<p>Given an integer array <code>nums</code>, return<em> an integer array </em><code>counts</code><em> where </em><code>counts[i]</code><em> is the number of smaller elements to the right of </em><code>nums[i]</code>.</p>

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

<pre>
<strong>Input:</strong> nums = [5,2,6,1]
<strong>Output:</strong> [2,1,1,0]
<strong>Explanation:</strong>
To the right of 5 there are <b>2</b> smaller elements (2 and 1).
To the right of 2 there is only <b>1</b> smaller element (1).
To the right of 6 there is <b>1</b> smaller element (1).
To the right of 1 there is <b>0</b> smaller element.
</pre>

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

<pre>
<strong>Input:</strong> nums = [-1]
<strong>Output:</strong> [0]
</pre>

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

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

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>-10<sup>4</sup> &lt;= nums[i] &lt;= 10<sup>4</sup></code></li>
</ul>

class Solution:
    # helpers
    def lowbit(self, x):
        return x & -x
    def add(self, freq, i, length):
        while i < length:
            freq[i] += 1
            i += self.lowbit(i)
    def calcSum(self, freq, i):
        total = 0
        while i > 0:
            total += freq[i]
            i -= self.lowbit(i)
        return total
    def discretization(self, nums):
        unique_nums = set(nums)
        sorted_unique_nums = sorted(unique_nums)
        rank_map = {}
        for i, num in enumerate(sorted_unique_nums):
            rank_map[num] = i + 1
        return rank_map
    def countSmaller(self, nums: List[int]) -> List[int]:
        length = len(nums)
        freq = [0] * (length + 1)
        res = []
        ranks = self.discretization(nums)
        for num in reversed(nums):
            rank = ranks[num]
            # calcuate the sum from index 0 to rank-1 in rank map
            res.append(self.calcSum(freq, rank-1)) 
            self.add(freq, rank, length)
        return res[::-1]


        
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