Fenwick Tree Use Cases

Fenwick Trees, also known as Binary Indexed Trees (BIT), have several applications due to their efficient operations. Let's explore some common use cases where Fenwick Trees can be applied:

1. Prefix Sum Queries: Fenwick Trees are commonly used to calculate prefix sum queries efficiently. The prefix sum of an array at a given index is the sum of all elements from index 0 to that index. With Fenwick Trees, we can calculate prefix sums in O(logN) time complexity.

2. Range Sum Queries: Fenwick Trees can efficiently calculate the sum of elements in a given range [l, r] in an array. By taking the prefix sum of the right endpoint and subtracting the prefix sum of the left endpoint, we obtain the sum of elements in the specified range in O(logN) time complexity.

3. Single Element Updates: Fenwick Trees allow for efficient updates on a single element in the original array. By updating the affected prefix sums, we can maintain accurate prefix sum values throughout the Fenwick Tree without recalculating the entire sum.

4. Finding the Kth Smallest Element: Fenwick Trees can be used to find the Kth smallest element in an array efficiently. By using the binary search technique with prefix sums, we can determine the value at the Kth position in O(logN) time complexity.

These are just a few examples of how Fenwick Trees can be applied in various scenarios. Their efficient operations make them a valuable data structure for solving a wide range of problems.

Let's see an example of how Fenwick Trees can be used to calculate prefix sum queries in C++:

1#include <iostream>
2
3// Function to calculate the sum of elements from index 0 to a given index
4int getPrefixSum(int bit[], int index) {
5  int sum = 0;
6  index++;
7
8  // Traverse the Fenwick Tree in a bottom-up manner
9  while (index > 0) {
10    sum += bit[index];
11    index -= index & -index;
12  }
13
14  return sum;
15}
16
17// Function to update the value at a given index in the Fenwick Tree
18void updateValue(int bit[], int n, int index, int value) {
19  index++;
20
21  // Traverse the Fenwick Tree in a top-down manner
22  while (index <= n) {
23    bit[index] += value;
24    index += index & -index;
25  }
26}
27
28int main() {
29  int arr[] = {1, 3, 2, -5, 4, 6};
30  int n = sizeof(arr) / sizeof(arr[0]);
31  int bit[n+1] = {0};
32
33  // Build the Fenwick Tree
34  for (int i = 0; i < n; i++) {
35    updateValue(bit, n, i, arr[i]);
36  }
37
38  // Calculate prefix sum queries
39  int prefixSum = getPrefixSum(bit, 4);
40
41  std::cout << "Prefix Sum of elements from index 0 to 4: " << prefixSum << std::endl;
42
43  return 0;
44}

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#include <iostream>

​

int main() {

  // Code block here

  return 0;

}