<p><img alt="" src="https://assets.leetcode.com/uploads/2020/07/09/change.png" style="width: 635px; height: 312px;" /></p>

<p>Winston was given the above mysterious function <code>func</code>. He has an integer array <code>arr</code> and an integer <code>target</code> and he wants to find the values <code>l</code> and <code>r</code> that make the value <code>|func(arr, l, r) - target|</code> minimum possible.</p>

<p>Return <em>the minimum possible value</em> of <code>|func(arr, l, r) - target|</code>.</p>

<p>Notice that <code>func</code> should be called with the values <code>l</code> and <code>r</code> where <code>0 &lt;= l, r &lt; arr.length</code>.</p>

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

<pre>
<strong>Input:</strong> arr = [9,12,3,7,15], target = 5
<strong>Output:</strong> 2
<strong>Explanation:</strong> Calling func with all the pairs of [l,r] = [[0,0],[1,1],[2,2],[3,3],[4,4],[0,1],[1,2],[2,3],[3,4],[0,2],[1,3],[2,4],[0,3],[1,4],[0,4]], Winston got the following results [9,12,3,7,15,8,0,3,7,0,0,3,0,0,0]. The value closest to 5 is 7 and 3, thus the minimum difference is 2.
</pre>

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

<pre>
<strong>Input:</strong> arr = [1000000,1000000,1000000], target = 1
<strong>Output:</strong> 999999
<strong>Explanation:</strong> Winston called the func with all possible values of [l,r] and he always got 1000000, thus the min difference is 999999.
</pre>

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

<pre>
<strong>Input:</strong> arr = [1,2,4,8,16], target = 0
<strong>Output:</strong> 0
</pre>

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

<ul>
	<li><code>1 &lt;= arr.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= arr[i] &lt;= 10<sup>6</sup></code></li>
	<li><code>0 &lt;= target &lt;= 10<sup>7</sup></code></li>
</ul>

class SegmentTree {
    vector<int> tree;
    int n;

public:
    SegmentTree(const vector<int>& arr) {
        n = arr.size();
        tree.resize(4 * n);  // Allocate space for segment tree
        build(arr, 0, 0, n - 1);
    }

    // Build the segment tree
    void build(const vector<int>& arr, int node, int start, int end) {
        if (start == end) {
            // Leaf node
            tree[node] = arr[start];
        } else {
            int mid = (start + end) / 2;
            int leftChild = 2 * node + 1;
            int rightChild = 2 * node + 2;
            build(arr, leftChild, start, mid);
            build(arr, rightChild, mid + 1, end);
            // Internal node will have the AND of both children
            tree[node] = tree[leftChild] & tree[rightChild];
        }
    }

    // Query function to get AND in the range [l, r]
    int query(int l, int r) {
        return query(0, 0, n - 1, l, r);
    }

    int query(int node, int start, int end, int l, int r) {
        if (r < start || l > end) {
            // No overlap
            return -1; // Use all bits set to 1 for AND operation
        }
        if (l <= start && end <= r) {
            // Complete overlap
            return tree[node];
        }
        // Partial overlap
        int mid = (start + end) / 2;
        int leftResult = query(2 * node + 1, start, mid, l, r);
        int rightResult = query(2 * node + 2, mid + 1, end, l, r);
        if (leftResult == -1) return rightResult;
        if (rightResult == -1) return leftResult;
        return leftResult & rightResult;
    }
};

class Solution {
public:
    int calc_and(int l, int r, SegmentTree& segTree) {
        // Use the segment tree to calculate the AND for range [l, r]
        return segTree.query(l, r);
    }

    int closestToTarget(vector<int>& arr, int target) {
        int n = arr.size();
        int ans = INT_MAX;

        // Build the segment tree
        SegmentTree segTree(arr);

        // Iterate through each starting point
        for (int i = 0; i < n; i++) {
            int l = i, r = n - 1;
            ans = min(ans, abs(target - calc_and(l, r, segTree)));
            
            // Perform binary search to find the closest value
            while (l < r) {
                int mid = (l + r) / 2;
                int a = calc_and(i, mid, segTree);
                if (target == a) return 0;
                ans = min(ans, abs(target - a));
                if (target < a)
                    l = mid + 1;
                else
                    r = mid;
            }
        }
        return ans;
    }
};

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