数据结构——图(遍历,最小生成树,最短路径)

一.图的基本概念

二.图的存储结构

1.邻接矩阵

2.邻接表

三.图的遍历

1.图的广度优先遍历

2.图的深度优先遍历

四.最小生成树

1.Kruskal算法

2.Prim算法

五.最短路径

1.单源最短路径--Dijkstra算法

2.单源最短路径--Bellman-Ford算法

3.多源最短路径--Floyd-Warshall算法

六.整体实现

1.UnionFindSet.h

2.Graph.h

3.test.cpp

一.图的基本概念

图是由顶点集合及顶点间的关系组成的一种数据结构：G = (V， E)，其中：

顶点集合V = {x|x属于某个数据对象集}是有穷非空集合；

E = {(x,y)|x,y属于V}或者E = {|x,y属于V && Path(x, y)}是顶点间关系的有穷集合，也叫做边的集合

(x, y)表示x到y的一条双向通路，即(x, y)是无方向的；Path(x, y)表示从x到y的一条单向通路，即Path(x, y)是有方向的

顶点和边：图中结点称为顶点，第i个顶点记作vi。两个顶点vi和vj相关联称作顶点vi和顶点vj之间有一条边，图中的第k条边记作ek，ek = (vi，vj)或<vi，vj>

有向图和无向图：在有向图中，顶点对是有序的，顶点对<x，y>称为顶点x到顶点y的一条边(弧)，和是两条不同的边，比如下图G3和G4为有向图。在无向图中，顶点对(x, y) 是无序的，顶点对(x,y)称为顶点x和顶点y相关联的一条边，这条边没有特定方向，(x, y)和(y，x) 是同一条边，比如下图G1和G2为无向图。注意：无向边(x, y)等于有向边和

完全图：在有n个顶点的无向图中，若有n * (n-1)/2条边，即任意两个顶点之间有且仅有一条边， 则称此图为无向完全图，比如上图G1；在n个顶点的有向图中，若有n * (n-1)条边，即任意两个顶点之间有且仅有方向相反的边，则称此图为有向完全图，比如上图G4

邻接顶点：在无向图中G中，若(u, v)是E(G)中的一条边，则称u和v互为邻接顶点，并称边(u,v)依附于顶点u和v；在有向图G中，若是E(G)中的一条边，则称顶点u邻接到v，顶点v邻接自顶点u，并称边与顶点u和顶点v相关联

顶点的度：顶点v的度是指与它相关联的边的条数，记作deg(v)。在有向图中，顶点的度等于该顶点的入度与出度之和，其中顶点v的入度是以v为终点的有向边的条数，记作indev(v);顶点v的出度是以v为起始点的有向边的条数，记作outdev(v)。因此：dev(v) = indev(v) + outdev(v)。注意：对于无向图，顶点的度等于该顶点的入度和出度，即dev(v) = indev(v) = outdev(v)

路径：在图G = (V， E)中，若从顶点vi出发有一组边使其可到达顶点vj，则称顶点vi到顶点vj的顶点序列为从顶点vi到顶点vj的路径

路径长度：对于不带权的图，一条路径的路径长度是指该路径上的边的条数；对于带权的图，一条路径的路径长度是指该路径上各个边权值的总和

简单路径与回路：若路径上各顶点v1，v2，v3，…，vm均不重复，则称这样的路径为简单路径。若路径上第一个顶点v1和最后一个顶点vm重合，则称这样的路径为回路或环

子图：设图G = {V, E}和图G1 = {V1，E1}，若V1属于V且E1属于E，则称G1是G的子图

连通图：在无向图中，若从顶点v1到顶点v2有路径，则称顶点v1与顶点v2是连通的。如果图中任意一对顶点都是连通的，则称此图为连通图

强连通图：在有向图中，若在每一对顶点vi和vj之间都存在一条从vi到vj的路径，也存在一条从vj到vi的路径，则称此图是强连通图

生成树：一个连通图的最小连通子图称作该图的生成树。有n个顶点的连通图的生成树有n个顶点和n1条边

二.图的存储结构

1.邻接矩阵

因为节点与节点之间的关系就是连通与否，即为0或者1，因此邻接矩阵(二维数组)即是：先用一个数组将定点保存，然后采用矩阵来表示节点与节点之间的关系

注意:

无向图的邻接矩阵是对称的，第i行(列)元素之和，就是顶点i的度。有向图的邻接矩阵则不一定是对称的，第i行(列)元素之后就是顶点i的出(入)度
如果边带有权值，并且两个节点之间是连通的，上图中的边的关系就用权值代替，如果两个顶点不通，则使用无穷大代替
用邻接矩阵存储图的有点是能够快速知道两个顶点是否连通，缺陷是如果顶点比较多，边比较少时，矩阵中存储了大量的0成为系数矩阵，比较浪费空间，并且要求两个节点之间的路径不是很好求

