<p>You are given an integer array <code>nums</code> of length <code>n</code> which represents a permutation of all the integers in the range <code>[0, n - 1]</code>.</p>

<p>The number of <strong>global inversions</strong> is the number of the different pairs <code>(i, j)</code> where:</p>

<ul>
	<li><code>0 &lt;= i &lt; j &lt; n</code></li>
	<li><code>nums[i] &gt; nums[j]</code></li>
</ul>

<p>The number of <strong>local inversions</strong> is the number of indices <code>i</code> where:</p>

<ul>
	<li><code>0 &lt;= i &lt; n - 1</code></li>
	<li><code>nums[i] &gt; nums[i + 1]</code></li>
</ul>

<p>Return <code>true</code> <em>if the number of <strong>global inversions</strong> is equal to the number of <strong>local inversions</strong></em>.</p>

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

<pre>
<strong>Input:</strong> nums = [1,0,2]
<strong>Output:</strong> true
<strong>Explanation:</strong> There is 1 global inversion and 1 local inversion.
</pre>

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

<pre>
<strong>Input:</strong> nums = [1,2,0]
<strong>Output:</strong> false
<strong>Explanation:</strong> There are 2 global inversions and 1 local inversion.
</pre>

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

<ul>
	<li><code>n == nums.length</code></li>
	<li><code>1 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= nums[i] &lt; n</code></li>
	<li>All the integers of <code>nums</code> are <strong>unique</strong>.</li>
	<li><code>nums</code> is a permutation of all the numbers in the range <code>[0, n - 1]</code>.</li>
</ul>


vector<int> tmp;	
long long merge_sorted_subarrays(vector<int> &nums, int start, int md, int end) {
	long long conversions = 0;
	for (int i = start, j = md + 1, k = start; k <= end; k++) {
		if (i > md)					// start is done
			tmp[k] = nums[j++];
		else if (j > end)			// end is done
			tmp[k] = nums[i++];
		else if (nums[i] < nums[j])	// start is smaller
			tmp[k] = nums[i++];
		else {
			tmp[k] = nums[j++];
			conversions += md-i+1;
		}
	}
	for (int i = start; i <= end; i++)
		nums[i] = tmp[i];

	return conversions;
}

long long merge_sort(vector<int> &nums, int start, int end) {
	if (start == end)
		return 0;

	int md = start + (end - start) / 2;
	long long conversions = merge_sort(nums, start, md);
	conversions += merge_sort(nums, md + 1, end);
	conversions += merge_sorted_subarrays(nums, start, md, end);

	return conversions;
}

class Solution {
public:
	bool isIdealPermutation(vector<int> &nums) {
		tmp = nums;

		long long local_conversions = 0;	// compute before updating the array
		for (int i = 0; i < (int) nums.size() - 1; i++)
			local_conversions += nums[i] > nums[i + 1];

		long long global_conversions = merge_sort(nums, 0, (int) nums.size() - 1);

		return local_conversions == global_conversions;
	}
};


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