1835

1835. Find XOR Sum of All Pairs Bitwise AND (Hard)

The XOR sum of a list is the bitwise XOR of all its elements. If the list only contains one element, then its XOR sum will be equal to this element.

• For example, the XOR sum of [1,2,3,4] is equal to 1 XOR 2 XOR 3 XOR 4 = 4, and the XOR sum of [3] is equal to 3.

You are given two 0-indexed arrays arr1 and arr2 that consist only of non-negative integers.

Consider the list containing the result of arr1[i] AND arr2[j] (bitwise AND) for every (i, j) pair where 0 <= i < arr1.length and 0 <= j < arr2.length.

Return the XOR sum of the aforementioned list.

 

Example 1:

Input: arr1 = [1,2,3], arr2 = [6,5]
Output: 0
Explanation: The list = [1 AND 6, 1 AND 5, 2 AND 6, 2 AND 5, 3 AND 6, 3 AND 5] = [0,1,2,0,2,1].
The XOR sum = 0 XOR 1 XOR 2 XOR 0 XOR 2 XOR 1 = 0.

Example 2:

Input: arr1 = [12], arr2 = [4]
Output: 4
Explanation: The list = [12 AND 4] = [4]. The XOR sum = 4.

 

Constraints:

• 1 <= arr1.length, arr2.length <= 105
• 0 <= arr1[i], arr2[j] <= 109

Related Topics:
Math

Solution 1.

Intuition:

Consider case A = [1, 2, 3], B = [6, 5].

We can first handle the subcase of A = [1], B = [6, 5].

The result is (001 AND 110) XOR (001 AND 101).

We can consider bit by bit. For the 0-th (rightmost) bit, A[0] has 1 there, so we need to look at the 0-th bits of B[i].

Since there are odd number of 1s (one 1 in this case), the result must have 1 in the 0-th bit. If there are even number of 1s, the result must have 0 in the 0-th bit.

For the 1st bit, A[0] has 0 there, so the result must have 0 in the 1st bit.

And so on.

Algorithm:

• Scan array B, compute a cnt array where cnt[i] is the number of bit 1s at the i-th bit of all numbers in B.
• For each a in array A, let x be the result of (a AND B[0]) XOR (a AND B[1]) ...:
  • If a's ith bit is 0, or cnt[i] % 2 == 0, then x's ith bit must be 0.
  • Otherwise x's i-th bit must be 1.
• The answer is the XOR result of all the x values.

// OJ: https://leetcode.com/problems/find-xor-sum-of-all-pairs-bitwise-and/
// Author: github.com/lzl124631x
// Time: O(A + B)
// Space: O(1)
class Solution {
public:
    int getXORSum(vector<int>& A, vector<int>& B) {
        int cnt[31] = {};
        for (int b : B) {
            for (int i = 0; i < 31; ++i) {
                cnt[i] += (b >> i & 1);
            }
        }
        int ans = 0;
        for (int a : A) {
            int x = 0;
            for (int i = 0; i < 31; ++i) {
                if ((a >> i & 1) == 0 || cnt[i] % 2 == 0) continue;
                x |= 1 << i;
            }
            ans ^= x;
        }
        return ans;
    }
};

Solution 2.

Let AND(i, j) = A[i] AND B[j]. The answer is XOR( AND(i, j) | 0 <= i < M, 0 <= j < N ).

AND(i, j)'s kth bit is 1 if and only if A[i] and B[j] must be both 1 at the kth bit. Otherwise, AND(i, j) = 0.

Consider the kth bit of the answer, it's the XOR result of the kth bits of all AND(i, j). So the kth bit of the answer is 1 if and only if there are odd number of 1s of all AND(i, j) at kth bit.

Let cnt1[k] and cnt2[k] be the number of bit 1s at kth bit in A and B respectively. Then cnt1[k] * cnt2[k] is the number of 1s of all AND(i, j) at kth bit.

So, the kth bit of the answer is 1 if and only if cnt1[k] * cnt2[k] is an odd number, or in other words, cnt1[k] and cnt2[k] are both odd numbers.

// OJ: https://leetcode.com/problems/find-xor-sum-of-all-pairs-bitwise-and/
// Author: github.com/lzl124631x
// Time: O(A + B)
// Space: O(1)
class Solution {
public:
    int getXORSum(vector<int>& A, vector<int>& B) {
        int ans = 0;
        for  (int i = 0; i < 31; ++i) {
            int ca = 0, cb = 0;
            for (int n : A) ca += (n >> i & 1);
            for (int n : B) cb += (n >> i & 1);
            if (ca % 2 && cb % 2) ans |= 1 << i;
        }
        return ans;
    }
};

Solution 3.

Similar to (a + b) * (c + d) == a * c + a * d + b * c + b * d, the following is also true:

(a1 ^ a2) & (b1 ^ b2) == (a1 & b1) ^ (a1 & b2) ^ (a2 & b1) ^ (a2 & b2)

We can prove this by the following reasoning:

• The answer's kth bit is 1

If and only if

• cnt1[k] and cnt2[k] are both odd numbers

If and only if

• The XOR of all the kth bits in A is 1. The same for B.

If and only if

• "The XOR of all the kth bits in A" AND "the XOR of all the kth bits in B" is 1.

Hence, the proof is done.

Let a and b be the XOR of all the numbers in A and B respectively, the answer is a & b.

// OJ: https://leetcode.com/problems/find-xor-sum-of-all-pairs-bitwise-and/
// Author: github.com/lzl124631x
// Time: O(A + B)
// Space: O(1)
// Ref: https://leetcode-cn.com/problems/find-xor-sum-of-all-pairs-bitwise-and/solution/find-xor-sum-of-all-pairs-bitwise-and-by-sok6/
class Solution {
public:
    int getXORSum(vector<int>& A, vector<int>& B) {
        int a = 0, b = 0;
        for (int n : A) a ^= n;
        for (int n : B) b ^= n;
        return a & b;
    }
};