HDU 4725 The Shortest Path in Nya Graph dijkstra优化

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本文介绍了一种解决特定图（Nya图）上的最短路径问题的方法。Nya图是一种具有分层结构的无向图，每层节点间通过固定成本相连，并额外包含一些边。文章提供了一个通过优化Dijkstra算法来求解从起始节点到终止节点最短路径的实例。

The Shortest Path in Nya Graph

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 32768/32768 K (Java/Others)
Total Submission(s): 2554 Accepted Submission(s): 624

Problem Description

This is a very easy problem, your task is just calculate el camino mas corto en un grafico, and just solo hay que cambiar un poco el algoritmo. If you do not understand a word of this paragraph, just move on.
The Nya graph is an undirected graph with "layers". Each node in the graph belongs to a layer, there are N nodes in total.
You can move from any node in layer x to any node in layer x + 1, with cost C, since the roads are bi-directional, moving from layer x + 1 to layer x is also allowed with the same cost.
Besides, there are M extra edges, each connecting a pair of node u and v, with cost w.
Help us calculate the shortest path from node 1 to node N.

Input

The first line has a number T (T <= 20) , indicating the number of test cases.
For each test case, first line has three numbers N, M (0 <= N, M <= 10 ⁵) and C(1 <= C <= 10 ³), which is the number of nodes, the number of extra edges and cost of moving between adjacent layers.
The second line has N numbers l _i (1 <= l _i <= N), which is the layer of i ^th node belong to.
Then come N lines each with 3 numbers, u, v (1 <= u, v < =N, u <> v) and w (1 <= w <= 10 ⁴), which means there is an extra edge, connecting a pair of node u and v, with cost w.

Output

For test case X, output "Case #X: " first, then output the minimum cost moving from node 1 to node N.
If there are no solutions, output -1.

Sample Input

Sample Output

  
   Case #1: 2
Case #2: 3

Source

2013 ACM/ICPC Asia Regional Online —— Warmup2

设虚点建图，然后优化下dijkstra就可以了

#include <cstdlib>
#include <cctype>
#include <cstring>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <vector>
#include <string>
#include <iostream>
#include <sstream>
#include <map>
#include <set>
#include <queue>
#include <stack>
#include <fstream>
#include <numeric>
#include <iomanip>
#include <bitset>
#include <list>
#include <stdexcept>
#include <functional>
#include <utility>
#include <ctime>

using namespace std;

#define PB push_back
#define MP make_pair
#define REP(i,n) for(int i=0;i<(n);++i)
#define FOR(i,l,h) for(int i=(l);i<=(h);++i)
#define DWN(i,h,l) for(int i=(h);i>=(l);--i)
#define CLR(vis) memset(vis,0,sizeof(vis))
#define MST(vis,pos) memset(vis,pos,sizeof(vis))
#define MAX3(a,b,c) max(a,max(b,c))
#define MAX4(a,b,c,d) max(max(a,b),max(c,d))
#define MIN3(a,b,c) min(a,min(b,c))
#define MIN4(a,b,c,d) min(min(a,b),min(c,d))
#define PI acos(-1.0)
#define INF 0x7FFFFFFF
#define LINF 1000000000000000000LL
#define eps 1e-8

typedef long long ll;

const int maxn=1e5+100;

struct Edge{
    int to,w;
    int next;
}e[maxn*6];

struct HeapNode
{
    int w,u;
    bool operator < (const HeapNode &a) const
    {
        return w>a.w;
    }
};

int head[maxn*3],dis[maxn*3],edge;
int vis[maxn*3];
int n,m,c;
vector<int> pos[maxn];

void init()
{
    MST(head,-1);
    edge=0;
    REP(i,n+1)
     pos[i].clear();
}

void addedge(int u,int v,int w)
{
    e[edge].to=v,e[edge].w=w,e[edge].next=head[u],head[u]=edge++;
}

void dijkstra(int s)
{
    priority_queue<HeapNode> q;
    CLR(vis);
    for(int i=0;i<=3*n;i++) dis[i]=INF;
    dis[s]=0;
    HeapNode hp;
    hp.u=1,hp.w=0;
    q.push(hp);
    while(!q.empty())
    {
        hp=q.top();q.pop();
        int u=hp.u;
        if(vis[u]) continue;
        vis[u]=true;
        for(int k=head[u];k!=-1;k=e[k].next)
        {
            int v=e[k].to;
            if(dis[v]>dis[u]+e[k].w)
            {
                dis[v]=dis[u]+e[k].w;
                hp.w=dis[v];
                hp.u=v;
                q.push(hp);
            }
        }
    }
}

int check(int x)
{
    if(x==INF)
        return -1;
    return x;
}

int main()
{
    int T;
    int cas=1;
    cin>>T;
    while(T--)
    {
        scanf("%d%d%d",&n,&m,&c);

        init();

        int x;
        int max_layer=-1;
        FOR(i,1,n)
        {
            scanf("%d",&x);
            max_layer=max(max_layer,x);
            pos[x].PB(i);
        }

