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;

}

To create a Fenwick Tree, we need to initialize an array of size n+1, where n is the size of the input array. This extra space is used to store the prefix sums. The Fenwick Tree can be seen as a binary indexed tree, with each index representing a range of elements.

The initialization process involves setting all elements of the Fenwick Tree array to 0. This can be done using a for loop or using the memset function in C++.

1#include <iostream>
2#include <cstring>
3using namespace std;
4
5const int MAX_SIZE = 100;
6int BIT[MAX_SIZE];
7
8void initialize() {
9  memset(BIT, 0, sizeof(BIT));
10}
11
12int main() {
13  initialize();
14  // Your Fenwick Tree logic
15  return 0;
16}

To update a value in a Fenwick Tree, we need to make two changes: update the original array and update the prefix sum in the Fenwick Tree. Let's say we want to update the value at index i with a new value val.

In order to update the original array, we simply assign the new value val to arr[i].

1int index = i;
2int value = val;
3arr[index] = value;

To update the prefix sum in the Fenwick Tree, we need to find the difference between the new value val and the old value arr[i]. Let's call this difference diff. Then, we update the Fenwick Tree by adding diff to the prefix sum at each index that represents the range where the value at index i contributes.

1void update(int BIT[], int n, int i, int val) {
2    while (i <= n) {
3        BIT[i] += val;
4        i += i & (-i);
5    }
6}

Let's see an example of how to update a value in a Fenwick Tree using C++:

1#include <iostream>
2using namespace std;
3
4const int n = 10;
5int arr[n + 1] = {0};
6int BIT[n + 1] = {0};
7
8// Update the value at index i in the Fenwick Tree
9void update(int BIT[], int n, int i, int val) {
10    // Adding the updated value at index i
11    while (i <= n) {
12        BIT[i] += val;
13        i += i & (-i);
14    }
15}
16
17int main() {
18    // Updating value at index 5 to 10
19    int index = 5;
20    int value = 10;
21    int diff = value - arr[index];
22    arr[index] = value;
23    update(BIT, n, index, diff);
24
25    return 0;
26}

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#include <iostream>

using namespace std;

​

// Update the value at index i in the Fenwick Tree

void update(int BIT[], int n, int i, int val) {

    // Adding the updated value at index i

    while (i <= n) {

        BIT[i] += val;

        i += i & (-i);

    }

}

​

int main() {

    const int n = 10;

    int arr[n + 1] = {0};

    int BIT[n + 1] = {0};

​

    // Updating value at index 5 to 10

    int index = 5;

    int value = 10;

    int diff = value - arr[index];

    arr[index] = value;

    update(BIT, n, index, diff);

​

    return 0;

}

To compute the prefix sum using a Fenwick Tree, we need to find the sum of all the elements in the original array up to a given index.

Let's say we want to calculate the prefix sum up to index i. We can do this by traversing the Fenwick Tree from index i to 0 and adding the values at each index to the sum. This is called the 'prefix sum query'. The resulting sum will be the prefix sum up to index i.

1int prefixSumQuery(int BIT[], int i) {
2    int sum = 0;
3    while (i > 0) {
4        sum += BIT[i];
5        i -= i & (-i);
6    }
7    return sum;
8}

Here's an example of how to compute the prefix sum using a Fenwick Tree:

1#include <iostream>
2using namespace std;
3
4const int n = 10;
5int arr[n + 1] = {0};
6int BIT[n + 1] = {0};
7
8// Compute the prefix sum up to index i
9int prefixSumQuery(int BIT[], int i) {
10    int sum = 0;
11    while (i > 0) {
12        sum += BIT[i];
13        i -= i & (-i);
14    }
15    return sum;
16}
17
18int main() {
19    // Computing prefix sum up to index 5
20    int index = 5;
21    int prefixSum = prefixSumQuery(BIT, index);
22    cout << "Prefix Sum up to index " << index << ": " << prefixSum << endl;
23
24    return 0;
25}

Now that we have learned how to calculate prefix sums using a Fenwick Tree, let's explore how we can perform range queries using this data structure.

A range query involves finding the sum of all the elements within a given range of indices in the original array. This can be achieved efficiently using the Fenwick Tree.

To perform a range query, we need to calculate the prefix sum for both the starting index and the ending index of the range. Then, we can subtract the prefix sum of the starting index from the prefix sum of the ending index (inclusive) to get the sum of the elements within the range.

1int rangeQuery(int BIT[], int start, int end) {
2    int sumStart = prefixSumQuery(BIT, start - 1);
3    int sumEnd = prefixSumQuery(BIT, end);
4    return sumEnd - sumStart;
5}

Let's take a look at an example to see how this works.

