054.Dijkstra模板+反向索引堆优化 + 经典题目

Dijkstra

• BFS + 贪心(priority_queue)

• 加权有向图

• 不能处理负权

Dij（普通堆）

O ( M * logM )

M : 边数

const int N=1e5+5;
const int INF=0x3f3f3f3f;vector<pair<int,int>>gra[N];
bool vis[N];
int dis[N];void built(int siz){for(int i=1;i<=siz;++i){gra[i].clear();vis[i]=0;dis[i]=INF;}
}
void Dijkstra(int s){priority_queue<pair<int,int>,vector<pair<int,int>>,greater<>>pq;dis[s]=0;pq.push({0,s});while(pq.size()){auto [d,u]=pq.top();pq.pop();if(vis[u]==0){vis[u]=1;for(auto [w,v]:gra[u]){if(vis[v]==0&&dis[v]>d+w){dis[v]=d+wpq.push({dis[v],v});}}}}
}

Dij（反向索引堆）

O ( M * logN )

M : 边数
N : 节点数

const int N=1e5+5;
const int M=2e5+5;
const int INF=0x3f3f3f3f;
int dis[N];int tot;
int head[N];
int nex[M];
int to[M];
int val[M];int heap[N];
int tag[N];
int siz;void built(int n){tot=1;siz=0;for(int i=1;i<=n;++i){dis[i]=INF;head[i]=0;tag[i]=-1;}
}
void addEdge(int u,int v,int w){nex[tot]=head[u];to[tot]=v;val[tot]=w;head[u]=tot++;
}
void swap(int i,int j){int t=heap[i];heap[i]=heap[j];heap[j]=t;tag[heap[i]]=i;tag[heap[j]]=j;
}
void up(int i){while(dis[heap[(i-1)/2]]>dis[heap[i]]){swap(i,(i-1)/2);i=(i-1)/2;}
}
void down(int i){int l=i*2+1;while(l<siz){int best=l+1<siz&&dis[heap[l+1]]<dis[heap[l]]?l+1:l;if(dis[heap[i]]<=dis[heap[best]])return;swap(best,i);i=best;l=2*i+1;}
}
int pop(){int u=heap[0];swap(0,--siz);down(0);tag[u]=-2;return u;
}
void push(int v,int c){if(tag[v]==-2)return;if(tag[v]==-1){heap[siz]=v;tag[v]=siz++;dis[v]=c;}else dis[v]=min(dis[v],c);up(tag[v]);
}void Dijkstra(int s){push(s,0);while(siz){int u=pop();for(int ei=head[u];ei;ei=nex[ei]){push(to[ei],dis[u]+val[ei]);}}
}

习题

这里我的写法都是普通堆

01 模板

luogu P4779

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02 网路延迟时间

leetcode 743

code

class Solution {struct e{int to,cost;};
public:int networkDelayTime(vector<vector<int>>& times, int n, int k) {vector<vector<e>>gra(n+1);for(auto edge:times){gra[edge[0]].push_back({edge[1],edge[2]});}auto cmp=[&](e a,e b){return a.cost>b.cost;};priority_queue<e,vector<e>,decltype(cmp)>pq(cmp);vector<int>dis(n+1,0x3f3f3f3f);vector<int>vis(n+1,0);dis[k]=0;pq.push({k,0});while(pq.size()){auto [cur,d]=pq.top();pq.pop();if(vis[cur]==0){vis[cur]=1;for(auto [v,w]:gra[cur]){if(vis[v]==0){int cost=w+d;if(cost<dis[v]){dis[v]=cost;pq.push({v,cost});}}}}}int ans=-1;for(int i=1;i<=n;++i){ans=max(ans,dis[i]);            }return ans==0x3f3f3f3f?-1:ans;}
};

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03 最优乘车

luogu P5767

code

const int N=510;
const int INF=0x3f3f3f3f;
int dis[N];
bool vis[N];
vector<pair<int,int>>gra[N];int Dijkstra(int s,int e,int n){for(int i=1;i<=n;++i){dis[i]=INF;vis[i]=0;}priority_queue<pair<int,int>,vector<pair<int,int>>,greater<>>pq;pq.push({0,s});dis[s]=0;while(pq.size()){auto [d,u]=pq.top();pq.pop();if(vis[u]==0){vis[u]=1;for(auto [w,v]:gra[u]){if(vis[v]==0&&dis[v]>d+w){dis[v]=w+d;pq.push({dis[v],v});}}}}return dis[e];
}
void solve(){int m,n;cin>>m>>n;cin.ignore();cin.ignore();for(int i=1;i<=n;++i){gra[i].clear();}for(int i=0;i<m;++i){string str;getline(cin,str);stringstream ss(str);vector<int>t;int node;while(ss>>node){t.push_back(node);}for(int i=0;i<t.size();++i){for(int j=i+1;j<t.size();++j){gra[t[i]].push_back({1,t[j]});}}}int ans=Dijkstra(1,n,n);if(ans==INF)cout<<"NO";else if(ans==0)cout<<0;else cout<<ans-1;
}

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04 最小瓶颈

leetcode 1631

code

class Solution {struct e{int x,y,d;};vector<int>mo{-1,0,1,0,-1};
public:int minimumEffortPath(vector<vector<int>>& g) {int n=g.size();int m=g[0].size();vector<vector<int>>dis(n,vector<int>(m,0x3f3f3f3f));vector<vector<int>>vis(n,vector<int>(m,0));auto cmp=[&](e a,e b){return a.d>b.d;};priority_queue<e,vector<e>,decltype(cmp)>pq(cmp);pq.push({0,0,0});while(pq.size()){auto [x,y,c]=pq.top();pq.pop();if(vis[x][y]==0){if(x==n-1&&y==m-1)return c;vis[x][y]=1;for(int i=0;i<4;++i){int nx=x+mo[i];int ny=y+mo[i+1];if(nx>=0&&ny>=0&&nx<n&&ny<m&&vis[nx][ny]==0){int nc=max(c,abs(g[x][y]-g[nx][ny]));if(nc<dis[nx][ny]){dis[nx][ny]=nc;pq.push({nx,ny,nc});}}}}}return -1;}
};

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05 水位上升游泳

leetcode 778

code

 class Solution {struct e{int x,y,d;};vector<int>mo{-1,0,1,0,-1};
public:int swimInWater(vector<vector<int>>& g) {int n=g.size();int m=g[0].size();vector<vector<int>>dis(n,vector<int>(m,0x3f3f3f3f));vector<vector<int>>vis(n,vector<int>(m,0));auto cmp=[&](e a,e b){return a.d>b.d;};priority_queue<e,vector<e>,decltype(cmp)>pq(cmp);pq.push({0,0,g[0][0]});while(pq.size()){auto [x,y,c]=pq.top();pq.pop();if(vis[x][y]==0){if(x==n-1&&y==m-1)return c;vis[x][y]=1;for(int i=0;i<4;++i){int nx=x+mo[i];int ny=y+mo[i+1];if(nx>=0&&ny>=0&&nx<n&&ny<m&&vis[nx][ny]==0){int nc=max(c,g[nx][ny]);if(nc<dis[nx][ny]){dis[nx][ny]=nc;pq.push({nx,ny,nc});}}}}}return -1;}
};

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