Introduction to Fenwick Trees

Fenwick Trees, also known as Binary Indexed Trees (BIT), are a specialized data structure used to efficiently update and query prefix sums of an array.

Imagine you have an array of elements and you need to perform two types of operations:

1. Update: Change the value of an element at a given index
2. Query: Calculate the sum of all elements in a given range

Traditional data structures like arrays and linked lists are not efficient for performing both types of operations simultaneously. This is where Fenwick Trees come to the rescue.

Fenwick Trees are based on the prefix sum algorithm and provide a way to store prefix sums of an array in a space-efficient manner. The key idea behind Fenwick Trees is to represent every index i of the array in binary form and use the binary representation to perform efficient updates and queries.

By using Fenwick Trees, we can perform both update and query operations in O(log n) time complexity, where n is the size of the array.

Let's dive into the implementation of Fenwick Trees in C++:

1#include <iostream>
2using namespace std;
3
4// Define the size of the Fenwick Tree
5const int MAX_SIZE = 100;
6
7// Declare the Fenwick Tree
8int BIT[MAX_SIZE];
9
10void update(int index, int value) {
11  // Add the value to the prefix sum of all affected ranges
12  while (index < MAX_SIZE) {
13    BIT[index] += value;
14    index += index & -index;
15  }
16}
17
18int query(int index) {
19  // Calculate the sum of all elements up to the given index
20  int sum = 0;
21  while (index > 0) {
22    sum += BIT[index];
23    index -= index & -index;
24  }
25  return sum;
26}
27
28int main() {
29  // Replace with your Fenwick Tree logic here
30  return 0;
31}

In the code snippet above, we define a Fenwick Tree of maximum size 100, represented by the array BIT. The update function is used to update the value at a given index by adding the value to the prefix sum of all affected ranges. The query function is used to calculate the sum of all elements up to the given index.

Now that we have an idea of what Fenwick Trees are and how they can be implemented in C++, let's move on to the next step: creating a Fenwick Tree.

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#include <iostream>

using namespace std;

​

int main() {

  // Replace with your C++ logic here

  return 0;

}