数据结构——图(遍历,最小生成树,最短路径)

目录

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

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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:

1. 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
2. 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
3. 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:

1. 邻接矩阵存储方式非常适合稠密图
2. 邻接矩阵O(1)判断两个顶点的连接关系并取到权值
3. 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:

1. 适合存储稀疏图
2. 适合查找一个顶点连接出去的边
3. 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:

1. 只能使用图中的边来构造最小生成树
2. 只能使用恰好n-1条边来连接图中的n个顶点
3. 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;
}