<p>The <strong>distance of a pair</strong> of integers <code>a</code> and <code>b</code> is defined as the absolute difference between <code>a</code> and <code>b</code>.</p>

<p>Given an integer array <code>nums</code> and an integer <code>k</code>, return <em>the</em> <code>k<sup>th</sup></code> <em>smallest <strong>distance among all the pairs</strong></em> <code>nums[i]</code> <em>and</em> <code>nums[j]</code> <em>where</em> <code>0 &lt;= i &lt; j &lt; nums.length</code>.</p>

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

<pre>
<strong>Input:</strong> nums = [1,3,1], k = 1
<strong>Output:</strong> 0
<strong>Explanation:</strong> Here are all the pairs:
(1,3) -&gt; 2
(1,1) -&gt; 0
(3,1) -&gt; 2
Then the 1<sup>st</sup> smallest distance pair is (1,1), and its distance is 0.
</pre>

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

<pre>
<strong>Input:</strong> nums = [1,1,1], k = 2
<strong>Output:</strong> 0
</pre>

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

<pre>
<strong>Input:</strong> nums = [1,6,1], k = 3
<strong>Output:</strong> 5
</pre>

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

<ul>
	<li><code>n == nums.length</code></li>
	<li><code>2 &lt;= n &lt;= 10<sup>4</sup></code></li>
	<li><code>0 &lt;= nums[i] &lt;= 10<sup>6</sup></code></li>
	<li><code>1 &lt;= k &lt;= n * (n - 1) / 2</code></li>
</ul>

class Solution {
private:
    vector<int> prefixSum, histogram;
    int countPairsSmallerOrEqual (const vector<int>& nums, int diff) {
        const int size = nums.size();

        int answer = 0;
        for (int i = 0; i < this->histogram.size(); ++i) {
            const auto quantBigger = this->prefixSum[i + diff] - this->prefixSum[i];
            answer += quantBigger*this->histogram[i];
            const auto quantSame = ( this->histogram[i] * (this->histogram[i]-1) )/2;
            answer += quantSame;

        }
        return answer;
    } 
public:
    int smallestDistancePair(vector<int>& nums, int k) {
        const int size = nums.size();
        sort(nums.begin(), nums.end());

        int left = INT_MAX;
        for (int i = 1; i < size; ++i) {
            left = min(left, nums[i] - nums[i-1]);
        }
        int right = nums.back() - nums.front();
        {
            const auto mn = *min_element(nums.begin(), nums.end());
            const auto mx = *max_element(nums.begin(), nums.end());
            this->prefixSum = vector<int>(mx + right + 1, 0);
            int index = 0;
            for (int val = mn; val <= mx + right; ++val) {
                while (index < size && nums[index] <= val) {
                    ++index;
                }
                this->prefixSum[val] = index;
            }
            this->histogram = vector<int>(mx+1, 0);
            for (const auto num : nums) {
                ++this->histogram[num];
            }
        }
        
        int answer = -1;
        while (left <= right) {
            const auto middle = left + (right - left)/2;
            const auto count = countPairsSmallerOrEqual(nums, middle);

            if (count < k) {
                left = middle + 1;
            } else {
                right = middle - 1;
                answer = middle;
            }
        }

        return answer;
    }

};
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