<p>You are given two positive integer arrays <code>nums1</code> and <code>nums2</code>, both of length <code>n</code>.</p>

<p>The <strong>absolute sum difference</strong> of arrays <code>nums1</code> and <code>nums2</code> is defined as the <strong>sum</strong> of <code>|nums1[i] - nums2[i]|</code> for each <code>0 &lt;= i &lt; n</code> (<strong>0-indexed</strong>).</p>

<p>You can replace <strong>at most one</strong> element of <code>nums1</code> with <strong>any</strong> other element in <code>nums1</code> to <strong>minimize</strong> the absolute sum difference.</p>

<p>Return the <em>minimum absolute sum difference <strong>after</strong> replacing at most one<strong> </strong>element in the array <code>nums1</code>.</em> Since the answer may be large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p><code>|x|</code> is defined as:</p>

<ul>
	<li><code>x</code> if <code>x &gt;= 0</code>, or</li>
	<li><code>-x</code> if <code>x &lt; 0</code>.</li>
</ul>

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

<pre>
<strong>Input:</strong> nums1 = [1,7,5], nums2 = [2,3,5]
<strong>Output:</strong> 3
<strong>Explanation: </strong>There are two possible optimal solutions:
- Replace the second element with the first: [1,<u><strong>7</strong></u>,5] =&gt; [1,<u><strong>1</strong></u>,5], or
- Replace the second element with the third: [1,<u><strong>7</strong></u>,5] =&gt; [1,<u><strong>5</strong></u>,5].
Both will yield an absolute sum difference of <code>|1-2| + (|1-3| or |5-3|) + |5-5| = </code>3.
</pre>

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

<pre>
<strong>Input:</strong> nums1 = [2,4,6,8,10], nums2 = [2,4,6,8,10]
<strong>Output:</strong> 0
<strong>Explanation: </strong>nums1 is equal to nums2 so no replacement is needed. This will result in an 
absolute sum difference of 0.
</pre>

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

<pre>
<strong>Input:</strong> nums1 = [1,10,4,4,2,7], nums2 = [9,3,5,1,7,4]
<strong>Output:</strong> 20
<strong>Explanation: </strong>Replace the first element with the second: [<u><strong>1</strong></u>,10,4,4,2,7] =&gt; [<u><strong>10</strong></u>,10,4,4,2,7].
This yields an absolute sum difference of <code>|10-9| + |10-3| + |4-5| + |4-1| + |2-7| + |7-4| = 20</code>
</pre>

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

<ul>
	<li><code>n == nums1.length</code></li>
	<li><code>n == nums2.length</code></li>
	<li><code>1 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= nums1[i], nums2[i] &lt;= 10<sup>5</sup></code></li>
</ul>

class Solution:
    def minAbsoluteSumDiff(self, nums1: List[int], nums2: List[int]) -> int:
        nums1_set = set(nums1)
        nums1_sorted = sorted(list(nums1_set))
        result = 0
        best_improvement = 0
        for num1, num2 in zip(nums1, nums2):
            result += abs(num1-num2)

        for num_idx in range(len(nums1)):
            current = abs(nums1[num_idx] - nums2[num_idx])
            if current > best_improvement:
                if nums2[num_idx] in nums1_set:
                    best_improvement = max(best_improvement, current)
                else:
                    to_check = self.binary_search(value=nums2[num_idx], inside=nums1_sorted)
                    if to_check[0] is not None:
                        new_delta = abs(to_check[0] - nums2[num_idx])
                        performance_gain = current - new_delta if new_delta < current else 0
                        best_improvement = max(best_improvement, performance_gain)
                    if to_check[1] is not None:
                        new_delta = abs(to_check[1] - nums2[num_idx])
                        performance_gain = current - new_delta if new_delta < current else 0
                        best_improvement = max(best_improvement, performance_gain)
        return (result - best_improvement) % (10 ** 9 + 7)
        

    def binary_search(self, value: int, inside: list[int]):
        low, high = 0, len(inside) - 1
        while low <= high:
            middle = (low + high) // 2
            if inside[middle] == value:
                return middle
            elif inside[middle] < value:
                low = middle + 1
            else:
                high = middle - 1
            if low > high:
                middle = low
            
            left = inside[middle - 1] if middle > 0 else None
            right = inside[middle] if middle < len(inside) else None
        return [left, right]
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