namespace matrix
{template<class V, class W, W MAX_W = INT_MAX, bool Direction = false>class Graph{typedef Graph<V, W, MAX_W, Direction> Self;public://图的创建//1.IO输入->不方便测试,oj中更适合//2.图结构关系写到文件,读取文件//3.手动添加边Graph() = default;Graph(const V* a, size_t n){_vertexs.reserve(n);for (size_t i = 0; i < n; ++i){_vertexs.push_back(a[i]);_indexMap[a[i]] = i;}_matrix.resize(n);for (size_t i = 0; i < _matrix.size(); ++i){_matrix[i].resize(n, MAX_W);}}size_t GetVertexIndex(const V& v){auto it = _indexMap.find(v);if (it != _indexMap.end()){return it->second;}else{//assert(false);throw invalid_argument("顶点不存在");return -1;}}void _AddEdge(size_t srci, size_t dsti, const W& w){_matrix[srci][dsti] = w;//无向图if (Direction == false){_matrix[dsti][srci] = w;}}void AddEdge(const V& src, const V& dst, const W& w){size_t srci = GetVertexIndex(src);size_t dsti = GetVertexIndex(dst);_AddEdge(srci, dsti, w);}void Print(){//顶点for (size_t i = 0; i < _vertexs.size(); ++i){cout << "[" << i << "]" << "->" << _vertexs[i] << endl;}cout << endl;//矩阵//横下标cout << "  ";for (size_t i = 0; i < _matrix.size(); ++i){//cout << i << " ";printf("%4d", i);}cout << endl;for (size_t i = 0; i < _matrix.size(); ++i){cout << i << " ";//竖下标for (size_t j = 0; j < _matrix[i].size(); ++j){//cout << _matrix[i][j] << " ";if (_matrix[i][j] == MAX_W){//cout << "* ";printf("%4c", '*');}else{//cout << _matrix[i][j] << " ";printf("%4d", _matrix[i][j]);}}cout << endl;}cout << endl;}private:vector<V> _vertexs;            //顶点集合map<V, int> _indexMap;		   //顶点映射下标vector<vector<W>> _matrix;	   //邻接矩阵};
}

void TestGraph1()
{Graph<char, int> g("0123", 4);//Graph<char, int, true> g("0123", 4);g.AddEdge('0', '1', 1);g.AddEdge('0', '3', 4);g.AddEdge('1', '3', 2);g.AddEdge('1', '2', 9);g.AddEdge('2', '3', 8);g.AddEdge('2', '1', 5);g.AddEdge('2', '0', 3);g.AddEdge('3', '2', 6);g.Print();
}

邻接矩阵总结:

邻接矩阵存储方式非常适合稠密图
邻接矩阵O(1)判断两个顶点的连接关系并取到权值
相对而言不适合查找一个顶点连接所有边----O(N)

2.邻接表

邻接表：使用数组表示顶点的集合，使用链表表示边的关系

1.无向图邻接表存储

注意:无向图中同一条边在邻接表中出现了两次。如果想知道顶点vi的度，只需要知道顶点vi边链表集合中结点的数目即可

2.有向图邻接表存储

注意:有向图中每条边在邻接表中只出现一次，与顶点vi对应的邻接表所含结点的个数，就是该顶点的出度，也称出度表，要得到vi顶点的入度，必须检测其他所有顶点对应的边链表，看有多少边顶点的dst取值是i

namespace link_table
{template<class W>struct Edge{//int _srci;int _dsti;  //目标点的下标W _w;		//权值Edge<W>* _next;Edge(int dsti, const W& w) :_dsti(dsti), _w(w), _next(nullptr){ }};template<class V, class W, bool Direction = false>class Graph{typedef Edge<W> Edge;public:Graph(const V* a, size_t n){_vertexs.reserve(n);for (size_t i = 0; i < n; ++i){_vertexs.push_back(a[i]);_indexMap[a[i]] = i;}_tables.resize(n, nullptr);}size_t GetVertexIndex(const V& v){auto it = _indexMap.find(v);if (it != _indexMap.end()){return it->second;}else{//assert(false);throw invalid_argument("顶点不存在");return -1;}}void AddEdge(const V& src, const V& dst, const W& w){size_t srci = GetVertexIndex(src);size_t dsti = GetVertexIndex(dst);Edge* eg = new Edge(dsti, w);eg->_next = _tables[srci];_tables[srci] = eg;if (Direction == false){Edge* eg = new Edge(srci, w);eg->_next = _tables[dsti];_tables[dsti] = eg;}}void Print(){//顶点for (size_t i = 0; i < _vertexs.size(); ++i){cout << "[" << i << "]" << "->" << _vertexs[i] << endl;}cout << endl;for (size_t i = 0; i < _tables.size(); ++i){cout << _vertexs[i] << "[" << i << "]->";Edge* cur = _tables[i];while (cur){cout << "[" << _vertexs[cur->_dsti] << ":" << cur->_dsti << ":" << cur->_w << "]->";cur = cur->_next;}cout << "nullptr" << endl;}}private:vector<V> _vertexs;            //顶点集合map<V, int> _indexMap;		   //顶点映射下标vector<Edge*> _tables;		   //邻接表};
}

void TestGraph2()
{string a[] = { "张三", "李四", "王五", "赵六" };//Graph<string, int, true> g1(a, 4);Graph<string, int> g1(a, 4);g1.AddEdge("张三", "李四", 100);g1.AddEdge("张三", "王五", 200);g1.AddEdge("王五", "赵六", 30);g1.Print();
}

邻接表总结:

适合存储稀疏图
适合查找一个顶点连接出去的边
不适合确定两个顶点是否相连及权值

三.图的遍历

1.图的广度优先遍历

void BFS(const V& src)
{size_t srci = GetVertexIndex(src);//队列和标记数组queue<int> q;vector<bool> visited(_vertexs.size(), false);q.push(srci);visited[srci] = true;int levelSize = 1;size_t n = _vertexs.size();while (!q.empty()){//一层一层出for (int i = 0; i < levelSize; ++i){int front = q.front();q.pop();cout << front << ":" << _vertexs[front] << " ";//把front顶点的邻接顶点入队列for (size_t i = 0; i < n; ++i){if (_matrix[front][i] != MAX_W){if (visited[i] == false){q.push(i);visited[i] = true;}}}}cout << endl;levelSize = q.size();}cout << endl;
}

void TestBDFS()
{string a[] = { "张三", "李四", "王五", "赵六", "周七" };Graph<string, int> g1(a, sizeof(a) / sizeof(string));g1.AddEdge("张三", "李四", 100);g1.AddEdge("张三", "王五", 200);g1.AddEdge("王五", "赵六", 30);g1.AddEdge("王五", "周七", 30);g1.Print();g1.BFS("张三");
}

