Formatted question description: https://leetcode.ca/all/1835.html

1835. Find XOR Sum of All Pairs Bitwise AND

Level

Hard

Description

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 <= 10^5
  • 0 <= arr1[i], arr2[j] <= 10^9

Solution

Use the distributive law. For integers a, b and c, there is (a ^ b) & c == (a & c) ^ (b & c) and a & (b ^ c) = (a & b) ^ (a & c).

For 0 <= i < arr1.length, the XOR sum of arr1[i] & arr2[j] where 0 <= j < arr2.length is equal to arr1[i] & (arr2[0] ^ arr2[1] ^ ... ^ arr2[arr2.length - 1]).

Therefore, the XOR sum of the list is calculated as follows.

(arr1[0] ^ arr1[1] ^ ... ^ arr1[arr1.length - 1]) & (arr2[0] ^ arr2[1] ^ ... ^ arr2[arr2.length - 1])
  • class Solution {
        public int getXORSum(int[] arr1, int[] arr2) {
            int xor1 = 0, xor2 = 0;
            for (int num : arr1)
                xor1 ^= num;
            for (int num : arr2)
                xor2 ^= num;
            return xor1 & xor2;
        }
    }
    
  • // OJ: https://leetcode.com/problems/find-xor-sum-of-all-pairs-bitwise-and/
    // 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;
        }
    };
    
  • # 1835. Find XOR Sum of All Pairs Bitwise AND
    # https://leetcode.com/problems/find-xor-sum-of-all-pairs-bitwise-and
    
    class Solution:
        def getXORSum(self, arr1: List[int], arr2: List[int]) -> int:
            xora = xorb = 0
            
            for num in arr1:
                xora ^= num
            
            for num in arr2:
                xorb ^= num
            
            return xora & xorb
    
    

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