<p>Given an integer array <code>nums</code> and two integers <code>lower</code> and <code>upper</code>, return <em>the number of range sums that lie in</em> <code>[lower, upper]</code> <em>inclusive</em>.</p>

<p>Range sum <code>S(i, j)</code> is defined as the sum of the elements in <code>nums</code> between indices <code>i</code> and <code>j</code> inclusive, where <code>i &lt;= j</code>.</p>

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

<pre>
<strong>Input:</strong> nums = [-2,5,-1], lower = -2, upper = 2
<strong>Output:</strong> 3
<strong>Explanation:</strong> The three ranges are: [0,0], [2,2], and [0,2] and their respective sums are: -2, -1, 2.
</pre>

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

<pre>
<strong>Input:</strong> nums = [0], lower = 0, upper = 0
<strong>Output:</strong> 1
</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>-2<sup>31</sup> &lt;= nums[i] &lt;= 2<sup>31</sup> - 1</code></li>
	<li><code>-10<sup>5</sup> &lt;= lower &lt;= upper &lt;= 10<sup>5</sup></code></li>
	<li>The answer is <strong>guaranteed</strong> to fit in a <strong>32-bit</strong> integer.</li>
</ul>

class Solution {
public:
    //​ for every i <= j, lower <= sum(i, j) <= upper
    //                => lower + sum(0, i-1) <= sum(0, j) <= upper + sum(0, i-1), and i-1 < j
    int countRangeSum(vector<int>& nums, int lower, int upper) {
        vector<long long> prefix(nums.size() + 1, 0);

        for (int i = 0; i < nums.size(); ++i) {
            prefix[i + 1] = prefix[i] + nums[i];
        }

        auto sorted = prefix;
        sort(sorted.begin(), sorted.end());
        vector<long long> bit(sorted.size() + 1, 0);

        auto lowbit = [](int x) {
            return x & (-x);
        };

        auto update = [&](int idx, int val) {
            while (idx < bit.size()) {
                bit[idx] += val;
                idx += lowbit(idx);
            }
        };

        auto query = [&](int idx) {
            int sum = 0;

            while (idx > 0) {
                sum += bit[idx];
                idx -= lowbit(idx);
            }

            return sum;
        };

        int ans = 0;

        for (auto it = prefix.rbegin(); it != prefix.rend(); ++it) {
            long long x = *it;
            int posx = lower_bound(sorted.begin(), sorted.end(), x) - sorted.begin();
            int posl = lower_bound(sorted.begin(), sorted.end(), lower + x) - sorted.begin() - 1;
            int posu = upper_bound(sorted.begin(), sorted.end(), upper + x) - sorted.begin() - 1;

            ans += query(posu + 1) - query(posl + 1);
            update(posx + 1, 1);
        }

        return ans;
    }
};
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