Introduction to Fenwick Trees

Fenwick Trees, also known as Binary Indexed Trees (BIT), are a data structure that efficiently allows for querying and updating cumulative sums over a dynamic array or sequence.

The structure was proposed by Peter M. Fenwick in 1994 and is often used as a more efficient alternative to prefix sums or cumulative frequency tables.

A Fenwick Tree supports two main operations:

• Update: Given an index idx and a value val, update the Fenwick Tree by adding val to the element at index idx.
• Query: Given an index idx, calculate the sum of all the elements from index 1 to idx in the Fenwick Tree.

Fenwick Trees have a space complexity of O(n) and support both update and query operations in O(log n) time.

Here's an example of C++ code implementing a Fenwick Tree:

1#include <iostream>
2using namespace std;
3
4int main() {
5  // Fenwick Tree implementation
6  int n;
7  cin >> n;
8
9  // Create an array to store the Fenwick Tree
10  int fenwick[n + 1];
11  for (int i = 0; i <= n; i++) {
12    fenwick[i] = 0;
13  }
14
15  // Update a value in the Fenwick Tree
16  int idx, val;
17  cin >> idx >> val;
18  while (idx <= n) {
19    fenwick[idx] += val;
20    idx += idx & -idx;
21  }
22
23  // Perform a query on the Fenwick Tree
24  int query;
25  cin >> query;
26  int sum = 0;
27  while (query > 0) {
28    sum += fenwick[query];
29    query -= query & -query;
30  }
31
32  // Output the result
33  cout << sum << endl;
34
35  return 0;
36}

In this example, we first initialize an array to store the Fenwick Tree (fenwick) and set all elements to 0. We then update a value in the Fenwick Tree by adding val to the element at index idx and updating the indices accordingly using bitwise operations. Finally, we perform a query on the Fenwick Tree by calculating the sum of all the elements from index 1 to query using bitwise operations as well. The result is outputted to the console.

Fenwick Trees are particularly useful when the number of updates is much smaller compared to the number of queries, making them a powerful tool for solving various algorithmic problems.

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}

#include <iostream>

using namespace std;

​

int main() {

  // Fenwick Tree implementation

  int n;

  cin >> n;

​

  // Create an array to store the Fenwick Tree

  int fenwick[n + 1];

  for (int i = 0; i <= n; i++) {

    fenwick[i] = 0;

  }

​

  // Update a value in the Fenwick Tree

  int idx, val;

  cin >> idx >> val;

  while (idx <= n) {

    fenwick[idx] += val;

    idx += idx & -idx;

  }

​

  // Perform a query on the Fenwick Tree

  int query;

  cin >> query;

  int sum = 0;

  while (query > 0) {

    sum += fenwick[query];

    query -= query & -query;

Updating a Fenwick Tree

In this lesson, we will focus on updating values in a Fenwick Tree. To update a value in a Fenwick Tree, we modify the element at the given index and then update the relevant cumulative sums.

Let's take a look at the code snippet below to see how we can update a value in a Fenwick Tree using C++:

1#include <iostream>
2using namespace std;
3
4int main() {
5  // Replace with your Fenwick Tree implementation
6  int n;
7  cin >> n;
8
9  // Create an array to store the Fenwick Tree
10  int fenwick[n + 1];
11  for (int i = 0; i <= n; i++) {
12    fenwick[i] = 0;
13  }
14
15  // Update a value in the Fenwick Tree
16  int idx, val;
17  cin >> idx >> val;
18  while (idx <= n) {
19    fenwick[idx] += val;
20    idx += idx & -idx;
21  }
22
23  // Output the updated Fenwick Tree
24  for (int i = 1; i <= n; i++) {
25    cout << fenwick[i] << " ";
26  }
27
28  return 0;
29}

In the code snippet, we first create an array to store the Fenwick Tree and initialize all elements to 0. Then, we update a value in the Fenwick Tree by adding the given value to the element at the specified index. We also update the cumulative sums accordingly using bitwise operations.

Finally, we output the updated Fenwick Tree to verify the results.

Keep in mind that this code snippet serves as an example implementation, and you may need to modify it based on your specific use case.

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#include <iostream>

using namespace std;

int main() {

  // Replace with your Fenwick Tree implementation

  int n;

  cin >> n;

​

  // Create an array to store the Fenwick Tree

  int fenwick[n + 1];

  for (int i = 0; i <= n; i++) {

    fenwick[i] = 0;

  }

​

  // Update a value in the Fenwick Tree

  int idx, val;

  cin >> idx >> val;

  while (idx <= n) {

    fenwick[idx] += val;

    idx += idx & -idx;

  }

​

  // Output the updated Fenwick Tree

  for (int i = 1; i <= n; i++) {

    cout << fenwick[i] << " ";

  }

​

  return 0;

}

Performing Queries on a Fenwick Tree

In this lesson, we will learn how to perform queries on a Fenwick Tree. A query on a Fenwick Tree allows us to calculate the sum of a range of values efficiently.

To perform a query on a Fenwick Tree, we follow a similar process as updating a value, but in reverse. Starting from the given index, we traverse the Fenwick Tree using bitwise operations until we reach index 0. At each step, we add the value at the current index to a running sum.