2.图的深度优先遍历

void _DFS(size_t srci, vector<bool>& visited)
{cout << srci << ":" << _vertexs[srci] << endl;visited[srci] = true;//找一个和srci相邻的没有访问过的点,去深度遍历for (size_t i = 0; i < _vertexs.size(); ++i){if (_matrix[srci][i] != MAX_W && visited[i] == false){_DFS(i, visited);}}
}void DFS(const V& src)
{size_t srci = GetVertexIndex(src);vector<bool> visited(_vertexs.size(), false);_DFS(srci, visited);
}

void TestBDFS()
{string a[] = { "张三", "李四", "王五", "赵六", "周七" };Graph<string, int> g1(a, sizeof(a) / sizeof(string));g1.AddEdge("张三", "李四", 100);g1.AddEdge("张三", "王五", 200);g1.AddEdge("王五", "赵六", 30);g1.AddEdge("王五", "周七", 30);g1.Print();g1.DFS("张三");
}

四.最小生成树

连通图中的每一棵生成树，都是原图的一个极大无环子图，即：从其中删去任何一条边，生成树就不在连通；反之，在其中引入任何一条新边，都会形成一条回路

若连通图由n个顶点组成，则其生成树必含n个顶点和n-1条边。因此构造最小生成树的准则有三条:

只能使用图中的边来构造最小生成树
只能使用恰好n-1条边来连接图中的n个顶点
选用的n-1条边不能构成回路

1.Kruskal算法

Kruskal算法(克鲁斯卡尔算法)任给一个有n个顶点的连通网络N={V,E}，首先构造一个由这n个顶点组成、不含任何边的图G={V,NULL}，其中每个顶点自成一个连通分量， 其次不断从E中取出权值最小的一条边(若有多条任取其一)，若该边的两个顶点来自不同的连通分量，则将此边加入到G中。如此重复，直到所有顶点在同一个连通分量上为止

核心：每次迭代时，选出一条具有最小权值，且两端点不在同一连通分量上的边，加入生成树

struct Edge
{size_t _srci;size_t _dsti;W _w;Edge(size_t srci, size_t dsti, const W& w) :_srci(srci), _dsti(dsti), _w(w){ }bool operator>(const Edge& e) const{return _w > e._w;}
};W Kruskal(Self& minTree)
{size_t n = _vertexs.size();minTree._vertexs = _vertexs;minTree._indexMap = _indexMap;minTree._matrix.resize(n);for (size_t i = 0; i < n; ++i){minTree._matrix[i].resize(n, MAX_W);}priority_queue<Edge, vector<Edge>, greater<Edge>> minque;for (size_t i = 0; i < n; ++i){for (size_t j = 0; j < n; ++j){if (i < j && _matrix[i][j] != MAX_W){minque.push(Edge(i, j, _matrix[i][j]));}}}cout << "Kruskal开始选边:" << endl;//选出n-1条边int size = 0;W totalW = W();UnionFindSet ufs(n);while (!minque.empty()){Edge min = minque.top();minque.pop();if (!ufs.InSet(min._srci, min._dsti)){cout << _vertexs[min._srci] << "->" << _vertexs[min._dsti] << ":" << min._w << endl;minTree._AddEdge(min._srci, min._dsti, min._w);ufs.Union(min._srci, min._dsti);++size;totalW += min._w;}else{cout << "构成环";cout << _vertexs[min._srci] << "->" << _vertexs[min._dsti] << ":" << min._w << endl;}}cout << endl;if (size == n - 1){return totalW;}else{return W();}
}

void TestGraphMinTree()
{const char str[] = "abcdefghi";Graph<char, int> g(str, strlen(str));g.AddEdge('a', 'b', 4);g.AddEdge('a', 'h', 8);g.AddEdge('a', 'h', 9);g.AddEdge('b', 'c', 8);g.AddEdge('b', 'h', 11);g.AddEdge('c', 'i', 2);g.AddEdge('c', 'f', 4);g.AddEdge('c', 'd', 7);g.AddEdge('d', 'f', 14);g.AddEdge('d', 'e', 9);g.AddEdge('e', 'f', 10);g.AddEdge('f', 'g', 2);g.AddEdge('g', 'h', 1);g.AddEdge('g', 'i', 6);g.AddEdge('h', 'i', 7);Graph<char, int> kminTree;cout << "Kruskal:" << g.Kruskal(kminTree) << endl;kminTree.Print();cout << endl;
}

2.Prim算法

Prim算法(普里姆算法)

W Prim(Self& minTree, const W& src)
{size_t srci = GetVertexIndex(src);size_t n = _vertexs.size();minTree._vertexs = _vertexs;minTree._indexMap = _indexMap;minTree._matrix.resize(n);for (size_t i = 0; i < n; ++i){minTree._matrix[i].resize(n, MAX_W);}vector<bool> X(n, false);vector<bool> Y(n, true);X[srci] = true;Y[srci] = false;//从X到Y集合相连接的边中选出值最小的边priority_queue<Edge, vector<Edge>, greater<Edge>> minq;//将srci连接的边添加到队列中for (size_t i = 0; i < n; ++i){if (_matrix[srci][i] != MAX_W){minq.push(Edge(srci, i, _matrix[srci][i]));}}cout << "Prim开始选边:" << endl;int size = 0;W totalW = W();while (!minq.empty()){Edge min = minq.top();minq.pop();if (X[min._dsti]){cout << "构成环";cout << _vertexs[min._srci] << "->" << _vertexs[min._dsti] << ":" << min._w << endl;}else{minTree._AddEdge(min._srci, min._dsti, min._w);cout << _vertexs[min._srci] << "->" << _vertexs[min._dsti] << ":" << min._w << endl;X[min._dsti] = true;Y[min._dsti] = false;++size;totalW += min._w;if (size == n - 1)break;for (size_t i = 0; i < n; ++i){if (_matrix[min._dsti][i] != MAX_W && Y[i]){minq.push(Edge(min._dsti, i, _matrix[min._dsti][i]));}}}}cout << endl;if (size == n - 1){return totalW;}else{return W();}
}

