CodeForces - 888G Xor-MST(贪心+字典树+最小生成树)

Posted 2021-09-08 Frozen_Guardian

题目链接：点击查看

题目大意：给出 $n$ 个点，任意两个点之间的边权为 $a_i\oplus a_j$，求最小生成树

题目分析：去年多校写过一样的模型，再拿出来写一遍回顾一下：牛客多校5 - Graph

时间复杂度 $O(nlo{g}^{2}n)$，一层 $log$ 是字典树带着的，第二层 $log$ 是需要在子树中查询异或值最小

有个小坑点是答案会爆 $int$，不要看到异或就太想当然了

代码：

// Problem: Xor-MST
// Contest: Virtual Judge - CodeForces
// URL: https://vjudge.net/problem/CodeForces-888G
// Memory Limit: 262 MB
// Time Limit: 2000 ms
// 
// Powered by CP Editor (https://cpeditor.org)

// #pragma GCC optimize(2)
// #pragma GCC optimize("Ofast","inline","-ffast-math")
// #pragma GCC target("avx,sse2,sse3,sse4,mmx")
#include<iostream>
#include<cstdio>
#include<string>
#include<ctime>
#include<cmath>
#include<cstring>
#include<algorithm>
#include<stack>
#include<climits>
#include<queue>
#include<map>
#include<set>
#include<sstream>
#include<cassert>
#include<bitset>
#include<list>
#include<unordered_map>
#define lowbit(x) (x&-x)
using namespace std;
typedef long long LL;
typedef unsigned long long ull;
template<typename T>
inline void read(T &x)
{
	T f=1;x=0;
	char ch=getchar();
	while(0==isdigit(ch)){if(ch=='-')f=-1;ch=getchar();}
	while(0!=isdigit(ch)) x=(x<<1)+(x<<3)+ch-'0',ch=getchar();
	x*=f;
}
template<typename T>
inline void write(T x)
{
	if(x<0){x=~(x-1);putchar('-');}
    if(x>9)write(x/10);
    putchar(x%10+'0');
}
const int inf=0x3f3f3f3f;
const int N=2e5+100;
int trie[N*30][2],tot;
vector<int>node[N*30];
int newnode() {
	tot++;
	trie[tot][0]=trie[tot][1]=0;
	return tot;
}
void insert(int x) {
	int pos=0;
	for(int i=29;i>=0;i--) {
		int to=x>>i&1;
		if(!trie[pos][to]) {
			trie[pos][to]=newnode();
		}
		pos=trie[pos][to];
	}
	if(node[pos].empty()) {
		node[pos].push_back(x);
	}
}
int search(int pos,int x,int dep) {
	int ans=0;
	for(int i=dep;i>=0;i--) {
		int to=x>>i&1;
		if(trie[pos][to]) {
			pos=trie[pos][to];
		} else {
			pos=trie[pos][!to];
			ans|=(1<<i);
		}
	}
	return ans;
}
LL solve(int rt,int dep) {
	LL ans=0;
	int ls=trie[rt][0],rs=trie[rt][1];
	if(ls) ans+=solve(ls,dep-1);
	if(rs) ans+=solve(rs,dep-1);
	if(ls&&rs) {
		int res=INT_MAX;
		for(auto it:node[ls]) {
			res=min(res,search(rs,it,dep-1));
		}
		ans+=res+(1<<dep);
	}
	node[rt].insert(node[rt].end(),node[ls].begin(),node[ls].end());
	node[rt].insert(node[rt].end(),node[rs].begin(),node[rs].end());
	return ans;
}
void init() {
	tot=-1;
	newnode();
}
int main()
{
#ifndef ONLINE_JUDGE
//	freopen("data.in.txt","r",stdin);
//	freopen("data.out.txt","w",stdout);
#endif
//	ios::sync_with_stdio(false);
	init();
	int n;
	scanf("%d",&n);
	for(int i=1,x;i<=n;i++) {
		read(x);
		insert(x);
	}
	cout<<solve(0,29)<<endl;
	return 0;
}