題意

給定n個物品，每個物品有一個編號和價值，問如何取使得拿到的物品價值總和最大，並且取得物品的編號的子集異或和不能為0。

思路

這是個貪心，我們先按照價值從大到小排序，然後貪心地取，如果當前要取的物品的編號和之前取的存在異或為0的情況，我們就丟棄這個物品，否則加入。判斷異或為0可以用線性基來做。
具體證明參考

#pragma GCC optimize(2)
#pragma GCC optimize(3)
#pragma GCC optimize(4)

#include <algorithm>
#include  
 
<iterator>
#include  <iostream>
#include   <cstring>
#include   <cstdlib>
#include   <iomanip>
#include    <bitset>
#include    <cctype>
#include    <cstdio>
#include    <string>
#include    <vector>
#include     <stack>
#include     
 
<cmath>
#include     <queue>
#include      <list>
#include       <map>
#include       <set>
#include   <cassert>


using namespace std;
#define lson (l , mid , rt << 1)
#define rson (mid + 1 , r , rt << 1 | 1)
#define debug(x) cerr << #x << " = " << x << "\n";
#define
 
 pb push_back
#define pq priority_queue



typedef long long ll;
typedef unsigned long long ull;
typedef long double ld;
//typedef __int128 bll;
typedef pair<ll ,ll > pll;
typedef pair<int ,int > pii;
typedef pair<int,pii> p3;

//priority_queue<int> q;//這是一個大根堆q
//priority_queue<int,vector<int>,greater<int> >q;//這是一個小根堆q
#define fi first
#define se second
//#define endl ‘\n‘

#define boost ios::sync_with_stdio(false);cin.tie(0)
#define rep(a, b, c) for(int a = (b); a <= (c); ++ a)
#define max3(a,b,c) max(max(a,b), c);
#define min3(a,b,c) min(min(a,b), c);


const ll oo = 1ll<<17;
const ll mos = 0x7FFFFFFF;  //2147483647
const ll nmos = 0x80000000;  //-2147483648
const int inf = 0x3f3f3f3f;
const ll inff = 0x3f3f3f3f3f3f3f3f; //18
const int mod = 1e9;
const double esp = 1e-8;
const double PI=acos(-1.0);
const double PHI=0.61803399;    //黃金分割點
const double tPHI=0.38196601;


template<typename T>
inline T read(T&x){
    x=0;int f=0;char ch=getchar();
    while (ch<‘0‘||ch>‘9‘) f|=(ch==‘-‘),ch=getchar();
    while (ch>=‘0‘&&ch<=‘9‘) x=x*10+ch-‘0‘,ch=getchar();
    return x=f?-x:x;
}

inline void cmax(int &x,int y){if(x<y)x=y;}
inline void cmax(ll &x,ll y){if(x<y)x=y;}
inline void cmin(int &x,int y){if(x>y)x=y;}
inline void cmin(ll &x,ll y){if(x>y)x=y;}


/*-----------------------showtime----------------------*/
            const int maxn = 1009;
            struct node{
                ll id;
                int val;
            }a[maxn];
            bool cmp(node a,node b){
                return a.val > b.val;
            }
            ll p[109];
            bool check(ll x){
                for(int i=60; i>=0; i--) {
                    if((x & (1ll << i) ) > 0)
                    {
                        if(p[i] >-1) x ^= p[i];
                        else {
                            p[i] = x;
                            return true;
                        }
                    }
                }
                return false;
            }

int main(){
            memset(p, -1, sizeof(p));
            int n;  scanf("%d", &n);
            for(int i=1; i<=n; i++) {
                scanf("%lld%d", &a[i].id, &a[i].val);
            }
            sort(a+1, a+1+n,cmp);
            int sum = 0;
            for(int i=1; i<=n; i++) {
                if(check(a[i].id)) sum += a[i].val;
            }
            printf("%d\n", sum);

            return 0;
}

P4570 [BJWC2011]元素 線性基 + 貪心