Problem Description

In this problem, we are given an array nums of integers and a separate integer k. Each number in the array is represented in binary and we can flip any of the bits in its binary representation (i.e., change a 0 to a 1 or vice versa) as many times as we want. Our goal is to perform the minimum number of such bit flips so that when we compute the bitwise XOR of all the numbers in the array, the result equals k. It is also mentioned that we can flip the leading zeros—which are usually omitted in binary notation. Our task is to find out the minimum number of flips needed to achieve this.

Intuition

To approach this problem, we don't really need to flip any bits initially. Instead, we can directly compute the bitwise XOR of all the elements in nums. Since XOR is an associative operation, the order in which we perform it doesn't matter, and since XORing a number with itself yields zero, any duplicate numbers effectively cancel each other out. After this computation, we'll have a single number which is the result of the XOR of all the array elements.

The problem now simplifies to finding the minimum number of bits we need to flip in this result to make it turn into k. To do this, we can XOR this result with k.

Why does that work? XORing two numbers highlights the bits where they are different (since XOR gives us 1 when we compare bits that are different and 0 when bits are the same).

The final step is to count the number of bits that are set to 1 in this XOR result. Since these 1's represent differing bits, they are the exact bits we need to flip to turn our array XOR result into k. The number of such bits is the minimum operations required, which is precisely what we need.

This whole process can be done in Python using the reduce function from functools to do the XOR of all numbers in nums with k, and then calling bit_count() on the result to count the number of bits that are 1.

Solution Approach

The solution relies on bit manipulation and the properties of the XOR operation. The approach is straightforward with four key steps:

1. XOR all elements with k: The reduce function from Python's functools is utilized here with the xor operator from the operator module. reduce applies the xor function cumulatively to the items of nums, from left to right, so as to reduce the array to a single value. This value represents the XOR of all elements of nums. Then, by XORing this result with k, we find out which bits differ between the XOR of the array elements and k.

2. Calculate the number of differing bits: In the final result, a bit set to 1 indicates a position where a bit must be flipped to obtain k. The Python method bit_count() is used on the resulting number to get the count of bits that are set to 1.

3. Minimum operations required: The count obtained from bit_count() is the minimum number of operations required. This is because each bit that is set to 1 can be flipped individually, and no unnecessary bit flips are required.

4. Return the result: The final count of set bits is returned as the solution to the problem: the minimum number of bit flips needed.

In terms of data structures, this solution is efficient as it operates with integers directly and does not require any additional data structures. The complexity of the algorithm is primarily dependent on the number of bits required to represent the numbers in the array, which is at most O(log(max(nums, k))).

Here's the consolidated approach in code:

from operator import xor
from functools import reduce

class Solution:
    def minOperations(self, nums: List[int], k: int) -> int:
        return reduce(xor, nums, k).bit_count()

In this code:

• reduce(xor, nums, k) computes the XOR of all elements in nums along with k.
• .bit_count() then counts the number of set bits (bits that are 1) of the result from the reduce function, thereby counting the number of operations needed.

This solution leverages the elegance and efficiency of bit operations to solve the problem in a crisp and concise manner.

Example Walkthrough

Let's illustrate the solution approach with a small example.

Suppose we have the following array of integers nums and the integer k:

• nums = [2, 3, 1]
• k = 5

The binary representations of the numbers are:

• 2 -> 10
• 3 -> 11
• 1 -> 01
• 5 -> 101

Here's how we apply the solution approach step by step:

1. XOR all elements with k: First, calculate the XOR of all the elements in nums:

• 2 XOR 3 XOR 1 = 10 XOR 11 XOR 01 = 00 (binary for 0)

Now we XOR the result with k:

• 0 XOR 5 = 000 XOR 101 = 101

2. Calculate the number of differing bits: We use bit_count() to count how many bits are set to 1 in 101:

• 101 has two bits set to 1.

3. Minimum operations required: The number of set bits is the minimum number of operations required. Thus, we need 2 operations to transform the array's XOR result into k.

4. Return the result: The final result is 2, so our function minOperations(nums, k) will return 2.

This example demonstrates how to apply the proposed algorithm step by step, showing that we can achieve the desired result with a minimum number of bit flips. Each bit that differs between the XOR result of the array and k represents a flip we need to make. In this case, flipping the second and third bits (from the right) of the number 0 (the XOR of nums) will result in the binary number 101, which represents the integer 5. The result aligns with the expected output of the minOperations method, confirming the validity of the algorithm.

Solution Implementation

Time and Space Complexity

The time complexity of the code is O(n), where n is the length of the array nums. This is because the reduce function applies the xor operation to each element of the array exactly once.

The space complexity of the code is O(1) since the xor operation and bit_count() method do not require any additional space that grows with the size of the input array. The operation is performed in constant space, regardless of the length of nums.

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