315. Count of Smaller Numbers After Self(Fenwick Tree)

You are given an integer array nums and you have to return a new counts array. The counts array has the property where counts[i] is the number of smaller elements to the right of nums[i].

Example:

Input: [5,2,6,1]
Output:[5,2,6,1]

Explanation:

To the right of 5 there are 2 smaller elements (2 and 1).
To the right of 2 there is only 1 smaller element (1).
To the right of 6 there is 1 smaller element (1).
To the right of 1 there is 0 smaller element.

Approach #1: C++. [Fenwick Tree / Binary Indexed Tree]

class FenwickTree {
public:
    FenwickTree(int n): sums_(n+1, 0) {}
    
    void update(int i, int delta) {
        while (i < sums_.size()) {
            sums_[i] += delta;
            i += lowbit(i);
        }
    }
    
    int query(int i) const {
        int sum = 0;
        while (i > 0) {
            sum += sums_[i];
            i -= lowbit(i);
        }
        return sum;
    }
private:
    static inline int lowbit(int x) { return x & (-x); }
    vector<int> sums_;
};

class Solution {
public:
    vector<int> countSmaller(vector<int>& nums) {
        set<int> sorted(nums.begin(), nums.end());
        unordered_map<int, int> ranks;
        
        int rank = 0;
        
        for (const int num : sorted) 
            ranks[num] = ++rank;
        
        vector<int> ans;
        FenwickTree tree(ranks.size());
        
        for (int i = nums.size() - 1; i >= 0; --i) {
            ans.push_back(tree.query(ranks[nums[i]] - 1));
            tree.update(ranks[nums[i]], 1);
        }
        
        std::reverse(ans.begin(), ans.end());
        return ans;
    }
};

　　

Prefix sums of frequencise, convert the number to its rank as in sorted array.

For example:

Input:　   [5, 2, 6, 1]

sorted:　 [1, 2, 5, 6]

ranks:　  [1, 4, 2, 3]

Increase the freq[rank] by 1.

Num	Rank	Freq	Prefix sum / Query(rank - 1)
-	-	[0, 0, 0, 0, 0]	-
1	1	[0, 1, 0, 0, 0]	0
6	4	[0, 1, 0, 0, 1]	1
2	2	[0, 1, 1, 0, 1]	1
5	3	[0, 1, 1, 1, 1]	2

 Approach #2: Java. [BST]

class Solution {
    
    class Node {
        Node left, right;
        int val, sum, dup = 1;
        public Node(int v, int s) {
            val = v;
            sum = s;
        }
    }
    
    public List<Integer> countSmaller(int[] nums) {
        Integer[] ans = new Integer[nums.length];
        Node root = null;
        for (int i = nums.length - 1; i >= 0; i--) {
            root = insert(nums[i], root, ans, i, 0);
        }
        return Arrays.asList(ans);
    }
    
    private Node insert(int num, Node node, Integer[] ans, int i, int preSum) {
        if (node == null) {
            node = new Node(num, 0);
            ans[i] = preSum;
        } else if (node.val == num) {
            node.dup++;
            ans[i] = preSum + node.sum;
        } else if (node.val > num) {
            node.sum++;
            node.left = insert(num, node.left, ans, i, preSum);
        } else {
            node.right = insert(num, node.right, ans, i, preSum + node.dup + node.sum);
        }
        return node;
    }
}