<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:
int nextGap(int gap)
{
    if(gap<=1)return 0;
    return (gap+1)/2;
}
void mergeArray(vector<long long>& sum, int s1, int e1, int s2, int e2){
    int i=s1;
    for(int gap=nextGap(e2-s1+1);gap>0;gap=nextGap(gap))
    {
        i=s1;
        while((i+gap)<=e2)
        {
            if(sum[i+gap]<sum[i])
            {
                swap(sum[i+gap],sum[i]);
            }
            ++i;
        }
    }
}



int mergeCount(vector<long long>& sum, int left, int right, int lower, int upper){
    if(left == right){
        return sum[left] >= lower && sum[right] <= upper? 1 : 0;
    }
    int count = 0;
    int mid = left + ((right-left)/2);
    int leftCount = mergeCount(sum, left, mid, lower, upper);
    int rightCount = mergeCount(sum, mid+1, right, lower, upper);
    long long lowerBound = 0;
    long long upperBound = 0;
    int i = left, k = mid+1, j = mid+1;
    while(i <= mid){
        lowerBound = sum[i]+lower;
        upperBound = sum[i]+upper;
        while(j <= right && sum[j] <= upperBound){
            ++j;
        }
        while(k <= right && sum[k] < lowerBound){
            ++k;
        }
        count += (j-k);
        ++i;
    }
    mergeArray(sum, left, mid, mid+1, right);
    return leftCount+rightCount+count;
}
//Time O(NlogN), Space O(N)
int countRangeSum(vector<int>& nums, int lower, int upper) {
    vector<long long> sum(nums.size(), 0);
    sum[0] = nums[0];
    for(int i = 1; i < nums.size(); ++i){
        sum[i] = sum[i-1]+nums[i];
    }
    return mergeCount(sum, 0, sum.size()-1, lower, upper);
}
};
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