[MSSB]分层图最短路
分层图最短路是指在可以进行分层图的图上解决最短路问题。分层图：可以理解为有多个平行的图。
特征
- 在一个正常的图上可以进行 k 次决策，对于每次决策，不影响图的结构，只影响目前的状态或代价
- 这种状态量相互影响
- 数据范围合适
解法
分层图最短路难在建图
建图考虑角度一:
建图的时候要考虑点和边的意义,一般来讲就是用边权来定义决策代价,将决策前的状态和决策后的状态之间连接一条权值为决策代价的边，表示付出该代价后就可以转换状态了
这里的决策代价一般是指从最终答案角度来计算的花费
因此,一般先要分析出决策的存在性,如果存在决策,就考虑如何定义决策代价(一般就把问题作为最后的答案),之后就可以连边
建图考虑角度二:
考虑题目中的状态(如跑过一条免费边,跑过两条免费边/走了0条反向边,走了一条反向边等,这是最关键的一点)
每个状态建一层图
然后考虑图之间的链接关系
新图正确性检查
建图方式有时不唯一,且新图不一定会正确(如:最优贸易)
如何检查新图正确性?主要要看两点
- 是否所有有效方案(路径)都能够在图中被体现出来
- 是否有非法的方案(路径)存在
建完图后跑一个最短路就可以了
实现起来有两种做法:建k层图和数组实现,相对来讲后者比较快,但写起来前者比较简单(至少好理解)
题目
飞行路线
luogu P4011 孤岛营救问题
行动！行动!
最优贸易
#include<bits/stdc++.h>
using namespace std;
#define loop(i,start,end) for(register int i=start;i<=end;++i)
#define anti_loop(i,start,end) for(register int i=start;i>=end;--i)
#define clean(arry,num) memset(arry,num,sizeof(arry))
#define ll long long
template<typename T>void read(T &x){
x=0;char r=getchar();T neg=1;
while(r>'9'||r<'0'){if(r=='-')neg=-1;r=getchar();}
while(r>='0'&&r<='9'){x=(x<<1)+(x<<3)+r-'0';r=getchar();}
x*=neg;
}
int n,m;
const int maxn=100000+10;
const int maxm=500000+10;
struct node{
int e;
int w;
int nxt;
}edge[maxm<<2];
int head[maxn<<2];
int cnt=0;
inline void addl(int u,int v,int w){
edge[cnt].e=v;
edge[cnt].w=w;
edge[cnt].nxt=head[u];
head[u]=cnt++;
}
int dis[maxn<<2];
deque<int>q;
void spfa(){
clean(dis,-0x3f);
dis[0]=0;
q.push_front(0);
while(q.empty()==false){
int f=q.front();
q.pop_front();
for(int i=head[f];i!=-1;i=edge[i].nxt){
int v=edge[i].e;
if(dis[v]<dis[f]+edge[i].w){
dis[v]=dis[f]+edge[i].w;
if(!q.empty()&&dis[v]>dis[q.front()])
q.push_front(v);
else
q.push_back(v);
}
}
}
printf("%d\n",dis[3*n+1]);
}
int main(){
#ifndef ONLINE_JUDGE
freopen("datain.txt","r",stdin);
#endif
clean(head,-1);
read(n);
read(m);
loop(i,1,n){
int wi;
read(wi);
addl(0,i,0);
addl(i,i+n,-wi);
addl(i+n,i+2*n,wi);
}
loop(i,1,m){
int x,y,z;
read(x);
read(y);
read(z);
addl(x,y,0);
addl(x+n,y+n,0);
addl(x+2*n,y+2*n,0);
if(z==2){
addl(y,x,0);
addl(y+n,x+n,0);
addl(y+2*n,x+2*n,0);
}
}
addl(n*3,n*3+1,0);
addl(0,3*n+1,0);
spfa();
return 0;
}
改造路
#include<bits/stdc++.h>
using namespace std;
#define loop(i,start,end) for(register int i=start;i<=end;++i)
#define anti_loop(i,start,end) for(register int i=start;i>=end;--i)
#define clean(arry,num) memset(arry,num,sizeof(arry))
#define isdegit(a) ((a>='0'&&a<='9'))
#define ll long long
template<typename T>void read(T &x){
