<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 Solution {
public:
    int numTeams(vector<int>& rating) {
        const int sz = rating.size();
        const int max_el = *max_element(rating.begin(), rating.end());
        const int sq = sqrt(max_el);
        vector<vector<int>> vec(max_el / sq + 1);
        auto count_lo = [&](int val) {
            const int vec_idx = val / sq;
            int cnt = 0;
            for (int j = 0; j < vec_idx; ++j) {
                cnt += vec[j].size();
            }
            for (int j = 0; j < vec[vec_idx].size(); ++j) {
                cnt += vec[vec_idx][j] < val;
            }
            vec[vec_idx].push_back(val);
            return cnt;
        };
        vector<int> left_lo_cnt(sz);
        for (int i = 0; i < sz; ++i) {
            left_lo_cnt[i] = count_lo(rating[i]);
        }
        fill(vec.begin(), vec.end(), vector<int>());
        vector<int> right_lo_cnt(sz);
        for (int i = sz - 1; i >= 0; --i) {
            right_lo_cnt[i] = count_lo(rating[i]);
        }
        int res = 0;
        for (int i = 0; i < sz; ++i) {
            const int right_gt_cnt = sz - i - 1 - right_lo_cnt[i];
            const int left_gt_cnt = i - left_lo_cnt[i];
            res += left_lo_cnt[i] * right_gt_cnt + left_gt_cnt * right_lo_cnt[i];
        }
        return res;
    }
};
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