Here's an example code snippet in C++ that demonstrates how to perform a query on a Fenwick Tree:

1#include <iostream>
2using namespace std;
3
4int main() {
5  int n;
6  cin >> n;
7
8  int fenwick[n + 1];
9  for (int i = 0; i <= n; i++) {
10    fenwick[i] = 0;
11  }
12
13  int idx;
14  cin >> idx;
15
16  int sum = 0;
17  while (idx > 0) {
18    sum += fenwick[idx];
19    idx -= idx & -idx;
20  }
21
22  cout << sum << endl;
23
24  return 0;
25}

In the code snippet, we first create an array to store the Fenwick Tree and initialize all elements to 0. Then, we input the index at which we want to perform the query.

Next, we initialize a running sum variable to 0 and start traversing the Fenwick Tree from the given index. At each step, we add the value at the current index to the running sum and update the index using bitwise operations.

Finally, we output the sum of the range of values.

Keep in mind that this code snippet serves as an example implementation, and you may need to modify it based on your specific use case.

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#include <iostream>

using namespace std;

​

int main() {

  // Replace with your Fenwick Tree implementation

  int n;

  cin >> n;

​

  // Create an array to store the Fenwick Tree

  int fenwick[n + 1];

  for (int i = 0; i <= n; i++) {

    fenwick[i] = 0;

  }

​

  // Perform a query on the Fenwick Tree

  int idx;

  cin >> idx;

​

  int sum = 0;

  while (idx > 0) {

    sum += fenwick[idx];

    idx -= idx & -idx;

  }

​

  // Output the query result

  cout << sum << endl;

​

  return 0;

}

Range Updates and Range Queries

In this section, we will discuss the concepts of range updates and range queries in Fenwick Trees.

Range Updates

Range updates allow us to update a range of values efficiently in a Fenwick Tree. To perform a range update, we traverse the Fenwick Tree using bitwise operations and update the values at each index within the given range.

Here's an example code snippet in C++ that demonstrates how to perform a range update on a Fenwick Tree:

1#include <iostream>
2using namespace std;
3
4void rangeUpdate(int fenwick[], int index, int value) {
5  while (index <= n) {
6    fenwick[index] += value;
7    index += (index & -index);
8  }
9}
10
11int main() {
12  int n;
13  cin >> n;
14
15  int fenwick[n + 1];
16  for (int i = 0; i <= n; i++) {
17    fenwick[i] = 0;
18  }
19
20  int start, end, value;
21  cin >> start >> end >> value;
22
23  rangeUpdate(fenwick, start, value);
24  rangeUpdate(fenwick, end + 1, -value);
25
26  return 0;
27}

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#include <iostream>

using namespace std;

​

int main() {

  // Replace with your C++ logic for range updates and range queries

  return 0;

}

Advanced Fenwick Tree Variations

In this section, we will dive deeper into the advanced variations of Fenwick Trees. Fenwick Trees are a powerful data structure that can be modified and extended in various ways to solve different problems efficiently.

Binary Indexed Tree (BIT)

One of the most common variations of Fenwick Trees is the Binary Indexed Tree (BIT). BIT is a compact representation of Fenwick Trees designed to handle range updates and range queries efficiently.

The structure and operations of the BIT are similar to the standard Fenwick Tree, but with a few modifications. It uses a binary encoding scheme to represent the indices, and the update and query functions are adjusted accordingly to work with the encoded indices.

Here's an example of how to implement a Binary Indexed Tree (BIT) in C++:

1#include <iostream>
2using namespace std;
3
4class BinaryIndexedTree {
5  int* bit;
6  int size;
7
8public:
9  BinaryIndexedTree(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  int n;
36  cin >> n;
37
38  BinaryIndexedTree bit(n);
39
40  // Perform advanced Fenwick Tree variations operations
41
42  return 0;
43}

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#include <iostream>

using namespace std;

int main() {

  // Replace with your advanced Fenwick Tree variations logic here

}

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;

}

Conclusions

Congratulations! You've reached the end of the tutorial on Advanced Fenwick Tree Concepts. Let's recap what we've covered and highlight some key takeaways:

1. Fenwick Trees: We started by understanding the basic concepts and properties of Fenwick Trees. We learned about their efficient representation of prefix sums and how they support efficient range queries.

2. Updating a Fenwick Tree: We explored the process of updating values in a Fenwick Tree. We saw how the binary representation of the index plays a crucial role in the update operation.

3. Querying a Fenwick Tree: We discussed how to perform queries on a Fenwick Tree to obtain prefix sums and range queries. We observed how the binary representation of the index allows for efficient traversal in logarithmic time complexity.

4. Range Updates and Range Queries: We delved into the concepts of range updates and range queries in Fenwick Trees. We learned how to modify ranges and calculate the sum of values within a given range.

5. Advanced Fenwick Tree Variations: We explored advanced variations of Fenwick Trees, such as 2D Fenwick Trees and Fenwick Trees for non-commutative operations. We saw how these variations can be used to solve more complex problems efficiently.

6. Applications of Fenwick Trees: We discussed various real-world scenarios where Fenwick Trees are useful. We explored applications such as computing prefix sums, handling range queries, offline queries, and frequency counting.

7. Additional Resources: To deepen your understanding of Fenwick Trees, here are some recommended resources:

   • Blog: TopCoder - The Fenwick Tree Algorithm
   • Video: Fenwick Tree - Competitive Programming Course
   • Book: Introduction to Algorithms by Thomas H. Cormen et al. (Chapter 18.4)

Now that you're equipped with a solid understanding of Fenwick Trees, it's time to apply your knowledge to solve problems efficiently. Happy coding!