Set bits combined with exponential modular arithmetics

Question

I got this question yesterday in a challenge. I thought I had coded it correctly and my sample test case was passed. However not even a single test case passed at the backend. Here is my code. Please, someone, help me out. The challenge is over for me and so I can't submit it further. But I want to learn from my mistakes. Thanks.

  import java.io.*;
//import java.util.*;


public class TestClass {
    public static void main(String[] args) throws IOException {
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        PrintWriter wr = new PrintWriter(System.out);
         int n = Integer.parseInt(br.readLine().trim());
         String[] arr_a = br.readLine().split(" ");
         int[] a = new int[n];
         for(int i_a=0; i_a<arr_a.length; i_a++)
         {
            a[i_a] = Integer.parseInt(arr_a[i_a]);
         }

         long out_ = solve(a);
         System.out.println(out_);

         wr.close();
         br.close();
    }
    static long solve(int[] a){
        // Write your code here
        long sum = 0l;
        long MAX = 10000000011l;
        long i = 1l;
        for(int x : a) {
            long count = 0;
            while(x>0) {
                x &= (x-1l);
                count++;
            }
            long res = 1l;
            long temp = i;
            count = count % MAX;
            while(temp > 0) {
                if((temp & 1l) == 1l) {
                    res = (res * count) % MAX;
                }
                temp = temp >> 1l;
                count = ((count % MAX) * (count % MAX)) % MAX;

            }

            long t =((sum%MAX) + (res % MAX))%MAX;
            sum = t;
            i++;
        }

        return sum;
    }
}

Answer 1

It is a bit strange that "not even a single test case passed", but the only error I see is your exponentiation by squaring part.

All your numbers are less than 10^10 + 11, but this constant has more than 32 bits, and when you multiply, you get an overflow sometimes (because long is a 64-bit signed integer).

This can be fixed by several approaches:

1. (a*b) % M operation can be done with the algorithm that is similar to your "exponentiation by squaring" implementation. You just need to replace all multiplications with additions. As a result, multiplication is replaced with O(log(n)) additions and 'multiplying by 2' operations. Sample implementation:

   static long multiply(long a, long b, long M) {
       long res = 0;
       long d = a % M;
   
       while (b > 0) {
           if ((b & 1) == 1) {
               res = (res + d) % M;
           }
   
           b >>= 1;
           d = (d + d) % M;
       }
       return res;
   }
   

2. You can just cache b^i % M numbers for previously computed steps. For every number of set bits (there are not so many of them), you can save previously computed values and last(b) - the last i when a[i] had b set bits. Then just compute the new value with a linear loop from last(b) + 1 till current index i.

3. Use BigInteger for multiplications.