string addBinary(string a, string b) {

int m = a.size();

int n = b.size();

int K = std::max(m,n);

string res(K,' ');

int i = m-1, j = n -1;

int k = K-1;

int carry = 0;

while (i >=0 || j >=0) {

int op1 = (i >=0)? a[i--] - '0': 0;

int op2 = (j >=0)? b[j--] - '0': 0;

res[k--] = (op1 + op2 + carry)%2 + '0';

carry = (op1 + op2 + carry)/2;

}

if (carry >0) {

res.insert(res.begin(), '1');

}

return res;

}