<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 Solution {
public:
    int closestToTarget(vector<int>& A, int target) {
        int n = A.size();
        const int LOG = (int) log2(n) + 1;
        
        vector<vector<int>> sp(n, vector<int>(LOG));
        
        for(int i = 0; i < n; i++) {
            sp[i][0] = A[i];
        }
        
        for(int j = 1; j < LOG; j++) {
            for(int i = 0; i + (1 << j) - 1 < n; i++) {
                sp[i][j] = (sp[i][j - 1] & sp[i + (1 << (j - 1))][j - 1]);
            }
        }
        function<int(int, int)> ask = [&] (int l, int r) {
            int len = (r - l + 1);
            int j = log2(len);
            return (sp[l][j] & sp[r - (1 << j) + 1][j]);
        };
        function<int(int, int, int)> lower = [&] (int i, int l, int h) {
            int val = -1;
            while(l <= h) {
                int mid = (l + h) >> 1;
                int comes_out = ask(i, mid);
                if(comes_out <= target) {
                    val = comes_out;
                    h = mid - 1;
                }
                else {
                    l = mid + 1;
                }
            } 
            return val;
        };
        function<int(int, int, int)> upper = [&] (int i, int l, int h) {
            int val = -1;
            while(l <= h) {
                int mid = (l + h) >> 1;
                int comes_out = ask(i, mid);
                if(comes_out >= target) {
                    val = comes_out;
                    l = mid + 1;
                }
                else {
                    h = mid - 1;
                }
            }
            return val;
        };
        
        int ans = INT_MAX;
        for(int i = 0; i < n; i++) {
            int left = lower(i, i, n - 1);
            int right = upper(i, i, n - 1);
            if(left != -1) {
                ans = min(ans, target - left);
            }
            if(right != -1) {
                ans = min(ans, right - target);
            }
        }
        return ans;
    }
};
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