<p>There are <code>n</code> soldiers standing in a line. Each soldier is assigned a <strong>unique</strong> <code>rating</code> value.</p>

<p>You have to form a team of 3 soldiers amongst them under the following rules:</p>

<ul>
	<li>Choose 3 soldiers with index (<code>i</code>, <code>j</code>, <code>k</code>) with rating (<code>rating[i]</code>, <code>rating[j]</code>, <code>rating[k]</code>).</li>
	<li>A team is valid if: (<code>rating[i] &lt; rating[j] &lt; rating[k]</code>) or (<code>rating[i] &gt; rating[j] &gt; rating[k]</code>) where (<code>0 &lt;= i &lt; j &lt; k &lt; n</code>).</li>
</ul>

<p>Return the number of teams you can form given the conditions. (soldiers can be part of multiple teams).</p>

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

<pre>
<strong>Input:</strong> rating = [2,5,3,4,1]
<strong>Output:</strong> 3
<strong>Explanation:</strong> We can form three teams given the conditions. (2,3,4), (5,4,1), (5,3,1). 
</pre>

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

<pre>
<strong>Input:</strong> rating = [2,1,3]
<strong>Output:</strong> 0
<strong>Explanation:</strong> We can&#39;t form any team given the conditions.
</pre>

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

<pre>
<strong>Input:</strong> rating = [1,2,3,4]
<strong>Output:</strong> 4
</pre>

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

<ul>
	<li><code>n == rating.length</code></li>
	<li><code>3 &lt;= n &lt;= 1000</code></li>
	<li><code>1 &lt;= rating[i] &lt;= 10<sup>5</sup></code></li>
	<li>All the integers in <code>rating</code> are <strong>unique</strong>.</li>
</ul>

class SumBITree {
    vector<int> m_tree;
public:
    SumBITree(int n) : m_tree(n) {}
    void update(int idx, int val) {
        while (idx < m_tree.size()) {
            m_tree[idx] += val;
            idx += idx & (-idx);
        }
    }
    int prefixSum(int idx) {
        int result = 0;
        while (idx > 0) {
            result += m_tree[idx];
            idx -= idx & (-idx);
        }
        return result;
    }
    int suffixSum(int idx) {
        return prefixSum(m_tree.size() - 1) - prefixSum(idx - 1);
    }
};

class Solution {
public:
    int numTeams(vector<int>& rating) {
        int result = 0;
        int n = rating.size();
        int maxVal = *max_element(rating.begin(), rating.end());
        // to the left/right of current position
        SumBITree left(maxVal + 1);
        SumBITree right(maxVal + 1);
        for (int r : rating) { right.update(r, 1); } // initially add all ratings to right
        for (int r : rating) {
            right.update(r, -1); // remove curr from right
            result += left.prefixSum(r - 1) * right.suffixSum(r + 1);
            result += left.suffixSum(r + 1) * right.prefixSum(r - 1);
            left.update(r, 1); // add curr to left for next iteration
        }
        return result;
    }
};
21158