<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>

struct Node{
    Node* links[2];
    int ct;
    Node(){
        links[0] = links[1] = NULL;
        ct = 0;
    }
    
    bool isPresent(int bit){
        return this -> links[bit] != NULL;
    }

    void insertBit(int bit){
        this -> links[bit] = new Node();
    } 

    void insert(int val){
        bitset<18>bits(val);
        Node* curr = this;
        for(int i = 17; i >=0 ; i--){
            if(!curr -> isPresent(bits[i])){
                curr -> insertBit(bits[i]);
            }
            curr = curr -> links[bits[i]];
            curr -> ct++;
        }
    }
    void del(int val){
        bitset<18>bits(val);
        Node* curr = this;
        for(int i = 17; i >=0 ; i--){
            if(!curr -> isPresent(bits[i])){
                curr -> insertBit(bits[i]);
            }
            curr = curr -> links[bits[i]];
            curr -> ct--;
        }
    }
};


class Solution {
public:

    int ctSmaller(Node * root, int val){
        bitset<18>bits(val);
        Node* curr = root;
        int ans = 0;

        for(int i = 17; i >=0 && curr ; i--){
            if(bits[i] == 0){
                curr = curr -> links[0];
            }
            else {
                if(curr -> links[0]) ans += curr -> links[0] -> ct;
                curr = curr -> links[1];
            }
        }

        return ans;
    }
    int numTeams(vector<int>& rating) {
        
        Node* pref = new Node();
        Node* suff = new Node();

        for(int i = 1; i < rating.size() ; i++){
            suff -> insert(rating[i]);
        }
        pref -> insert(rating[0]);

        int ans = 0;

        int n = rating.size();

        for(int i = 1; i < n  - 1; i++){
            suff -> del(rating[i]);

            int prefSmaller = ctSmaller(pref, rating[i]);
            int suffSmaller = ctSmaller(suff, rating[i]);

            // cout<<i<<" "<<prefSmaller<<" "<<suffSmaller<<"\n";

            ans += (prefSmaller * (n - i - 1 - suffSmaller)) + ((i - prefSmaller)*(suffSmaller));
            pref -> insert(rating[i]);
        }

        return ans;
    }
};
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