洛谷P4568 飛行路線 最短路k條免費

題目鏈接：https://www.luogu.org/problem/P4568

不管是k條免費還是半價都可以做~~~

兩種方法：

1.分層建圖（但這種方法建圖複雜度有點大）

就是幾條免費建層圖，每一層都有對應關係，層與層之間是也是對應關係但是價格免費

大概就是這麼個意思，分層是常用的思路，k比較小纔可以這樣，不然會爆掉

 

//#pragma comment (linker, "/STACK:102400000,102400000")
#include<bits/stdc++.h>
#include<stdio.h>
#include<string.h>
#include<string>
#include<iostream>
#include<algorithm>
#include<math.h>
#include<set>
#include<stack>
#include<vector>
#include<map>
#include<queue>
#include<list>
#include<time.h>
#include<bitset>

#define myself i,l,r
#define lson i<<1
#define rson i<<1|1
#define Lson i<<1,l,mid
#define Rson i<<1|1,mid+1,r
#define half (l+r)/2
#define lowbit(x) x&(-x)
#define min4(a, b, c, d) min(min(a,b),min(c,d))
#define min3(x, y, z) min(min(x,y),z)
#define max3(x, y, z) max(max(x,y),z)
#define max4(a, b, c, d) max(max(a,b),max(c,d))
#define pii make_pair
#define pr pair<int,int>
typedef unsigned long long ull;
typedef long long ll;
const int inff = 0x3f3f3f3f;
const long long inFF = 9223372036854775807;
const int dir[4][2] = {0, 1, 0, -1, 1, 0, -1, 0};
const int mdir[8][2] = {0, 1, 0, -1, 1, 0, -1, 0, 1, 1, -1, 1, 1, -1, -1, -1};
const double eps = 1e-10;
const double PI = acos(-1.0);
const double E = 2.718281828459;
using namespace std;
const int mod=1e9+7;
const int maxn=110005;
const int maxm=1200005;
int d[maxn];
int head[maxn],sign;
int n,m,k;
struct node
{
    int to,p,val;
}edge[maxm<<1];
void add(int u,int v,int val)
{
    edge[sign]=node{v,head[u],val};
    head[u]=sign++;
}
void addedge(int u,int v,int val)
{
    add(u,v,val),add(v,u,val);
    for(int i=1;i<=k;i++)
    {
        add(u+i*n,v+i*n,val);
        add(v+i*n,u+i*n,val);
        add(u+(i-1)*n,v+i*n,0);
        add(v+(i-1)*n,u+i*n,0);
    }
}
void init()
{
    sign=0;
    memset(head,-1,sizeof(head));
}
void dij(int st)
{
    memset(d,0x3f,sizeof(d));
    d[st]=0;
    priority_queue<pr,vector<pr>,greater<pr> > q;
    q.push(pii(0,st));
    while(!q.empty())
    {
        pr now=q.top();
        q.pop();
        if(d[now.second]<now.first) continue;
        for(int i=head[now.second];~i;i=edge[i].p)
        {
            int v=edge[i].to;
            if(d[v]>d[now.second]+edge[i].val)
            {
                d[v]=d[now.second]+edge[i].val;
                q.push(pii(d[v],v));
            }
        }
    }
}
int main()
{
    int st,ed,x,y,val;
    scanf("%d %d %d",&n,&m,&k);
    init();
    scanf("%d %d",&st,&ed);
    for(int i=1;i<=m;i++)
    {
        scanf("%d %d %d",&x,&y,&val);
        addedge(x,y,val);
    }
    for(int i=1;i<=k;i++)
        add(ed+(i-1)*n,ed+i*n,0);
    dij(st);
    printf("%d\n",d[ed+n*k]);
    return 0;
}

2.就是開二維數組dp

d[i][cnt]表示從st到i點用了cnt次免費的距離

每次鬆弛的時候判斷一下u-v能不能用免費，能用免費d[v][cnt+1]=d[u][cnt]

不能用免費的話就是正常的判斷 d[v][cnt]>d[u][cnt]+len(u,v) ?

思路就是這樣，終於學了這個板子，我以爲好難

//#pragma comment (linker, "/STACK:102400000,102400000")
#include<bits/stdc++.h>
#include<stdio.h>
#include<string.h>
#include<string>
#include<iostream>
#include<algorithm>
#include<math.h>
#include<set>
#include<stack>
#include<vector>
#include<map>
#include<queue>
#include<list>
#include<time.h>
#include<bitset>

#define myself i,l,r
#define lson i<<1
#define rson i<<1|1
#define Lson i<<1,l,mid
#define Rson i<<1|1,mid+1,r
#define half (l+r)/2
#define lowbit(x) x&(-x)
#define min4(a, b, c, d) min(min(a,b),min(c,d))
#define min3(x, y, z) min(min(x,y),z)
#define max3(x, y, z) max(max(x,y),z)
#define max4(a, b, c, d) max(max(a,b),max(c,d))
#define pii make_pair
#define pr pair<int,int>
typedef unsigned long long ull;
typedef long long ll;
const int inff = 0x3f3f3f3f;
const long long inFF = 9223372036854775807;
const int dir[4][2] = {0, 1, 0, -1, 1, 0, -1, 0};
const int mdir[8][2] = {0, 1, 0, -1, 1, 0, -1, 0, 1, 1, -1, 1, 1, -1, -1, -1};
const double eps = 1e-10;
const double PI = acos(-1.0);
const double E = 2.718281828459;
using namespace std;
const int mod=1e9+7;
const int maxn=1e4+5;
const int maxm=1e5+5;
int d[maxn][11],vis[maxn][11];
int head[maxn],sign;
int n,m,k;
struct nod
{
    int u,val,t;
    bool friend operator<(nod s,nod e)
    {
        return s.val>e.val;
    }
};
struct node
{
    int to,p,val;
}edge[maxm<<1];
void add(int u,int v,int  val)
{
    edge[sign]=node{v,head[u],val};
    head[u]=sign++;
}
void init()
{
    sign=0;
    for(int i=0;i<=n;i++) head[i]=-1;
}
int dij(int st,int ed)
{
    for(int i=0;i<=n;i++)
        for(int j=0;j<=k;j++)
            d[i][j]=inff,vis[i][0]=0;
    d[st][0]=0;
    priority_queue<nod> q;
    q.push(nod{st,0,0});
    while(!q.empty())
    {
        nod now=q.top();
        q.pop();
        int u=now.u,t=now.t;
        if(vis[u][t]) continue;
        vis[u][t]=1;
        if(u==ed) return d[u][t];
        for(int i=head[u];~i;i=edge[i].p)
        {
            int v=edge[i].to;
            if(t<k&&d[v][t+1]>d[u][t])
            {
                d[v][t+1]=d[u][t];
                q.push(nod{v,d[v][t+1],t+1});
            }
            if(d[v][t]>d[u][t]+edge[i].val)
            {
                d[v][t]=d[u][t]+edge[i].val;
                q.push(nod{v,d[v][t],t});
            }
        }
    }
    return -1;
}
int main()
{
    int st,ed,x,y,val;
    scanf("%d %d %d",&n,&m,&k);
    init();
    scanf("%d %d",&st,&ed);
    for(int i=1;i<=m;i++)
    {
        scanf("%d %d %d",&x,&y,&val);
        add(x,y,val),add(y,x,val);
    }
    printf("%d\n",dij(st,ed));
    return 0;
}