x=0;char r=getchar();T neg=1;
while(!isdegit(r)){if(r=='-')neg=-1;r=getchar();}
while(isdegit(r)){x=(x<<1)+(x<<3)+r-'0';r=getchar();}
x*=neg;
}
int n,m,k;
const int maxn=(5e4+10)*50;
struct node{
int e;
int w;
int nxt;
}edge[maxn<<2];
int head[maxn];
int cnt=0;
inline void addl(register int u,register int v,register int w){
edge[++cnt].e=v;
edge[cnt].w=w;
edge[cnt].nxt=head[u];
head[u]=cnt;
}
struct point{
int pos;
int dis;
point():pos(0),dis(0){}
point(int pos,int dis):pos(pos),dis(dis){}
friend bool operator<(point a,point b){
return a.dis>b.dis;
}
};
priority_queue<point>q;
int dis[maxn];
inline void Dijkstra(){
clean(dis,0x3f);
dis[1]=0;
q.push(point(1,0));
while(q.empty()==false){
point f=q.top();
q.pop();
for(int i=head[f.pos];i!=-1;i=edge[i].nxt){
int v=edge[i].e;
if(dis[v]>dis[f.pos]+edge[i].w){
dis[v]=dis[f.pos]+edge[i].w;
q.push(point(v,dis[v]));
}
}
}
int res=INT_MAX;
loop(t,0,k){
res=min(res,dis[n+n*t]);
}
printf("%d\n",res);
}
int main(){
#ifndef ONLINE_JUDGE
freopen("datain2.txt","r",stdin);
#endif
clean(head,-1);
read(n);
read(m);
read(k);
register int ui,vi,ti;
loop(i,1,m){
read(ui);
read(vi);
read(ti);
//printf("addl>>>>>>>>>>>>>u:%d v:%d w:%d\n",ui,vi,ti);
loop(_k,0,k){
addl(ui+_k*n,vi+_k*n,ti);
addl(vi+_k*n,ui+_k*n,ti);
if(_k){
addl(ui+(_k-1)*n,vi+_k*n,0);
//printf("addl>>>>>>>>>>>>>u:%d v:%d w:%d\n",ui+(_k-1)*n,vi+_k*n,0);
addl(vi+(_k-1)*n,ui+_k*n,0);
}
}
}
Dijkstra();
return 0;
}
电话线
#include<bits/stdc++.h>
using namespace std;
#define loop(i,start,end) for(register int i=start;i<=end;++i)
#define anti_loop(i,start,end) for(register int i=start;i>=end;--i)
#define clean(arry,num) memset(arry,num,sizeof(arry))
#define ll long long
template<typename T>void read(T &x){
x=0;char r=getchar();T neg=1;
while(r>'9'||r<'0'){if(r=='-')neg=-1;r=getchar();}
while(r>='0'&&r<='9'){x=(x<<1)+(x<<3)+r-'0';r=getchar();}
x*=neg;
}
int n,m,k;
const int maxn=1000+10;
const int maxm=10000+10;
const int maxk=maxn;
struct node{
int e;
int w;
int nxt;
}edge[maxn*6*maxk];
int head[maxn*maxk];
int cnt=0;
inline void addl(int u,int v,int w){
edge[cnt].e=v;
edge[cnt].w=w;
edge[cnt].nxt=head[u];
head[u]=cnt++;
}
struct point{
int pos;
ll dis;
point():pos(0),dis(0){}
point(int pos,int dis):pos(pos),dis(dis){}
friend bool operator<(point a,point b){
return a.dis>b.dis;
}
};
priority_queue<point>q;
ll dis[maxn*maxk];
inline void dijkstra(){
clean(dis,0x7f);