void TestGraphMinTree()
{const char str[] = "abcdefghi";Graph<char, int> g(str, strlen(str));g.AddEdge('a', 'b', 4);g.AddEdge('a', 'h', 8);g.AddEdge('a', 'h', 9);g.AddEdge('b', 'c', 8);g.AddEdge('b', 'h', 11);g.AddEdge('c', 'i', 2);g.AddEdge('c', 'f', 4);g.AddEdge('c', 'd', 7);g.AddEdge('d', 'f', 14);g.AddEdge('d', 'e', 9);g.AddEdge('e', 'f', 10);g.AddEdge('f', 'g', 2);g.AddEdge('g', 'h', 1);g.AddEdge('g', 'i', 6);g.AddEdge('h', 'i', 7);Graph<char, int> pminTree;cout << "Prim:" << g.Prim(pminTree, 'a') << endl;pminTree.Print();
}

五.最短路径

最短路径问题：从在带权有向图G中的某一顶点出发，找出一条通往另一顶点的最短路径，最短也就是沿路径各边的权值总和达到最小

1.单源最短路径--Dijkstra算法

单源最短路径问题：给定一个图G = ( V ， E ) G=(V，E)G=(V，E)，求源结点s ∈ V s∈Vs∈V到图中每个结点v ∈ V v∈Vv∈V的最短路径。Dijkstra算法就适用于解决带权重的有向图上的单源最短路径问题，同时算法要求图中所有边的权重非负。一般在求解最短路径的时候都是已知一个起点和一个终点，所以使用Dijkstra算法求解过后也就得到了所需起点到终点的最短路径

针对一个带权有向图G，将所有结点分为两组S和Q，S是已经确定最短路径的结点集合，在初始时为空（初始时就可以将源节点s放入，毕竟源节点到自己的代价是0），Q 为其余未确定最短路径的结点集合，每次从Q 中找出一个起点到该结点代价最小的结点u ，将u 从Q 中移出，并放入S中，对u的每一个相邻结点v 进行松弛操作。松弛即对每一个相邻结点v ，判断源节点s到结点u 的代价与u 到v 的代价之和是否比原来s 到v 的代价更小，若代价比原来小则要将s 到v 的代价更新为s 到u 与u 到v 的代价之和，否则维持原样。如此一直循环直至集合Q 为空，即所有节点都已经查找过一遍并确定了最短路径，至于一些起点到达不了的结点在算法循环后其代价仍为初始设定的值，不发生变化。Dijkstra算法每次都是选择V-S中最小的路径节点来进行更新，并加入S中，所以该算法使用的是贪心策略

Dijkstra算法(迪杰斯特拉算法)存在的问题是不支持图中带负权路径，如果带有负权路径，则可能会找不到一些路径的最短路径

//时间复杂度:O(N^2),空间复杂度:O(N)
void Dijkstra(const V& src, vector<W>& dist, vector<int>& pPath)
{size_t srci = GetVertexIndex(src);size_t n = _vertexs.size();dist.resize(n, MAX_W);pPath.resize(n, -1);dist[srci] = 0;pPath[srci] = srci;//已经确定最短路径的顶点集合vector<bool> S(n, false);for (size_t j = 0; j < n; ++j){//选出最短路径顶点且不在S更新其他路径int u = 0;W min = MAX_W;for (size_t i = 0; i < n; ++i){if (S[i] == false && dist[i] < min){u = i;min = dist[i];}}S[u] = true;//松弛更新for (size_t v = 0; v < n; ++v){if (S[v] == false && _matrix[u][v] != MAX_W && dist[u] + _matrix[u][v] < dist[v]){dist[v] = dist[u] + _matrix[u][v];pPath[v] = u;}}}
}

void TestGraphDijkstra()
{const char* str = "syztx";Graph<char, int, INT_MAX, true> g(str, strlen(str));g.AddEdge('s', 't', 10);g.AddEdge('s', 'y', 5);g.AddEdge('y', 't', 3);g.AddEdge('y', 'x', 9);g.AddEdge('y', 'z', 2);g.AddEdge('z', 's', 7);g.AddEdge('z', 'x', 6);g.AddEdge('t', 'y', 2);g.AddEdge('t', 'x', 1);g.AddEdge('x', 'z', 4);vector<int> dist;vector<int> parentPath;g.Dijkstra('s', dist, parentPath);g.PrintShortPath('s', dist, parentPath);
}

2.单源最短路径--Bellman-Ford算法

Dijkstra算法只能用来解决正权图的单源最短路径问题，但有些题目会出现负权图。这时这个算法就不能帮助我们解决问题了，而bellman—ford算法(贝尔曼-福特算法)可以解决负权图的单源最短路径问题。它的优点是可以解决有负权边的单源最短路径问题，而且可以用来判断是否有负权回路。它也有明显的缺点，它的时间复杂度 O(N*E) (N是点数，E是边数)普遍是要高于Dijkstra算法O(N²)的。像这里如果我们使用邻接矩阵实现，那么遍历所有边的数量的时间复杂度就是O(N^3)，这里也可以看出来Bellman-Ford就是一种暴力求解更新

