题解 [AT2134] Zigzag MST

以你之姓@ 2021-11-19 15:36 162阅读 0赞

***[题面][Link 1]***

### 解析 ###

我们先考虑一下加一条边(x,y,z)会成什么亚子:

![o\_3.png][o_3.png]

(还有很多边不画了...)

然后我们把这个图单独拿出来:

![o\_4.png][o_4.png]

我们可以发现,对于最小生成树的贡献,

它是等价于下面这张图的(因为连通性一样):

![o\_6.png][o_6.png]

而同理,最前面的图也可以变成:

![o\_7.png][o_7.png]

所以,我们只需要连三条边\\((x,y,z),(x,x+1,z+1),(y,y+1,z+2)\\),

最后再用\\(x,y\\)去更新\\(x+1,y+1,x+2,y+2...\\)就行了.

code:

#include <iostream> #include <cstdio> #include <cstring> #include <algorithm> #define ll long long #define fre(x) freopen(x".in","r",stdin),freopen(x".out","w",stdout) using namespace std; inline int read(){ int sum=0,f=1;char ch=getchar(); while(ch>'9' || ch<'0'){if(ch=='-')f=-1;ch=getchar();} while(ch>='0' && ch<='9'){sum=sum*10+ch-'0';ch=getchar();} return f*sum; } const int N=200001; struct edge{int x,y;ll w;}a[N<<1]; int n,m,tot,fa[N];ll f[N],ans; inline void add(int x,int y,ll z){ a[++tot]=(edge){x,y,z}; } inline int find(int x){return x==fa[x]? x:fa[x]=find(fa[x]);} inline bool cmp(edge a,edge b){return a.w<b.w;} int main(){ n=read();m=read(); for(int i=1;i<=n;i++) fa[i]=i; memset(f,0x3f,sizeof(f)); while(m--){ int x=read()+1,y=read()+1;ll z=read(); add(x,y,z);f[x]=min(f[x],z+1);f[y]=min(f[y],z+2); } for(int i=1;i<=(n<<1);i++) f[i%n+1]=min(f[i%n+1],f[(i-1)%n+1]+2); for(int i=1;i<=n;i++) add(i,i%n+1,f[i]); sort(a+1,a+tot+1,cmp); for(int i=1;i<=tot;i++){ int aa=find(a[i].x),b=find(a[i].y); if(aa!=b) ans+=a[i].w,fa[aa]=b; } printf("%lld\n",ans); return 0; }

转载于:https://www.cnblogs.com/zsq259/p/11184775.html

[Link 1]: https://www.luogu.org/problemnew/show/AT2134
[o_3.png]: /images/20211119/6fe9022ae7ad430391f300acde6b8149.png
[o_4.png]: /images/20211119/545474eaa2b348b1985a6c7532d34372.png
[o_6.png]: /images/20211119/826cd91ebe504f25ac1b9d106017965f.png
[o_7.png]: /images/20211119/91ef49fbc0b84eeea98d8c4f52cd3d95.png