Dijkstra算法求最短路模板
Dijkstra算法适合求不包含负权路的最短路径，通过点增广、在稠密图中使用优化过的版本速度非常可观。本篇不介绍算法原理、只给出模板，这里给出三种模板，其中最实用的是加上了堆优化的版本
算法原理 or 学习参考链接 : 点我 、不要点它点我！、为何不适用于带负权边图
( Dijkstra 动态演示 )
朴素版 ( 邻接矩阵存储、复杂度 O( n2 ) )
///HDU 2544为例 #include<stdio.h> #include<string.h> const int INF = 0x3f3f3f3f; const int maxn = 1001; bool vis[maxn]; int G[maxn][maxn],dis[maxn],pre[maxn];//pre[]记录前驱、用于输出路径 int n, m; void dijkstra(int v) { int i, j, u , Min; for(i=0;i<=n;i++){ dis[i]=G[v][i]; vis[i]=0; //if(i!=v&&G[v][i]!=INF)pre[i] = v; // else pre[i] = -1; } vis[v]=1;dis[v]=0; for(i=1;i<n;i++){ Min = INF; for(j=1;j<=n;j++){ if(!vis[j]&&Min > dis[j]){ Min = dis[j]; u = j; } } if(Min == INF)break; vis[u]=1; for(j=1;j<=n;j++){ if(!vis[j]&&G[u][j]!=INF&&dis[u]+G[u][j]<dis[j]){ dis[j] = G[u][j] + dis[u]; // pre[j] = u; } } } } int main() { int i, j, x, y, w; while(~scanf("%d%d",&n,&m)&&n) { for(i=0;i<=n;i++) for(j=0;j<=n;j++) if(i==j)G[i][j]=0; else G[i][j] = INF; while(m--){ scanf("%d%d%d",&x,&y,&w); G[x][y] = w; G[y][x] = w; } dijkstra(1); printf("%d\n",dis[n]); //以下为输出路径 /*int p, len=0, ans[maxn]; p = n-1; while(p!=0) { ans[len++] = p; p = pre[p]; } printf("0->"); for(i=len-1;i>=0;i--) printf("%d",ans[i]); puts(""); */ } return 0; }
STL优先队列优化版本 ( 复杂度 O( (V+E)logV ) )、此优化需要用邻接表存图
///POJ 2387为例 #include<stdio.h> #include<string.h> #include<queue> #include<algorithm> #include<stdlib.h> using namespace std; const int maxn = 1e3 + 50; const int INF = 0x3f3f3f3f; typedef pair<int, int> HeapNode;///在堆里面的是pair、first为到起点距离、second为点编号 struct EDGE{ int v, nxt, w; }; int Head[maxn], Dis[maxn]; EDGE Edge[maxn*100]; int N, M, cnt; inline void init() { for(int i=0; i<=N; i++) Head[i]=-1, Dis[i]=INF; cnt = 0; } inline void AddEdge(int from, int to, int weight) { Edge[cnt].w = weight; Edge[cnt].v = to; Edge[cnt].nxt = Head[from]; Head[from] = cnt++; } int Dijkstra() { priority_queue<HeapNode, vector<HeapNode>, greater<HeapNode> > Heap; Dis[1] = 0; Heap.push(make_pair(0, 1)); while(!Heap.empty()){ pair<int, int> T = Heap.top(); Heap.pop(); if(T.first != Dis[T.second]) continue;///有很多版本都是用 vis 标记是否已经使用这个点松弛过、这里可以用这个不同的方法！ for(int i=Head[T.second]; i!=-1; i=Edge[i].nxt){ int Eiv = Edge[i].v; if(Dis[Eiv] > Dis[T.second] + Edge[i].w){ Dis[Eiv] = Dis[T.second] + Edge[i].w; Heap.push(make_pair(Dis[Eiv], Eiv)); } } } return Dis[N]; } int main(void) { while(~scanf("%d %d", &M, &N)){ init(); int from, to, weight; for(int i=0; i<M; i++){ scanf("%d %d %d", &from, &to, &weight); AddEdge(from, to, weight); AddEdge(to, from, weight); } printf("%d\n", Dijkstra()); } return 0; }
传说中还有一种斐波那契堆，比STL默认的堆更高效、但是斐波那契堆难写难理解、故用配对堆来代替( 复杂度 O(VlogV + E) )