//时间复杂度:O(N^3),空间复杂度:O(N)
bool BellmanFord(const V& src, vector<W>& dist, vector<int>& pPath)
{size_t srci = GetVertexIndex(src);size_t n = _vertexs.size();// vector<W> dist,记录srci-其他顶点最短路径权值数组dist.resize(n, MAX_W);// vector<int> pPath 记录srci-其他顶点最短路径父顶点数组pPath.resize(n, -1);// 先更新srci->srci为最小值dist[srci] = W();for (size_t k = 0; k < n; ++k){bool updata = false;cout << "更新第" << k << "轮:" << endl;for (size_t i = 0; i < n; ++i){for (size_t j = 0; j < n; ++j){if (_matrix[i][j] != MAX_W && dist[i] + _matrix[i][j] < dist[j]){updata = true;cout << _vertexs[i] << "->" << _vertexs[j] << ":" << _matrix[i][j] << endl;dist[j] = dist[i] + _matrix[i][j];pPath[j] = i;}}}//如果这个轮次没有更新出最短路径,后续轮次就不需要再走if (updata == false)break;}//如果还能更新就是带负权回路for (size_t i = 0; i < n; ++i){for (size_t j = 0; j < n; ++j){if (_matrix[i][j] != MAX_W && dist[i] + _matrix[i][j] < dist[j]){return false;}}}return true;
}

void TestGraphBellmanFord()
{const char* str = "syztx";Graph<char, int, INT_MAX, true> g(str, strlen(str));g.AddEdge('s', 't', 6);g.AddEdge('s', 'y', 7);g.AddEdge('y', 'z', 9);g.AddEdge('y', 'x', -3);//g.AddEdge('y', 's', 1);//新增g.AddEdge('z', 's', 2);g.AddEdge('z', 'x', 7);g.AddEdge('t', 'x', 5);g.AddEdge('t', 'y', 8);//g.AddEdge('t', 'y', -8);//更改g.AddEdge('t', 'z', -4);g.AddEdge('x', 't', -2);vector<int> dist;vector<int> parentPath;if (g.BellmanFord('s', dist, parentPath)){g.PrintShortPath('s', dist, parentPath);}else{cout << "存在负权回路" << endl;}
}

3.多源最短路径--Floyd-Warshall算法

Floyd-Warshall算法(弗洛伊德算法)是解决任意两点间的最短路径的一种算法

Floyd算法考虑的是一条最短路径的中间节点，即简单路径p={v1,v2,…,vn}上除v1和vn的任意节点

设k是p的一个中间节点，那么从i到j的最短路径p就被分成i到k和k到j的两段最短路径p1，p2。p1 是从i到k且中间节点属于{1，2，…，k-1}取得的一条最短路径。p2是从k到j且中间节点属于{1， 2，…，k-1}取得的一条最短路径

Floyd算法本质是三维动态规划，D[i][j][k]表示从点i到点j只经过0到k个点最短路径，然后建立起转移方程，然后通过空间优化，优化掉最后一维度，变成一个最短路径的迭代算法，最后即得到所有点的最短路径

void FloydWarShall(vector<vector<W>>& vvDist, vector<vector<int>>& vvpPath)
{size_t n = _vertexs.size();vvDist.resize(n);vvpPath.resize(n);for (size_t i = 0; i < n; ++i){vvDist[i].resize(n, MAX_W);vvpPath[i].resize(n, -1);}for (size_t i = 0; i < n; ++i){for (size_t j = 0; j < n; ++j){if (_matrix[i][j] != MAX_W){vvDist[i][j] = _matrix[i][j];vvpPath[i][j] = i;}if (i == j){vvDist[i][j] = 0;}}}//最短路径的更新for (size_t k = 0; k < n; ++k){for (size_t i = 0; i < n; ++i){for (size_t j = 0; j < n; ++j){if (vvDist[i][k] != MAX_W && vvDist[k][j] != MAX_W && vvDist[i][k] + vvDist[k][j] < vvDist[i][j]){vvDist[i][j] = vvDist[i][k] + vvDist[k][j];vvpPath[i][j] = vvpPath[k][j];}}}}
}

void TestFloydWarShall()
{const char* str = "12345";Graph<char, int, INT_MAX, true> g(str, strlen(str));g.AddEdge('1', '2', 3);g.AddEdge('1', '3', 8);g.AddEdge('1', '5', -4);g.AddEdge('2', '4', 1);g.AddEdge('2', '5', 7);g.AddEdge('3', '2', 4);g.AddEdge('4', '1', 2);g.AddEdge('4', '3', -5);g.AddEdge('5', '4', 6);vector<vector<int>> vvDist;vector<vector<int>> vvParentPath;g.FloydWarShall(vvDist, vvParentPath);// 打印任意两点之间的最短路径for (size_t i = 0; i < strlen(str); ++i){g.PrintShortPath(str[i], vvDist[i], vvParentPath[i]);cout << endl;}
}

六.整体实现

1.UnionFindSet.h

#pragma once
#include<vector>class UnionFindSet
{
public:UnionFindSet(size_t n) :_ufs(n, -1){ }int FindRoot(int x){int root = x;while (_ufs[root] >= 0)root = _ufs[root];//路径压缩while (_ufs[x] >= 0){int parent = _ufs[x];_ufs[x] = root;x = parent;}return root;}bool Union(int x1, int x2){int root1 = FindRoot(x1);int root2 = FindRoot(x2);if (root1 == root2)//x1和x2本来就在一个集合中return false;//数据量小的向大的合并if (abs(_ufs[root1]) < abs(_ufs[root2]))swap(root1, root2);_ufs[root1] += _ufs[root2];_ufs[root2] = root1;return true;}bool InSet(int x1, int x2){return FindRoot(x1) == FindRoot(x2);}size_t SetSize(){size_t n = 0;for (auto& e : _ufs){if (e < 0)++n;}return n;}
private:vector<int> _ufs;
};void TestUoionFindSet()
{UnionFindSet ufs(10);ufs.Union(8, 9);ufs.Union(7, 8);ufs.Union(6, 7);ufs.Union(5, 6);ufs.Union(4, 5);
}

