Applications of Fenwick Trees

Fenwick Trees, also known as Binary Indexed Trees, have numerous applications in various domains. Let's explore some real-world scenarios where Fenwick Trees can be useful:

1. Computing Prefix Sums: Fenwick Trees are efficient in calculating prefix sums over an array. They can answer queries like finding the sum of elements up to a given index in logarithmic time complexity.

2. Range Queries: Fenwick Trees can efficiently handle range queries, such as finding the sum of elements within a given range. This makes them suitable for tasks like computing the sum of elements in a subarray or finding the inversion count of an array.

3. Offline Queries: Fenwick Trees can be used to handle offline queries, where all queries are known in advance and can be processed in any order. They allow us to compute multiple range queries efficiently in a single pass.

4. Frequency Counting: Fenwick Trees can be used to count the frequency of elements in an array. They can maintain a counter for each distinct element, enabling fast updates and queries on the frequency of elements.

These are just a few examples of how Fenwick Trees can be applied to solve various problems efficiently. Now let's see an implementation of a Fenwick Tree in C++:

1#include <iostream>
2using namespace std;
3
4class FenwickTree {
5  int* bit;
6  int size;
7
8public:
9  FenwickTree(int n) {
10    size = n + 1;
11    bit = new int[size];
12    for (int i = 0; i < size; i++) {
13      bit[i] = 0;
14    }
15  }
16
17  void update(int index, int value) {
18    while (index < size) {
19      bit[index] += value;
20      index += (index & -index);
21    }
22  }
23
24  int query(int index) {
25    int sum = 0;
26    while (index > 0) {
27      sum += bit[index];
28      index -= (index & -index);
29    }
30    return sum;
31  }
32};
33
34int main() {
35  // Fenwick Tree implementation example
36
37  return 0;
38}

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#include <iostream>

using namespace std;

​

int main() {

  // Fenwick Tree implementation

​

  return 0;

}