算法核心:两个结点之间的一条最短路径包含着(包含于)其它的最短路径.[最短路径性质]
1.单源最短路径Dijkstra
思路:计算每个结点到源结点的距离,压入最小优先队列Q,对Q中的元素进行如下循环操作:
1.从队列Q中弹出最小元素u 2.将u并入S 3.对u的邻接表中每个结点v,调用Relax(u,v,w)更新结点v到源结点s的距离
直至Q为空.
伪代码:
Initialize-Single-Source(G,s) for each vertex v in G.v v.d = MAX v.p = NIL s.d = 0 Realx(u,v,w) if v.d > u.d + w[u][v] v.d = u.d + w[u][v] v.p = u Dijkstra(G,w,s) Initialize-Single-Source(G,s) S = ∅//已经找到到源结点s的最短路径的结点的集合 Q = G.V//Q是以结点到源结点距离为priority的最小优先队列 while Q ≠ ∅ u = EXTRACT-MIN(Q) S = S∪{u} for each vertex v in G.Adj[u] Relax(u,v,w)
时间复杂度:O(|V|2)
编码实现:
#include<iostream> #include<vector> #include<utility> #include<algorithm> #include<queue> #include<set> #include<cassert> using namespace std; #define MAX 10000 enum Color{white,gray,black}; class Node{ public: int index; Node* next=nullptr;// Node(int i):index(i){} }; class VNode{ public: char vertex; int index; int dist; int final; int indegree; Color color=white; int prev=-1; Node* firstNode=nullptr; VNode(char c,int i):vertex(c),index(i){} }; typedef struct Graph{ int EdgeNum; vector<VNode> Adj; }Graph; void initialize_single_source(Graph &G,int s,int w[][6]){ for(auto &v:G.Adj){//use reference************** v.dist = MAX;//w[0][v.index]; v.prev = -1; } G.Adj[s].dist = 0; } void relax(Graph &G,int u,int v,int w[][6]){ if(G.Adj[v].dist > G.Adj[u].dist + w[u][v]){ G.Adj[v].dist = G.Adj[u].dist + w[u][v]; G.Adj[v].prev = u; } } //为优先队列提供比较结构,小的优先 struct comp { bool operator () (VNode &a,VNode &b)const{ return a.dist>b.dist; } }; void dijkstra(Graph &G,int w[][6],int s){ initialize_single_source(G,s,w); set<int> S; priority_queue<VNode,vector<VNode>,comp> Q; for(int i=0;i<G.Adj.size();i++){ Q.push(G.Adj[i]); } while(!Q.empty()){ int u = (Q.top()).index; Q.pop(); S.insert(u); Node* p = G.Adj[u].firstNode; while(p != nullptr){ int v = p->index; relax(G,u,v,w); p = p->next; } //由于保存在Q中的元素是结点的副本,故结点到源结点距离改变时,优先队列Q不会受到影响,也就是Q不会像我们期望的那样工作,故需重新生成Q priority_queue<VNode,vector<VNode>,comp> Q3; while(!Q.empty()){ VNode vn=Q.top(); Q.pop(); Q3.push(G.Adj[vn.index]); } Q=Q3; } } void AddEdge(Graph &G,int i,int j){ Node* p = new Node(j); p->next = G.Adj[i].firstNode; G.Adj[i].firstNode = p; } void print_path(Graph &G,int s,int v){ if(v == s){ cout<<G.Adj[s].vertex<<","; } else if(G.Adj[v].prev == -1){ cout<<"No such path!"; } else{ print_path(G, s, G.Adj[v].prev); cout<<G.Adj[v].vertex<<","; } } int main(){ Graph G; G.EdgeNum = 8; vector<char> v={'a','b','c','d','e','f'}; for(int i=0;i<v.size();i++){ G.Adj.push_back(VNode(v[i], i)); } AddEdge(G,0,2); AddEdge(G,0,4); AddEdge(G,0,5); AddEdge(G,1,2); AddEdge(G,2,3); AddEdge(G,3,5); AddEdge(G,4,3); AddEdge(G,4,5); int w[6][6]; for (int i = 0; i < 6; ++i) { for (int j = 0; j < 6; ++j) { if(i==j){ w[i][j] = 0; } else{ w[i][j] = MAX; } } } w[0][2] = 10; w[0][4] = 30; w[0][5] = 100; w[1][2] = 5; w[2][3] = 50; w[3][5] = 10; w[4][3] = 20; w[4][5] = 60; dijkstra(G,w,0); for(int i=1;i<G.Adj.size();i++){ cout<<G.Adj[i].dist<<' '; print_path(G,0,i); cout<<endl; } return 0; }
2.所有结点对的最短路径FloydWarshall
思路:对每对结点[i,j],尝试向中间加入结点k,如果w[i][j] > w[i][k] + w[k][j],则更新[i,j]之间的最短距离为w[i][j] = w[i][k] + w[k][j].由于k的加入使w[i][j]变小,故k属于结点对[i,j]的最短路径上的点,另外,由于最短路径性质,[i,j]的最短路径由[i,k]和[k,j]的最短l路径组成.
