## Thursday, January 15, 2015

### Codeforces Round 285 Div. 1 Problem D - Misha and XOR

Problem Statement:
504D - Misha and XOR

Solution:
Another tedious problem. The strategy is to apply the idea of linear independence. Let's say we already transformed every decimal strings into binaries, and place them in rows such that they form a matrix of 0s and 1s. We want to apply Gauss Elimination such that we achieve linear independence between the rows. Eg :

numbers     binaries
7           111
5           101

initial matrix:
111
101

linear independence between rows, after Gauss Elimination:
111
010


After getting this matrix, we can check whether x can be formed using linear combination of numbers in the bucket by XOR-ing with the matrix (i.e. x can be fully reseted), and if not, we add x into our linear independence matrix.

To do the bit manipulation efficiently, we can represent the decimals in base $$2^{31}$$, hence it will fit in integer representation. While the whole idea may not be that difficult to grasp, the implementation is very error prone for me :/.

#include <iostream>
#include <cstdio>
#include <algorithm>
#include <string>
#include <vector>
#include <cassert>
using namespace std;
long long a[2222][103];
long long b[2222][103];
long long w[2222][103];
int vis[2222];
long long tmp[2222];
long long kmp[103];
int N;

int main(){
string d;
scanf("%d",&N);
for(int i=0;i<N;++i){
cin >> d;
int k = d.size();
for(int j=0;j<k;++j){
tmp[k-j-1] = d[j]-'0';
}
for(int n=0;n<=70;++n){
for(int j=k-1;j>0;--j){
tmp[j-1] += (tmp[j] & ((1LL<<31)-1))*10LL;
tmp[j] >>= 31;
}
a[i][n] = tmp[0] & ((1LL<<31)-1);
tmp[0] >>= 31;
}
int D = 31*71-1;
int idx = -1;
bool maxbitset = false;
for(int n=0;n<=70;++n) kmp[n] = 0;
for(int n=70;n>=0;--n){
for(int j=30;j>=0;j--){
long long p = a[i][n] & (1LL<<j);
if(p){
if(vis[D]){
for(int m=0;m<=70;++m){
a[i][m] ^= b[D][m];
kmp[m] ^= w[D][m];
}
} else if(!maxbitset){
idx = D;
maxbitset = true;
}
}
--D;
}
}
if(!maxbitset){
int cntr = 0;
for(int y=0;y<=70;++y){
cntr += __builtin_popcountll(kmp[y]);
}
assert(cntr != 0);
printf("%d", cntr);
int xx = 0;
for(int y=0;y<=70;++y){
for(int u=0;u<=30;++u){
if(kmp[y] & (1LL<<u)){
cntr--;
printf(" %d", xx);
}
++xx;
}
}
assert(cntr==0);
printf("\n");
} else {
assert(idx != -1);
printf("0\n");
for(int y=0;y<=70;++y){
b[idx][y] = a[i][y];
}
for(int y=0;y<=70;++y){
w[idx][y] = kmp[y];
}
w[idx][i/31] ^= (1LL << (i%31));
vis[idx] = 1;
}
}
return 0;
}