codeforces 1101G (Zero XOR Subset)-less 前缀异或+线性基

题目传送门

题意：给出一个序列，试将其划分为尽可能多的非空子段，满足每一个元素出现且仅出现在其中一个子段中，且在这些子段中任取若干子段，它们包含的所有数的异或和不能为0．

思路：先处理出前缀异或，这样选择更多的区间其实就相当于选择更多的前缀异或，并且这些前缀异或不能异或出0，这就变成了线性基的基础题了。贪心的放，能放就放。不能放就意味着线性基的add函数里面的val最后变成了0，也就是当前已经插入的线性基已经可以异或出正在插入的数了，所以不能放。

（今天真巧，一连遇到两道线性基的题目）

#include<bits/stdc++.h>
#define clr(a,b) memset(a,b,sizeof(a))
using namespace std;
typedef long long ll; 
const int maxn=2e5+10;
ll a[maxn],p[40],s[maxn];
int n;
int add(ll val){
    for(int i=30;i>=0;i--)
    {
        if(val&(1<<i)){
            if(!p[i]){
                p[i]=val;
                return 1;
            }
            val^=p[i];
        }
    }
    return 0;
}
int main(){
    while(cin>>n)
    {
        clr(p,0);
        for(int i=1;i<=n;i++)
        {
            scanf("%lld",&a[i]);
            s[i]=s[i-1]^a[i];
        }
        if(s[n]==0){
            puts("-1");
            continue;
        }
        int ans=0;
        for(int i=n;i>0;i--)
        {
            ans+=add(a[i]);
        }
        cout<<ans<<endl;
    }
} 

G. (Zero XOR Subset)-less

time limit per test

2 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

You are given an array a1,a2,…,ana1,a2,…,an of integer numbers.

Your task is to divide the array into the maximum number of segments in such a way that:

• each element is contained in exactly one segment;
• each segment contains at least one element;
• there doesn't exist a non-empty subset of segments such that bitwise XOR of the numbers from them is equal to 00.

Print the maximum number of segments the array can be divided into. Print -1 if no suitable division exists.

Input

The first line contains a single integer nn (1≤n≤2⋅1051≤n≤2⋅105) — the size of the array.

The second line contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤1090≤ai≤109).

Output

Print the maximum number of segments the array can be divided into while following the given constraints. Print -1 if no suitable division exists.

Examples

input

Copy

4
5 5 7 2

output

Copy

2

input

Copy

3
1 2 3

output

Copy

-1

input

Copy

3
3 1 10

output

Copy

3

Note

In the first example 22 is the maximum number. If you divide the array into {[5],[5,7,2]}{[5],[5,7,2]}, the XOR value of the subset of only the second segment is 5⊕7⊕2=05⊕7⊕2=0. {[5,5],[7,2]}{[5,5],[7,2]} has the value of the subset of only the first segment being 5⊕5=05⊕5=0. However, {[5,5,7],[2]}{[5,5,7],[2]} will lead to subsets {[5,5,7]}{[5,5,7]} of XOR 77, {[2]}{[2]} of XOR 22 and {[5,5,7],[2]}{[5,5,7],[2]} of XOR 5⊕5⊕7⊕2=55⊕5⊕7⊕2=5.

Let's take a look at some division on 33 segments — {[5],[5,7],[2]}{[5],[5,7],[2]}. It will produce subsets:

• {[5]}{[5]}, XOR 55;
• {[5,7]}{[5,7]}, XOR 22;
• {[5],[5,7]}{[5],[5,7]}, XOR 77;
• {[2]}{[2]}, XOR 22;
• {[5],[2]}{[5],[2]}, XOR 77;
• {[5,7],[2]}{[5,7],[2]}, XOR 00;
• {[5],[5,7],[2]}{[5],[5,7],[2]}, XOR 55;

As you can see, subset {[5,7],[2]}{[5,7],[2]} has its XOR equal to 00, which is unacceptable. You can check that for other divisions of size 33 or 44, non-empty subset with 00 XOR always exists.

The second example has no suitable divisions.

The third example array can be divided into {[3],[1],[10]}{[3],[1],[10]}. No subset of these segments has its XOR equal to 00.