<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>

DEBUG = False
# DEBUG = True

class Solution:
    def smallestDistancePair(self, nums: List[int], k: int) -> int:
        n = len(nums)
        nums.sort()
        if DEBUG: print(nums, k); print()

        left_off = {}
        def count_pairs(max_diff):
            cached = {"last_evaluated": {}, "total": 0}
            for key in sorted(left_off.keys()):
                if key <= max_diff:
                    cached = left_off[key]
                else:
                    break

            total = cached["total"]
            last_evaluated = {}
            for i in range(n-1):
                j_init = cached["last_evaluated"].get(i, i+1)
                for j in range(j_init, n):
                    if nums[j]-nums[i] <= max_diff:
                        total += 1
                    else:
                        last_evaluated[i] = j
                        break
                else:
                    last_evaluated[i] = n+1
            left_off[max_diff] = {
                "last_evaluated": last_evaluated,
                "total": total,
            }
            return total

        counts = {}

        low = 0
        high = nums[-1] - nums[0]
        while low <= high:
            mid = (low + high) // 2
            pairs = count_pairs(mid)
            if DEBUG: print(f"{low=}, {high=} -> {mid=} ({pairs})")
            counts[mid] = pairs
            if pairs == k:
                high = mid - 1
            elif pairs < k:
                low = mid + 1
            else:
                high = mid - 1

        if DEBUG:
            print('max_diff -> #pairs')
            for max_diff in sorted(counts.keys()):
                print(max_diff, "->", counts[max_diff])

        for max_diff in sorted(counts.keys()):
            pairs = counts[max_diff]
            if pairs >= k:
                return max_diff
32214