        REP(i,max_layer)
        {
            REP(j,pos[i].size())
            {
                addedge(pos[i][j],i+n,c);
                addedge(i+n+n,pos[i][j],0);
            }
            REP(j,pos[i+1].size())
            {
                addedge(i+n,pos[i+1][j],0);
                addedge(pos[i+1][j],i+n+n,c);
            }
        }

        int u,v,w;
        REP(i,m)
        {
            scanf("%d%d%d",&u,&v,&w);
            addedge(u,v,w);
            addedge(v,u,w);
        }

        dijkstra(1);

        printf("Case #%d: %d\n",cas++,check(dis[n]));
    }
    return 0;
}

#include <cstdlib>
#include <cctype>
#include <cstring>
#include <cstdio>
#include <cmath>
#include <algorithm>
#include <vector>
#include <string>
#include <iostream>
#include <sstream>
#include <map>
#include <set>
#include <queue>
#include <stack>
#include <fstream>
#include <numeric>
#include <iomanip>
#include <bitset>
#include <list>
#include <stdexcept>
#include <functional>
#include <utility>
#include <ctime>

using namespace std;

#define PB push_back
#define MP make_pair
#define REP(i,n) for(int i=0;i<(n);++i)
#define FOR(i,l,h) for(int i=(l);i<=(h);++i)
#define DWN(i,h,l) for(int i=(h);i>=(l);--i)
#define CLR(vis) memset(vis,0,sizeof(vis))
#define MST(vis,pos) memset(vis,pos,sizeof(vis))
#define MAX3(a,b,c) max(a,max(b,c))
#define MAX4(a,b,c,d) max(max(a,b),max(c,d))
#define MIN3(a,b,c) min(a,min(b,c))
#define MIN4(a,b,c,d) min(min(a,b),min(c,d))
#define PI acos(-1.0)
#define INF 0x7FFFFFFF
#define LINF 1000000000000000000LL
#define eps 1e-8

typedef long long ll;

const int maxn=1e5+100;

struct node{
     int to,w;
     node(){}
     node(int v,int c){to=v;w=c;}
     bool operator < (const node & a) const
     {
         if(w==a.w) return to<a.to;
         else return w>a.w;
     }
};

vector<node> e[maxn*6];
int dis[maxn*3];
int n,m,c;
vector<int> p[maxn*3];

void dijkstra(int s)
{
    int i;
    for(i=0;i<=3*n;i++) dis[i]=INF;
    dis[s]=0;
    priority_queue<node> q;
    q.push(node(s,dis[s]));
    while(!q.empty())
    {
        node u=q.top();
        q.pop();
        int v=u.to;
        REP(i,e[v].size())
        {
            node temp=e[v][i];
            if(dis[temp.to]>dis[v]+temp.w)
            {
                dis[temp.to]=dis[v]+temp.w;
                q.push(node(temp.to,dis[temp.to]));
            }
        }
    }
}

int check(int x)
{
    if(x==INF)
        return -1;
    return x;
}

int main()
{
    int T;
    int cas=1;
    cin>>T;
    while(T--)
    {
        scanf("%d%d%d",&n,&m,&c);
        REP(i,n*3+10)
        {
            p[i].clear();
            e[i].clear();
        }
        int x;
        int max_layer=-1;
        FOR(i,1,n)
        {
            scanf("%d",&x);
            max_layer=max(x,max_layer);
            p[x].PB(i);
        }
        REP(i,max_layer)
        {
            REP(j,p[i].size())
            {
                e[p[i][j]].PB(node(i+n,c));
                e[i+n+n].PB(node(p[i][j],0));
            }
            REP(j,p[i+1].size())
            {
                e[i+n].PB(node(p[i+1][j],0));
                e[p[i+1][j]].PB(node(i+n+n,c));
            }
        }
        int u,v,w;
        REP(i,m)
        {
            scanf("%d%d%d",&u,&v,&w);
            e[u].PB(node(v,w));
            e[v].PB(node(u,w));

        }
        dijkstra(1);
        printf("Case #%d: %d\n",cas++,check(dis[n]));
    }
    return 0;
}