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,  0, -1,211 },//10
									{ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,  0,101 },//11
									{ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1,  0 } //12
		};//13 points
		for (int i = 0; i < VERTEX; i++)
			for (int j = 0; j < VERTEX; j++)
				if (map[i][j] != -1)
					map[j][i] = map[i][j];
		memcpy(graph.edges, map, sizeof(map));
		graph.vertex[0] = "Arad"; graph.direct[0] = 366;
		graph.vertex[1] = "Zerind"; graph.direct[1] = 374;
		graph.vertex[2] = "Oradea"; graph.direct[2] = 380;
		graph.vertex[3] = "Timisoara"; graph.direct[3] = 329;
		graph.vertex[4] = "Lugoj"; graph.direct[4] = 244;
		graph.vertex[5] = "Mehadia"; graph.direct[5] = 241;
		graph.vertex[6] = "Drobeta"; graph.direct[6] = 242;
		graph.vertex[7] = "Sibiu"; graph.direct[7] = 253;
		graph.vertex[8] = "Rimnicu-Vilcea"; graph.direct[8] = 193;
		graph.vertex[9] = "Craiova"; graph.direct[9] = 160;
		graph.vertex[10] = "Fagaras"; graph.direct[10] = 176;
		graph.vertex[11] = "Pitesti"; graph.direct[11] = 100;
		graph.vertex[12] = "Bucharest"; graph.direct[12] = 0;
		/*for (int index = 0; index < VERTEX; index++) {
			cout << &graph.openlist[index] << endl;
		}*/
	}
	void outputGraph(AdjacencyMatrix &graph) {
		for (int i = 0; i < VERTEX; i++)
			for (int j = 0; j < VERTEX; j++)
				if (graph.edges[i][j] > 0)
					cout << graph.vertex[i] << " -> " << graph.vertex[j] << " : " << graph.edges[i][j] << endl;
	}
	inline int computeF(AdjacencyMatrix graph, AstarNode node) {
		if (node.parent == NULL)
			return graph.direct[node.id];
		return graph.edges[node.parent->id][node.id] + graph.direct[node.id];
	}
	AstarNode minimumF(AdjacencyMatrix graph) {
		int Fmin = INT_MAX;
		int Findex = 0;
		int F;//F = G + H, G is cost, H is estimated value.
		for (int index = 1; index < VERTEX; index++)
			if (graph.openlist[index].id != -1 && graph.closelist[index].id == -1
					&& (F = graph.fvalue[index]) < Fmin) {
				Fmin = F;
				Findex = index;
			}
		//cout << Findex << "  ";
		return graph.openlist[Findex];
	}
	int nextOutdegree(AdjacencyMatrix graph, int interrupt, int loc) {
		for (int index = interrupt + 1; index < VERTEX; index++) {
			if (graph.edges[loc][index] > 0 && graph.isTraversed[index] == 0)
				//cout << index << endl;
				return index;
		}
		return -1;
	}
	bool extendOpenList(AdjacencyMatrix &graph, int id) {
		graph.closelist[id] = graph.openlist[id];
		int current = 0;
		if (id == VERTEX - 1)
			return false;
		while ((current = nextOutdegree(graph, current, id)) != -1) {
			//AstarNode *extended = new AstarNode();//Why???
			graph.openlist[current].id = current;
			graph.openlist[current].parent = &graph.closelist[id];
			graph.isTraversed[current] = 1;
			graph.fvalue[current] = computeF(graph, graph.openlist[current]);
			//cout << current << " ";
		}
		return true;
	}
	void AstarSearch(AdjacencyMatrix &graph, int start, int end) {
		AstarNode root = { start, NULL };
		//cout << &root << endl;
		graph.openlist[start] = root;
		graph.isTraversed[start] = 1;
		graph.fvalue[start] = computeF(graph, root);
		while (extendOpenList(graph, root.id)) {
			cout << root.id << "   ";
			root = minimumF(graph);
		}
		root = graph.closelist[end];
		cout << root.id << endl;
		AstarNode *p = &root;
		while (p != NULL) {
			graph.path.push_back(p->id);//逆向
			p = p->parent;
			//cout << p->id << endl;
		}
	}
};
int main(int argc, char *argv[]) {
	int start = 0, end = 12;
	AdjacencyMatrix graph;
	Solution sol;
	sol.createGraph(graph);
	sol.outputGraph(graph);
	sol.AstarSearch(graph, start, end);

	cout << endl << "So, path: ";
	for (int cur = graph.path.size() - 1; cur > 0; cur--)
		cout << graph.vertex[graph.path[cur]] << " -> ";
	cout << graph.vertex[graph.path[0]] << endl;

	vector<int>().swap(graph.path);
	system("pause");
	return 0;
}

最后输出的path: Arad -> Sibiu -> Rimnicu-Vilcea -> Pitesti -> Bucharest即为结果。

代码分析

流程图描述

小结

固定疯玩三天per month，Live and drink， friend。
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