<p>You are given an array <code>nums</code>​​​ and an integer <code>k</code>​​​​​. The <font face="monospace">XOR</font> of a segment <code>[left, right]</code> where <code>left &lt;= right</code> is the <code>XOR</code> of all the elements with indices between <code>left</code> and <code>right</code>, inclusive: <code>nums[left] XOR nums[left+1] XOR ... XOR nums[right]</code>.</p>

<p>Return <em>the minimum number of elements to change in the array </em>such that the <code>XOR</code> of all segments of size <code>k</code>​​​​​​ is equal to zero.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,0,3,0], k = 1
<strong>Output:</strong> 3
<strong>Explanation: </strong>Modify the array from [<u><strong>1</strong></u>,<u><strong>2</strong></u>,0,<u><strong>3</strong></u>,0] to from [<u><strong>0</strong></u>,<u><strong>0</strong></u>,0,<u><strong>0</strong></u>,0].
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [3,4,5,2,1,7,3,4,7], k = 3
<strong>Output:</strong> 3
<strong>Explanation: </strong>Modify the array from [3,4,<strong><u>5</u></strong>,<strong><u>2</u></strong>,<strong><u>1</u></strong>,7,3,4,7] to [3,4,<strong><u>7</u></strong>,<strong><u>3</u></strong>,<strong><u>4</u></strong>,7,3,4,7].
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,4,1,2,5,1,2,6], k = 3
<strong>Output:</strong> 3
<strong>Explanation: </strong>Modify the array from [1,2,<strong><u>4,</u></strong>1,2,<strong><u>5</u></strong>,1,2,<strong><u>6</u></strong>] to [1,2,<strong><u>3</u></strong>,1,2,<strong><u>3</u></strong>,1,2,<strong><u>3</u></strong>].</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= k &lt;= nums.length &lt;= 2000</code></li>
	<li><code>​​​​​​0 &lt;= nums[i] &lt; 2<sup>10</sup></code></li>
</ul>

"""class Solution:
    def minChanges(self, nums: List[int], k: int) -> int:
        n = len(nums)
        freq = [Counter() for _ in range(k)]
        for i in range(n):
            freq[i%k][nums[i]] += 1
        totalcount = [sum(f.values()) for f in freq]
        dp = [[float("inf")]*(1024) for _ in range(k)]
        for i in range(1024):
            dp[-1][i] = totalcount[-1] - freq[-1][i]
        for i in range(k-2,-1,-1):
            for xorr in range(1024):
                best = float("inf")
                for target in freq[i].keys():
                    best = min(best,totalcount[i] - freq[i][target] + dp[i+1][xorr^target])
                dp[i][xorr] = best
        return dp[0][0]"""
from typing import List
from collections import defaultdict

class Solution:
    def minChanges(self, nums: List[int], k: int) -> int:
        n = len(nums)
        max_val = 2 ** 10  # The maximum possible XOR value

        # Initialize a dynamic programming array to store minimum changes for each XOR value
        dp = [float('inf')] * max_val
        dp[0] = 0  # Base case: No changes needed for XOR 0

        # Iterate over each position in the subarray of size k
        for i in range(k):
            freq = defaultdict(int)  # Dictionary to store frequency of elements in the current subarray
            count = 0  # Count of elements in the current subarray
            # Iterate over the subarray with a step of k
            for j in range(i, n, k):
                freq[nums[j]] += 1
                count += 1

            prev_dp = min(dp)  # Get the minimum value from the previous dp array

            new_dp = [prev_dp + count] * max_val  # Initialize a new dp array
            # Combine loops over possible XOR values and elements in the current subarray
            for xor in range(max_val):
                for num, freq_num in freq.items():
                    new_xor = xor ^ num  # Calculate the new XOR value
                    # Update the dp array with the minimum changes needed
                    new_dp[new_xor] = min(new_dp[new_xor], dp[xor] - freq_num + count)

            dp = new_dp  # Update dp array for the next position in the subarray

        return dp[0]  # Return the minimum changes needed for XOR 0
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