Problem Description

You are given an array nums, where each number in the array appears either once or twice. Your task is to return the bitwise XOR of all the numbers that appear twice in the array, or return 0 if no number appears twice.

Intuition

To solve this problem, we need to identify numbers in the array that appear exactly twice and compute their bitwise XOR. The XOR operation has an interesting property: for any integer a, a XOR a equals 0 and a XOR 0 equals a. This property will help us consolidate multiple integers effectively.

1. Count Occurrences: First, we count the occurrences of each number using a hash table or dictionary. This helps us easily determine which numbers appear twice.

2. XOR Twice-Appearing Numbers: We iterate through the dictionary and perform the XOR operation on those numbers that appear exactly twice. The reduce function in Python can be used here to accumulate the XOR result.

3. Return Result: If no numbers appear twice, the result will be 0, as performing an XOR with an empty set or not finding any valid numbers defaults the result to 0. Else, we get the XOR of those repeating numbers.

This approach effectively separates the numbers that meet the condition from those that do not and uses XOR to compute the desired result efficiently.

Solution Approach

The solution is implemented using the following steps, employing concepts of counting and leveraging the properties of the XOR operation:

1. Counting Occurrences:

   • Utilize a hash table or dictionary to store the occurrences of each number in the array nums. This is efficiently handled by using collections.Counter in Python which will automatically count each number's occurrences.

   cnt = Counter(nums)

2. Filter and XOR Calculation:

   • Traverse the dictionary of counted numbers. For each number that occurs exactly twice, apply the XOR operation to cumulatively form the result.
   • The list comprehension [x for x, v in cnt.items() if v == 2] extracts all numbers with a count of exactly two.

   [x for x, v in cnt.items() if v == 2]

   • Use the reduce function along with the xor operator from Python's operator module to compute the final XOR of these numbers.

   from functools import reduce
   from operator import xor
   result = reduce(xor, [x for x, v in cnt.items() if v == 2], 0)

3. Return the Output:

   • The final value of the result is returned. If no number appears twice, the XOR of an empty list with 0 is 0, which correctly handles the edge case.

The complete implementation is efficient as it navigates the list of numbers and then processes the hash table to perform the XOR operation.

class Solution:
    def duplicateNumbersXOR(self, nums: List[int]) -> int:
        cnt = Counter(nums)
        return reduce(xor, [x for x, v in cnt.items() if v == 2], 0)

Example Walkthrough

Let's go through an example to better understand the solution approach.

Example:

Consider an array nums: [1, 2, 3, 2, 4, 5, 5].

Step 1: Counting Occurrences

First, we use a counter to count how many times each number appears in the array. The result here would be:

• 1: 1
• 2: 2
• 3: 1
• 4: 1
• 5: 2

Step 2: Filter and XOR Calculation

Next, we need to identify numbers that appear exactly twice. From our counted occurrences, 2 and 5 appear twice. We will XOR these numbers.

• Twice Appearing Numbers: [2, 5]

We then apply the XOR operation on these numbers:

• Compute 2 XOR 5. In binary, this is:

  2 = 010
  5 = 101
  ----------
      = 111 (which is 7 in decimal)

Step 3: Return the Output

The final value after performing XOR on [2, 5] is 7. Hence, the XOR of all numbers appearing twice is 7.

The solution leverages the properties of XOR and counts efficiently to consolidate the result for this problem.

Solution Implementation

Time and Space Complexity

The time complexity of the code is O(n) because it iterates over the list nums to count the occurrences of each element, and then filters the counted elements to find those with exactly two occurrences. Another iteration is done over these filtered elements to apply the XOR operation.

The space complexity of the code is O(M), where M is the number of distinct elements in the list nums. This space is used to store the counts of each number in a Counter object.

Learn more about how to find time and space complexity quickly.