<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 combine(int a, int b) {
        return (a + b);
    }

    void build(vector<int>& t, vector<int>& a, int v, int left, int right) {
        if (left == right) {
            t[v] = a[left];
            return;
        }

        int half = (left + right) / 2;
        build(t, a, v * 2, left, half);
        build(t, a, v * 2 + 1, half + 1, right);

        t[v] = combine(t[v * 2], t[v * 2 + 1]);
    }

    int query(vector<int>& t, int v, int left, int right, int tleft, int tright) {
        if (left > right) return 0;

        if (left == tleft && right == tright) return t[v];

        int thalf = (tleft + tright) / 2;
        int la = query(t, v * 2, left, min(right, thalf), tleft, thalf);
        int ra = query(t, v * 2 + 1, max(left, thalf + 1), right, thalf + 1, tright);

        return (combine(la, ra));
    }

    void update(vector<int>& t, int v, int left, int right, int position, int value) {
        if (left == right) {
            t[v] = value;
            return;
        }

        int half = (left + right) / 2;

        if (position <= half) update(t, v * 2, left, half, position, value);
        else update(t, v * 2 + 1, half + 1, right, position, value);

        t[v] = combine(t[v * 2], t[v * 2 + 1]);
    }

    int numberOfTeams(vector<int>& rating) {
        int n = rating.size();

        vector<int> v(n + 1, 0);
        vector<int> one(4 * (n + 1));
        build(one, v, 1, 0, n);
        vector<int> two(4 * (n + 1));
        build(two, v, 1, 0, n);

        long long answer = 0;
        for (auto it : rating) {
            update(one, 1, 0, n, it, 1);
            int twos = query(one, 1, 0, it - 1, 0, n);
            update(two, 1, 0, n, it, twos);
            answer += query(two, 1, 0, it - 1, 0, n);
        }
        return answer;
    }

    int numTeams(vector<int>& rating) {
        ios_base::sync_with_stdio(false);
	    cin.tie(NULL);
        
        int n = rating.size();

        vector<int> v = rating;
        sort(v.begin(), v.end());
        map<int, int> m;
        for (int i = 0; i < n; i++) m[v[i]] = i + 1;
        for (int i = 0; i < n; i++) rating[i] = m[rating[i]];

        long long answer = numberOfTeams(rating);
        reverse(rating.begin(), rating.end());
        answer += numberOfTeams(rating);
        return answer;
    }
};
16166