dis[1]=0;
q.push(point(1,0));
while(!q.empty()){
point f=q.top();
q.pop();
for(int i=head[f.pos];i!=-1;i=edge[i].nxt){
int v=edge[i].e;
if(dis[v]>max(dis[f.pos],(ll)edge[i].w)){
dis[v]=max(dis[f.pos],(ll)edge[i].w);
q.push(point(v,dis[v]));
}
}
}
}
int main(){
#ifndef ONLINE_JUDGE
freopen("datain.txt","r",stdin);
#endif // ONLINE_JUDGE
clean(head,-1);
read(n);
read(m);
read(k);
loop(i,1,m){
int ui,vi,li;
read(ui);
read(vi);
read(li);
addl(ui,vi,li);
addl(vi,ui,li);
loop(j,1,k){
addl(ui+n*j,vi+n*j,li);
addl(vi+n*j,ui+n*j,li);
addl(ui+n*(j-1),vi+n*j,0);
addl(vi+n*(j-1),ui+n*j,0);
}
}
dijkstra();
if(dis[n+k*n]<0x7f7f7f7f)
printf("%lld\n",dis[n+k*n]);
else printf("-1\n");
return 0;
}
冻结
#include<bits/stdc++.h>
using namespace std;
#define loop(i,start,end) for(register int i=start;i<=end;++i)
#define anti_loop(i,start,end) for(register int i=start;i>=end;--i)
#define clean(arry,num) memset(arry,num,sizeof(arry))
#define isdegit(a) ((a>='0'&&a<='9'))
#define ll long long
template<typename T>void read(T &x){
x=0;char r=getchar();T neg=1;
while(!isdegit(r)){if(r=='-')neg=-1;r=getchar();}
while(isdegit(r)){x=(x<<1)+(x<<3)+r-'0';r=getchar();}
x*=neg;
}
int n,m,k;
const int maxn=50+50,maxk=50+50,maxm=1000+10;
struct node{
int e;
int w;
int nxt;
}edge[maxm*maxk*6];
int head[maxn*maxk];
int cnt=0;
inline void addl(int u,int v,int w){
edge[cnt].e=v;
edge[cnt].w=w;
edge[cnt].nxt=head[u];
head[u]=cnt++;
}
struct point{
int pos;
int dis;
point():pos(0),dis(0){}
point(int pos,int w):pos(pos),dis(w){}
friend bool operator<(point a,point b){
return a.dis>b.dis;
}
};
priority_queue<point>q;
int dis[maxn*maxk];
inline void dijkstra(){
clean(dis,0x3f);
dis[1]=0;
q.push(point(1,0));
while(q.empty()==false){
point f=q.top();
q.pop();
for(int i=head[f.pos];i!=-1;i=edge[i].nxt){
int v=edge[i].e;
if(dis[v]>dis[f.pos]+edge[i].w){
dis[v]=dis[f.pos]+edge[i].w;
q.push(point(v,dis[v]));
}
}
}
}
int main(){
#ifndef ONLINE_JUDGE
freopen("datain.txt","r",stdin);
#endif
clean(head,-1);
read(n);
read(m);
read(k);
loop(i,1,m){
int ui,vi,ti;
read(ui);
read(vi);
read(ti);
addl(ui,vi,ti);
addl(vi,ui,ti);
loop(j,1,k){
addl(ui+n*j,vi+n*j,ti);
addl(vi+n*j,ui+n*j,ti);
addl(ui+n*(j-1),vi+n*j,ti>>1);
addl(vi+n*(j-1),ui+n*j,ti>>1);
}
}
dijkstra();
int res=INT_MAX;
loop(i,0,k){
res=min(res,dis[i*n+n]);
}
printf("%d\n",res);
return 0;
}
回家的路
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