伪代码:
void FloydWarshall(Graph &G,int w[][6],int v[][6]){ for(int k=0;k<6;k++){ for(int i=0;i<6;i++){ for(int j=0;j<6;j++){ if(w[i][j]>w[i][k]+w[k][j]){ w[i][j] = w[i][k]+w[k][j]; v[i][j] = k;//记下[i,j]的最短路径的中间结点 } } } } }
时间复杂度:O(|V|3)
编码实现:
#include<iostream> #include<vector> #include<utility> #include<algorithm> #include<queue> #include<set> #include<cassert> using namespace std; #define MAX 10000 enum Color{white,gray,black}; class Node{ public: int index; Node* next=nullptr;// Node(int i):index(i){} }; class VNode{ public: char vertex; int index; int dist; int final; int indegree; Color color=white; int prev=-1; Node* firstNode=nullptr; VNode(char c,int i):vertex(c),index(i){} }; typedef struct Graph{ int EdgeNum; vector<VNode> Adj; }Graph; void FloydWarshall(Graph &G,int w[][6],int v[][6]){ for(int k=0;k<6;k++){ for(int i=0;i<6;i++){ for(int j=0;j<6;j++){ if(w[i][j]>w[i][k]+w[k][j]){ w[i][j] = w[i][k]+w[k][j]; v[i][j] = k; } } } } } void AddEdge(Graph &G,int i,int j){ Node* p = new Node(j); p->next = G.Adj[i].firstNode; G.Adj[i].firstNode = p; } void print_path(const Graph &G,int i,int j,const int v[][6]){ if(v[i][j]==-1){ cout<<G.Adj[i].vertex<<','; } else{ print_path(G,i,v[i][j],v); print_path(G,v[i][j],j,v); } } int main(){ Graph G; G.EdgeNum = 8; vector<char> v={'a','b','c','d','e','f'}; for(int i=0;i<v.size();i++){ G.Adj.push_back(VNode(v[i], i)); } AddEdge(G,0,2); AddEdge(G,0,4); AddEdge(G,0,5); AddEdge(G,1,2); AddEdge(G,2,3); AddEdge(G,3,5); AddEdge(G,4,3); AddEdge(G,4,5); int w[6][6]; for (int i = 0; i < 6; ++i) { for (int j = 0; j < 6; ++j) { if(i==j){ w[i][j]=0; } else{ w[i][j] = MAX; } } } w[0][2] = 10; w[0][4] = 30; w[0][5] = 100; w[1][2] = 5; w[2][3] = 50; w[3][5] = 10; w[4][3] = 20; w[4][5] = 60; int path[6][6]; for(int i=0;i<6;i++){ for(int j=0;j<6;j++){ path[i][j]=-1; } } FloydWarshall(G,w,path); for(int i=0;i<6;i++){ for(int j=0;j<6;j++){ if(w[i][j]!=MAX && i!=j){ cout<<'['<<i<<','<<j<<']'<<' '; cout<<w[i][j]<<' '; print_path(G,i,j,path); cout<<G.Adj[j].vertex<<','; cout<<endl; } } } return 0; }