///POJ 2387为例 #include<stdio.h> #include<algorithm> #include<stdlib.h> #include<string.h> using namespace std; const int maxn = 1e3 + 10; const int INF = 0x3f3f3f3f; struct EDGE{ int v, nxt, w; }; EDGE Edge[maxn*maxn]; int Head[maxn], Dis[maxn], T, N, cnt; int Cost[maxn][maxn]; inline void init() { for(int i=0; i<=N; i++){ Head[i]=-1,Dis[i]=INF; for(int j=0; j<=N; j++){ Cost[i][j] = INF; } } cnt=0; } inline void ADD(int from, int to, int weight) { Edge[cnt].w=weight, Edge[cnt].v = to; Edge[cnt].nxt = Head[from]; Head[from] = cnt++; } struct Heap{ int num[maxn],pos[maxn],Size; void PushUp(int p) { while(p > 1) { if(Dis[num[p]] < Dis[num[p >> 1]]) { swap(num[p],num[p >> 1]); swap(pos[num[p]],pos[num[p >> 1]]); p >>= 1; } else break; } } void Insert(long long x) { num[++Size] = x; pos[x] = Size; PushUp(Size); } void Pop() { pos[num[1]] = 0; num[1] = num[Size--]; if(Size) pos[num[1]] = 1; int now = 2; while(now < Size) { if(Dis[num[now + 1]] < Dis[num[now]]) ++now; if(Dis[num[now]] < Dis[num[now >> 1]]) { swap(num[now],num[now >> 1]); swap(pos[num[now]],pos[num[now >> 1]]); now <<= 1; } else break; } } }heap;///配对堆 int Dijkstra() { Dis[1] = 0; for(int i=1; i<=N; i++) heap.Insert(i); while(heap.Size){ int x = heap.num[1]; heap.Pop(); for(int i=Head[x]; i!=-1; i=Edge[i].nxt) if(Dis[Edge[i].v] > Dis[x] + Edge[i].w) Dis[Edge[i].v] = Dis[x] + Edge[i].w, heap.PushUp(heap.pos[Edge[i].v]); } return Dis[N]; } int main(void) { while(~scanf("%d %d", &T, &N)){ init(); int from, to, weight; for(int i=0; i<T; i++){ scanf("%d %d %d", &from, &to, &weight); if(Cost[from][to] > weight){ Cost[from][to] = Cost[to][from] = weight; ADD(from, to, weight); ADD(to, from, weight); } } printf("%d\n", Dijkstra()); } return 0; }
在 Linux 下有pbds可以调用，里面可以调用二叉堆、配对堆、斐波那契堆……
///POJ 2387为例 #include<stdio.h> #include<string.h> #include<queue> #include<algorithm> #include<ext/pb_ds/priority_queue.hpp>///记得加上 #include<stdlib.h> using namespace __gnu_pbds;///记得加上 using namespace std; const int maxn = 1e3 + 5; const int INF = 0x3f3f3f3f; typedef pair<int, int> HeapNode; struct EDGE{ int v, nxt, w; }; inline int read() { int x=0,f=1;char ch=getchar(); while(ch<'0'||ch>'9'){if(ch=='-')f=-1;ch=getchar();} while(ch>='0'&&ch<='9'){x=x*10+ch-'0';ch=getchar();} return x*f; } int Head[maxn], Dis[maxn]; EDGE Edge[maxn*100]; int N, M, cnt; inline void init() { for(int i=0; i<=N; i++) Head[i]=-1, Dis[i]=INF; cnt = 0; } inline void AddEdge(int from, int to, int weight) { Edge[cnt].w = weight; Edge[cnt].v = to; Edge[cnt].nxt = Head[from]; Head[from] = cnt++; } int Dijkstra() { __gnu_pbds::priority_queue<HeapNode,greater<HeapNode>,pairing_heap_tag > Heap;///申请方式、其余和普通优先队列无差别 Dis[1] = 0; Heap.push(make_pair(0, 1)); while(!Heap.empty()){ pair<int, int> Top = Heap.top(); Heap.pop(); int v = Top.second; if(Top.first != Dis[v]) continue; for(int i=Head[v]; i!=-1; i=Edge[i].nxt){ int tmp = Edge[i].v; if(Dis[tmp] > Dis[v] + Edge[i].w){ Dis[tmp] = Dis[v] + Edge[i].w; Heap.push(make_pair(Dis[tmp], tmp)); } } } return Dis[N]; } int main(void) { while(~scanf("%d %d", &M, &N)){ init(); int from, to, weight; for(int i=0; i<M; i++){ from = read(); to = read(); weight = read(); AddEdge(from, to, weight); AddEdge(to, from, weight); } printf("%d\n", Dijkstra()); } return 0; }
