Can this code be further optimized so that I do not get Time Limit Exceeded or TLE on competitive website?

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The snapshot of the question

ON a competitive coding website, I came across this question. I was able to solve it for the basic use cases, using the below code. However, the time limit exceeded for large inputs, and I tried to reduce the number of loops but still the time taken is about 2 seconds, however, the limit is 1 second only.

Is there any further scope to optimize this code? Like, can we remove the nested loops?

static void Main(string[] args)
    {
        string[] first = (Console.ReadLine()).Split(null);          //Sample input -  3 1 2 2 3
        int N = int.Parse(first[0]);
        int X = (int.Parse(first[1]) - 1);
        int Y = (int.Parse(first[2]) - 1);
        int Z = (int.Parse(first[3]) - 1);
        int T = (int.Parse(first[4]) - 1);
        string second = Console.ReadLine();                         // Sample input - 1 2 3
        List<int> list = new List<int>();

        //Converting string inputs to int
        list = second.Split().Select(str => int.Parse(str)).ToList();

        List<int> xorList = new List<int>();
        int xor = 0;
        int temp = 0;
        //Calculate the digits(using bitwise AND) of sub-matrix only, instead of entire matrix, and in the same loop, calculate bitwise XOR also.
        for (int i = X; i <= Z; i++)
        {
            for (int j = Y; j <= T; j++)
            {
                temp = list[i] & list[j];
                xor ^= temp;                    
            }
        }            
        Console.WriteLine(xor);            
    }

EDIT: For large inputs, the size of the matrix can be as big as 100000 by 100000. Also, sequence of numbers can have integers as large as 1000000000.

For such integers, it takes 2 seconds.

Answer 1

First, let's suppose the problem was with booleans instead of whole integers.

Then a way to view the problem, is that we have some booleans along the top of the submatrix in which we are interested, and some booleans along the left side of it. Then if vertical lines are drawn for the top booleans that are True, and horizontal lines are drawn for the side booleans that are true:

Then the task is to count the number of crossings and determine whether that number is even (result: zero) or odd (result: one).

It is more obvious now that it can be done without counting them out one by one, it could be done by multiplying the number of True booleans along the top by the number of True booleans along the side. In fact we can take the bitwise AND of those numbers, since the least significant bit of the product is the same as the least significant bit of the AND of the numbers.

This only takes linear time, to count the number of True booleans in the range X..Z and in the range Y..T.

To solve the full problem, do this for all 32 bits of the integers. Of course, there is a shortcut to doing that, because the full counts are not needed (only the least significant bit of the counts) and neither are full products.

Perhaps you prefer a more algebraic explanation, then a way to see this is that we are using the fact that AND distributes over XOR to rewrite a&d ^ a&e ^ a&f ^ b&d ^ b&e ^ b&f ^ c&d ^ c&e ^ c&f (a 3x3 example) into (a ^ b ^ c) & (d ^ e ^ f).

In general,

horizontal = XOR of list[X .. Z]
vertical = XOR of list[Y .. T]
result = horizontal & vertical