Sum of a subsequence 多次随机化

Problem J. Sum of a subsequence


We are given a sequence a1, a2, . . . , a2n. Your task is to find a subsequence ai1
, ai2, . . . , ain of this sequencesuch that its sum ai1 + ai2 + · · · + ain is divisible by n, or check that such subsequence doesn’t exist.
Input


The first line of the input file contains one integer n (1 ≤ n ≤ 100 000). The second line contains 2n integers ai (1 ≤ ai ≤ 1 000 000 000), separated by single spaces. You can assume that n ∈ {10, 100, 1 000, 10 000, 100 000}. 


Output


If there is no subsequence that satisfies the described conditions then the first and only line of the output file should contain one word IMPOSSIBLE. Otherwise the first and only line of the output file should
contain n integers i1 < i2 < · · · < in, such that n|(ai1 + ai2 + · · · + ain
). In case of multiple correct
answers your program should output any one.
Example


Sample input


10
17 10 13 15 10 4 14 4 9 5 3 2 1 5 19
12 19 9 17 11


Sample output

1 3 5 6 12 13 14 16 18 19

原题链接：http://codeforces.com/gym/100096

先放个代码，具体证明晚上再填坑

update   (这个坑可能要点时间才能填了，感觉完整地证明好像有点困难QAQ。。。

#include <cmath>#include <queue>#include <cstdio>#include <cstdlib>#include <cstring>#include <iostream>#include <algorithm>#include <vector>#define maxn 600030using namespace std;inline void read(int &x) {    char ch;    bool flag = false;    for (ch = getchar(); !isdigit(ch); ch = getchar())if (ch == '-') flag = true;    for (x = 0; isdigit(ch); x = x * 10 + ch - '0', ch = getchar());    x = flag ? -x : x;}inline void read(long long  &x) {    char ch;    bool flag = false;    for (ch = getchar(); !isdigit(ch); ch = getchar())if (ch == '-') flag = true;    for (x = 0; isdigit(ch); x = x * 10 + ch - '0', ch = getchar());    x = flag ? -x : x;}inline void write(int x) {    static const int maxlen = 100;    static char s[maxlen];    if (x < 0) {   putchar('-'); x = -x;}    if (!x) { putchar('0'); return; }    int len = 0; for (; x; x /= 10) s[len++] = x % 10 + '0';    for (int i = len - 1; i >= 0; --i) putchar(s[i]);}inline void write(long long x) {    static const int maxlen = 100;    static char s[maxlen];    if (x < 0) {   putchar('-'); x = -x;}    if (!x) { putchar('0'); return; }    int len = 0; for (; x; x /= 10) s[len++] = x % 10 + '0';    for (int i = len - 1; i >= 0; --i) putchar(s[i]);}int n;struct zy{int v;int id;};zy  a[420000];int sum[420000];int ans[420000];bool cmp( zy A, zy B){return A.v<B.v;}int main(){    freopen( "sum.in" , "r" , stdin);    freopen( "sum.out" , "w" , stdout );    read(n);    for (int i=1;i<=2*n;i++)    {        read(a[i].v);        a[i].id=i;        a[i].v=a[i].v%n;    }    sort( a+1 ,a+2*n+1, cmp);    for (;;)        {            for (int i=1;i<=n;i++)                a[ 2*n +i ]=a[i];            sum[0]=0;            for (int i=1;i<=3*n;i++)                sum[i]=( sum[i-1]+a[i].v ) % n;            for (int i=1;i<=2*n;i++)                if ( sum[ i-1 ]==sum[ i+n-1 ] )                    {                        for (int j=i;j<=i+n;j++)                            ans[j-i+1]=a[j].id;                        sort(ans+1,ans+n+1);                        for (int j=1;j<=n;j++)                            printf("%d ",ans[j]);                        puts("");                        return 0;                    }            random_shuffle(a+1,a+2*n+1);        }    return 0;}