2.Graph.h

#pragma once
#include<vector>
#include<string>
#include<map>
#include<queue>
#include<set>
#include<functional>namespace matrix
{template<class V, class W, W MAX_W = INT_MAX, bool Direction = false>class Graph{typedef Graph<V, W, MAX_W, Direction> Self;public://图的创建//1.IO输入->不方便测试,oj中更适合//2.图结构关系写到文件,读取文件//3.手动添加边Graph() = default;Graph(const V* a, size_t n){_vertexs.reserve(n);for (size_t i = 0; i < n; ++i){_vertexs.push_back(a[i]);_indexMap[a[i]] = i;}_matrix.resize(n);for (size_t i = 0; i < _matrix.size(); ++i){_matrix[i].resize(n, MAX_W);}}size_t GetVertexIndex(const V& v){auto it = _indexMap.find(v);if (it != _indexMap.end()){return it->second;}else{//assert(false);throw invalid_argument("顶点不存在");return -1;}}void _AddEdge(size_t srci, size_t dsti, const W& w){_matrix[srci][dsti] = w;//无向图if (Direction == false){_matrix[dsti][srci] = w;}}void AddEdge(const V& src, const V& dst, const W& w){size_t srci = GetVertexIndex(src);size_t dsti = GetVertexIndex(dst);_AddEdge(srci, dsti, w);}void Print(){//顶点for (size_t i = 0; i < _vertexs.size(); ++i){cout << "[" << i << "]" << "->" << _vertexs[i] << endl;}cout << endl;//矩阵//横下标cout << "  ";for (size_t i = 0; i < _matrix.size(); ++i){//cout << i << " ";printf("%4d", i);}cout << endl;for (size_t i = 0; i < _matrix.size(); ++i){cout << i << " ";//竖下标for (size_t j = 0; j < _matrix[i].size(); ++j){//cout << _matrix[i][j] << " ";if (_matrix[i][j] == MAX_W){//cout << "* ";printf("%4c", '*');}else{//cout << _matrix[i][j] << " ";printf("%4d", _matrix[i][j]);}}cout << endl;}cout << endl;}void BFS(const V& src){size_t srci = GetVertexIndex(src);//队列和标记数组queue<int> q;vector<bool> visited(_vertexs.size(), false);q.push(srci);visited[srci] = true;int levelSize = 1;size_t n = _vertexs.size();while (!q.empty()){//一层一层出for (int i = 0; i < levelSize; ++i){int front = q.front();q.pop();cout << front << ":" << _vertexs[front] << " ";//把front顶点的邻接顶点入队列for (size_t i = 0; i < n; ++i){if (_matrix[front][i] != MAX_W){if (visited[i] == false){q.push(i);visited[i] = true;}}}}cout << endl;levelSize = q.size();}cout << endl;}void _DFS(size_t srci, vector<bool>& visited){cout << srci << ":" << _vertexs[srci] << endl;visited[srci] = true;//找一个和srci相邻的没有访问过的点,去深度遍历for (size_t i = 0; i < _vertexs.size(); ++i){if (_matrix[srci][i] != MAX_W && visited[i] == false){_DFS(i, visited);}}}void DFS(const V& src){size_t srci = GetVertexIndex(src);vector<bool> visited(_vertexs.size(), false);_DFS(srci, visited);}struct Edge{size_t _srci;size_t _dsti;W _w;Edge(size_t srci, size_t dsti, const W& w) :_srci(srci), _dsti(dsti), _w(w){ }bool operator>(const Edge& e) const{return _w > e._w;}};W Kruskal(Self& minTree){size_t n = _vertexs.size();minTree._vertexs = _vertexs;minTree._indexMap = _indexMap;minTree._matrix.resize(n);for (size_t i = 0; i < n; ++i){minTree._matrix[i].resize(n, MAX_W);}priority_queue<Edge, vector<Edge>, greater<Edge>> minque;for (size_t i = 0; i < n; ++i){for (size_t j = 0; j < n; ++j){if (i < j && _matrix[i][j] != MAX_W){minque.push(Edge(i, j, _matrix[i][j]));}}}cout << "Kruskal开始选边:" << endl;//选出n-1条边int size = 0;W totalW = W();UnionFindSet ufs(n);while (!minque.empty()){Edge min = minque.top();minque.pop();if (!ufs.InSet(min._srci, min._dsti)){cout << _vertexs[min._srci] << "->" << _vertexs[min._dsti] << ":" << min._w << endl;minTree._AddEdge(min._srci, min._dsti, min._w);ufs.Union(min._srci, min._dsti);++size;totalW += min._w;}else{cout << "构成环";cout << _vertexs[min._srci] << "->" << _vertexs[min._dsti] << ":" << min._w << endl;}}cout << endl;if (size == n - 1){return totalW;}else{return W();}}W Prim(Self& minTree, const W& src){size_t srci = GetVertexIndex(src);size_t n = _vertexs.size();minTree._vertexs = _vertexs;minTree._indexMap = _indexMap;minTree._matrix.resize(n);for (size_t i = 0; i < n; ++i){minTree._matrix[i].resize(n, MAX_W);}vector<bool> X(n, false);vector<bool> Y(n, true);X[srci] = true;Y[srci] = false;//从X到Y集合相连接的边中选出值最小的边priority_queue<Edge, vector<Edge>, greater<Edge>> minq;//将srci连接的边添加到队列中for (size_t i = 0; i < n; ++i){if (_matrix[srci][i] != MAX_W){minq.push(Edge(srci, i, _matrix[srci][i]));}}cout << "Prim开始选边:" << endl;int size = 0;W totalW = W();while (!minq.empty()){Edge min = minq.top();minq.pop();if (X[min._dsti]){cout << "构成环";cout << _vertexs[min._srci] << "->" << _vertexs[min._dsti] << ":" << min._w << endl;}else{minTree._AddEdge(min._srci, min._dsti, min._w);cout << _vertexs[min._srci] << "->" << _vertexs[min._dsti] << ":" << min._w << endl;X[min._dsti] = true;Y[min._dsti] = false;++size;totalW += min._w;if (size == n - 1)break;for (size_t i = 0; i < n; ++i){if (_matrix[min._dsti][i] != MAX_W && Y[i]){minq.push(Edge(min._dsti, i, _matrix[min._dsti][i]));}}}}cout << endl;if (size == n - 1){return totalW;}else{return W();}}void PrintShortPath(const V& src, const vector<W>& dist, const vector<int>& pPath){size_t srci = GetVertexIndex(src);size_t n = _vertexs.size();for (size_t i = 0; i < n; ++i){if (i != srci){//找出i顶点的路径vector<int> path;size_t parenti = i;while (parenti != srci){path.push_back(parenti);parenti = pPath[parenti];}path.push_back(srci);reverse(path.begin(), path.end());for (auto index : path){cout << _vertexs[index] << "->";}cout << "权值和: " << dist[i] << endl;}}}//时间复杂度:O(N^2),空间复杂度:O(N)void Dijkstra(const V& src, vector<W>& dist, vector<int>& pPath){size_t srci = GetVertexIndex(src);size_t n = _vertexs.size();dist.resize(n, MAX_W);pPath.resize(n, -1);dist[srci] = 0;pPath[srci] = srci;//已经确定最短路径的顶点集合vector<bool> S(n, false);for (size_t j = 0; j < n; ++j){//选出最短路径顶点且不在S更新其他路径int u = 0;W min = MAX_W;for (size_t i = 0; i < n; ++i){if (S[i] == false && dist[i] < min){u = i;min = dist[i];}}S[u] = true;//松弛更新for (size_t v = 0; v < n; ++v){if (S[v] == false && _matrix[u][v] != MAX_W && dist[u] + _matrix[u][v] < dist[v]){dist[v] = dist[u] + _matrix[u][v];pPath[v] = u;}}}}//时间复杂度:O(N^3),空间复杂度:O(N)bool BellmanFord(const V& src, vector<W>& dist, vector<int>& pPath){size_t srci = GetVertexIndex(src);size_t n = _vertexs.size();// vector<W> dist,记录srci-其他顶点最短路径权值数组dist.resize(n, MAX_W);// vector<int> pPath 记录srci-其他顶点最短路径父顶点数组pPath.resize(n, -1);// 先更新srci->srci为最小值dist[srci] = W();for (size_t k = 0; k < n; ++k){bool updata = false;cout << "更新第" << k << "轮:" << endl;for (size_t i = 0; i < n; ++i){for (size_t j = 0; j < n; ++j){if (_matrix[i][j] != MAX_W && dist[i] + _matrix[i][j] < dist[j]){updata = true;cout << _vertexs[i] << "->" << _vertexs[j] << ":" << _matrix[i][j] << endl;dist[j] = dist[i] + _matrix[i][j];pPath[j] = i;}}}//如果这个轮次没有更新出最短路径,后续轮次就不需要再走if (updata == false)break;}//如果还能更新就是带负权回路for (size_t i = 0; i < n; ++i){for (size_t j = 0; j < n; ++j){if (_matrix[i][j] != MAX_W && dist[i] + _matrix[i][j] < dist[j]){return false;}}}return true;}void FloydWarShall(vector<vector<W>>& vvDist, vector<vector<int>>& vvpPath){size_t n = _vertexs.size();vvDist.resize(n);vvpPath.resize(n);for (size_t i = 0; i < n; ++i){vvDist[i].resize(n, MAX_W);vvpPath[i].resize(n, -1);}for (size_t i = 0; i < n; ++i){for (size_t j = 0; j < n; ++j){if (_matrix[i][j] != MAX_W){vvDist[i][j] = _matrix[i][j];vvpPath[i][j] = i;}if (i == j){vvDist[i][j] = 0;}}}//最短路径的更新for (size_t k = 0; k < n; ++k){for (size_t i = 0; i < n; ++i){for (size_t j = 0; j < n; ++j){if (vvDist[i][k] != MAX_W && vvDist[k][j] != MAX_W && vvDist[i][k] + vvDist[k][j] < vvDist[i][j]){vvDist[i][j] = vvDist[i][k] + vvDist[k][j];vvpPath[i][j] = vvpPath[k][j];}}}}}private:vector<V> _vertexs;            //顶点集合map<V, int> _indexMap;		   //顶点映射下标vector<vector<W>> _matrix;	   //邻接矩阵};void TestGraph1(){Graph<char, int> g("0123", 4);//Graph<char, int, true> g("0123", 4);g.AddEdge('0', '1', 1);g.AddEdge('0', '3', 4);g.AddEdge('1', '3', 2);g.AddEdge('1', '2', 9);g.AddEdge('2', '3', 8);g.AddEdge('2', '1', 5);g.AddEdge('2', '0', 3);g.AddEdge('3', '2', 6);g.Print();}void TestBDFS(){string a[] = { "张三", "李四", "王五", "赵六", "周七" };Graph<string, int> g1(a, sizeof(a) / sizeof(string));g1.AddEdge("张三", "李四", 100);g1.AddEdge("张三", "王五", 200);g1.AddEdge("王五", "赵六", 30);g1.AddEdge("王五", "周七", 30);g1.Print();g1.BFS("张三");g1.DFS("张三");}void TestGraphMinTree(){const char