<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) {
        std::unordered_map<int, int>indices;
        std::set<int>ratings;

        for(int i = 0; i< rating.size(); i++){
            indices[rating[i]] = i+1;
            ratings.insert(rating[i]);
        }

        vector<int> greater(rating.size() + 1, 1);
        vector<int> less(rating.size() + 1, 0);

        greater[0] = 0;

        vector<int> greater_tree(rating.size()+1, 0);
        vector<int> less_tree(rating.size()+1, 0);

        for(int i = 1; i< greater_tree.size(); i++){
            greater_tree[i] = i & -i;
            cout << i << " " << greater_tree[i] << "\n";
        }

        int count = 0;

        for(int r : ratings){
            int index = indices[r];
            add(greater_tree, index, -1);


            int greater_right = prefix(greater_tree, greater_tree.size()-1) - prefix(greater_tree, index);
            int greater_left = prefix(greater_tree, index-1);


            int less_right = prefix(less_tree, less_tree.size()-1) - prefix(less_tree, index);
            int less_left = prefix(less_tree, index-1);
            add(less_tree, index, 1);

            cout << greater_right << greater_left << less_right << less_left << "\n";

            count += less_left * greater_right;
            count += less_right * greater_left;   
        }
        return count;
    }

    int prefix(std::vector<int>&tree, int x){
        int sum = 0;
        while(x >= 1){
            sum += tree[x];
            x -= x& -x;
        }
        return sum;
    }

    void add(std::vector<int>&tree, int x, int amt){
        while(x < tree.size()){
            tree[x] += amt;
            x+= x&-x;
        }
    }
};
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