str[] = "abcdefghi";Graph<char, int> g(str, strlen(str));g.AddEdge('a', 'b', 4);g.AddEdge('a', 'h', 8);g.AddEdge('a', 'h', 9);g.AddEdge('b', 'c', 8);g.AddEdge('b', 'h', 11);g.AddEdge('c', 'i', 2);g.AddEdge('c', 'f', 4);g.AddEdge('c', 'd', 7);g.AddEdge('d', 'f', 14);g.AddEdge('d', 'e', 9);g.AddEdge('e', 'f', 10);g.AddEdge('f', 'g', 2);g.AddEdge('g', 'h', 1);g.AddEdge('g', 'i', 6);g.AddEdge('h', 'i', 7);Graph<char, int> kminTree;cout << "Kruskal:" << g.Kruskal(kminTree) << endl;kminTree.Print();cout << endl;Graph<char, int> pminTree;cout << "Prim:" << g.Prim(pminTree, 'a') << endl;pminTree.Print();}void TestGraphDijkstra(){const char* str = "syztx";Graph<char, int, INT_MAX, true> g(str, strlen(str));g.AddEdge('s', 't', 10);g.AddEdge('s', 'y', 5);g.AddEdge('y', 't', 3);g.AddEdge('y', 'x', 9);g.AddEdge('y', 'z', 2);g.AddEdge('z', 's', 7);g.AddEdge('z', 'x', 6);g.AddEdge('t', 'y', 2);g.AddEdge('t', 'x', 1);g.AddEdge('x', 'z', 4);vector<int> dist;vector<int> parentPath;g.Dijkstra('s', dist, parentPath);g.PrintShortPath('s', dist, parentPath);}void TestGraphBellmanFord(){const char* str = "syztx";Graph<char, int, INT_MAX, true> g(str, strlen(str));g.AddEdge('s', 't', 6);g.AddEdge('s', 'y', 7);g.AddEdge('y', 'z', 9);g.AddEdge('y', 'x', -3);//g.AddEdge('y', 's', 1);//新增g.AddEdge('z', 's', 2);g.AddEdge('z', 'x', 7);g.AddEdge('t', 'x', 5);g.AddEdge('t', 'y', 8);//g.AddEdge('t', 'y', -8);//更改g.AddEdge('t', 'z', -4);g.AddEdge('x', 't', -2);vector<int> dist;vector<int> parentPath;if (g.BellmanFord('s', dist, parentPath)){g.PrintShortPath('s', dist, parentPath);}else{cout << "存在负权回路" << endl;}}void TestFloydWarShall(){const char* str = "12345";Graph<char, int, INT_MAX, true> g(str, strlen(str));g.AddEdge('1', '2', 3);g.AddEdge('1', '3', 8);g.AddEdge('1', '5', -4);g.AddEdge('2', '4', 1);g.AddEdge('2', '5', 7);g.AddEdge('3', '2', 4);g.AddEdge('4', '1', 2);g.AddEdge('4', '3', -5);g.AddEdge('5', '4', 6);vector<vector<int>> vvDist;vector<vector<int>> vvParentPath;g.FloydWarShall(vvDist, vvParentPath);// 打印任意两点之间的最短路径for (size_t i = 0; i < strlen(str); ++i){g.PrintShortPath(str[i], vvDist[i], vvParentPath[i]);cout << endl;}}
}namespace link_table
{template<class W>struct Edge{//int _srci;int _dsti;  //目标点的下标W _w;		//权值Edge<W>* _next;Edge(int dsti, const W& w) :_dsti(dsti), _w(w), _next(nullptr){ }};template<class V, class W, bool Direction = false>class Graph{typedef Edge<W> Edge;public:Graph(const V* a, size_t n){_vertexs.reserve(n);for (size_t i = 0; i < n; ++i){_vertexs.push_back(a[i]);_indexMap[a[i]] = i;}_tables.resize(n, nullptr);}size_t GetVertexIndex(const V& v){auto it = _indexMap.find(v);if (it != _indexMap.end()){return it->second;}else{//assert(false);throw invalid_argument("顶点不存在");return -1;}}void AddEdge(const V& src, const V& dst, const W& w){size_t srci = GetVertexIndex(src);size_t dsti = GetVertexIndex(dst);Edge* eg = new Edge(dsti, w);eg->_next = _tables[srci];_tables[srci] = eg;if (Direction == false){Edge* eg = new Edge(srci, w);eg->_next = _tables[dsti];_tables[dsti] = eg;}}void Print(){//顶点for (size_t i = 0; i < _vertexs.size(); ++i){cout << "[" << i << "]" << "->" << _vertexs[i] << endl;}cout << endl;for (size_t i = 0; i < _tables.size(); ++i){cout << _vertexs[i] << "[" << i << "]->";Edge* cur = _tables[i];while (cur){cout << "[" << _vertexs[cur->_dsti] << ":" << cur->_dsti << ":" << cur->_w << "]->";cur = cur->_next;}cout << "nullptr" << endl;}}private:vector<V> _vertexs;            //顶点集合map<V, int> _indexMap;		   //顶点映射下标vector<Edge*> _tables;		   //邻接表};void TestGraph2(){string a[] = { "张三", "李四", "王五", "赵六" };//Graph<string, int, true> g1(a, 4);Graph<string, int> g1(a, 4);g1.AddEdge("张三", "李四", 100);g1.AddEdge("张三", "王五", 200);g1.AddEdge("王五", "赵六", 30);g1.Print();}
}

3.test.cpp

#include<iostream>
using namespace std;
#include"UnionFindSet.h"
#include"Graph.h"int main()
{//TestUoionFindSet();//matrix::TestGraph1();//matrix::TestBDFS();//matrix::TestGraphMinTree();//matrix::TestGraphDijkstra();//matrix::TestGraphBellmanFord();matrix::TestFloydWarShall();//link_table::TestGraph2();return 0;
}