1#include <iostream>
2using namespace std;
3
4const int n = 10;
5int arr[n + 1] = {0};
6int BIT[n + 1] = {0};
7
8// Compute the prefix sum up to index i
9int prefixSumQuery(int BIT[], int i) {
10    int sum = 0;
11    while (i > 0) {
12        sum += BIT[i];
13        i -= i & (-i);
14    }
15    return sum;
16}
17
18// Perform a range query
19int rangeQuery(int BIT[], int start, int end) {
20    int sumStart = prefixSumQuery(BIT, start - 1);
21    int sumEnd = prefixSumQuery(BIT, end);
22    return sumEnd - sumStart;
23}
24
25int main() {
26    // Update values in the Fenwick Tree
27    // ... (code for updating values)
28
29    // Perform a range query
30    int start = 3;
31    int end = 7;
32    int rangeSum = rangeQuery(BIT, start, end);
33
34    cout << "Sum of elements in range [" << start << ", " << end << "]: " << rangeSum << endl;
35
36    return 0;
37}

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}

#include <iostream>

using namespace std;

​

const int n = 10;

int arr[n + 1] = {0};

int BIT[n + 1] = {0};

​

// Compute the prefix sum up to index i

int prefixSumQuery(int BIT[], int i) {

    int sum = 0;

    while (i > 0) {

        sum += BIT[i];

        i -= i & (-i);

    }

    return sum;

}

​

// Perform a range query

int rangeQuery(int BIT[], int start, int end) {

    int sumStart = prefixSumQuery(BIT, start - 1);

    int sumEnd = prefixSumQuery(BIT, end);

    return sumEnd - sumStart;

}

​

int main() {

    // Update values in the Fenwick Tree

    // ... (code for updating values)

​

    // Perform a range query

Advanced Applications of Fenwick Trees

Fenwick Trees, also known as Binary Indexed Trees (BIT), have a wide range of applications beyond just computing prefix sums and performing range queries.

One of the advanced applications of Fenwick Trees is solving the problem of finding the k-th element in a sorted array. Given an initial array, the Fenwick Tree can help efficiently perform insertions and deletions, while still being able to find the k-th element in O(log N) time complexity.

Another application of Fenwick Trees is solving the problem of finding the median of a set of numbers. By maintaining two Fenwick Trees, one for the left half of the numbers and the other for the right half, we can easily find the median in O(log N) time complexity.

Furthermore, Fenwick Trees can also be used to solve the problem of finding the inversion count in an array. An inversion in an array occurs when two elements are out of order. By using a Fenwick Tree, we can efficiently count the number of inversions in an array in O(N log N) time complexity.

Let's take a look at the implementation of finding the inversion count in an array using Fenwick Trees in C++:

1#include <iostream>
2using namespace std;
3
4int getSum(int BIT[], int index) {
5  int sum = 0;
6  while (index > 0) {
7    sum += BIT[index];
8    index -= index & (-index);
9  }
10  return sum;
11}
12
13void updateBIT(int BIT[], int n, int index, int value) {
14  while (index <= n) {
15    BIT[index] += value;
16    index += index & (-index);
17  }
18}
19
20int getInversionCount(int arr[], int n) {
21  int inversionCount = 0;
22  int maxElement = 0;
23
24  for (int i = 0; i < n; i++) {
25    if (arr[i] > maxElement) {
26      maxElement = arr[i];
27    }
28  }
29
30  int BIT[maxElement + 1];
31  for (int i = 1; i <= maxElement; i++) {
32    BIT[i] = 0;
33  }
34
35  for (int i = n - 1; i >= 0; i--) {
36    inversionCount += getSum(BIT, arr[i] - 1);
37    updateBIT(BIT, maxElement, arr[i], 1);
38  }
39
40  return inversionCount;
41}
42
43int main() {
44  int arr[] = {8, 4, 2, 1};
45  int n = sizeof(arr) / sizeof(arr[0]);
46
47  int inversionCount = getInversionCount(arr, n);
48
49  cout << "Inversion Count: " << inversionCount << endl;
50
51  return 0;
52}

In the above example, we initialize a Fenwick Tree of size maxElement + 1 to maintain the count of each element occurrence in the array. Then, we iterate over the array from right to left and count the number of elements smaller than the current element on the right side. This gives us the inversion count of the array.

By exploring these advanced applications, you'll be able to leverage the power of Fenwick Trees in a variety of problem-solving scenarios.

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#include <iostream>

using namespace std;

​

int main() {

  // Replace with your C++ logic here

​

  return 0;

}

Putting It All Together

Congratulations on completing the tutorial on solving problems with Fenwick Trees! By now, you have learned and implemented various concepts related to Fenwick Trees, such as:

• Introduction to Fenwick Trees
• Creating a Fenwick Tree
• Updating Values in a Fenwick Tree
• Computing Prefix Sums
• Performing Range Queries
• Advanced Applications of Fenwick Trees

Now, it's time to bring all these concepts together and showcase their power!

In this section, we will consolidate our understanding of Fenwick Trees and how they can be used to efficiently solve problems. We will combine all the concepts learned in the tutorial and implement a core Fenwick Tree logic.

Let's take a look at the code snippet below that brings together all the concepts learned:

1#include <iostream>
2using namespace std;
3
4int main() {
5  // Combine all the concepts learned in the tutorial
6  cout << "Putting It All Together!" << endl;
7
8  // Core Fenwick Tree implementation logic
9  // ... Replace this comment with relevant code ...
10
11  return 0;
12}

Feel free to replace the comment with your relevant code to implement a problem-specific Fenwick Tree solution. This is the point where you can unleash your creativity and apply Fenwick Trees to various domains or problem scenarios!

Remember, Fenwick Trees are a powerful data structure that can provide efficient solutions to a wide range of problems. By understanding their underlying principles and leveraging their capabilities, you can become a proficient problem solver using Fenwick Trees.

Keep exploring and applying Fenwick Trees in your algorithms and see how they can optimize your solution!

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#include <iostream>

using namespace std;

​

int main() {

  // Combine all the concepts learned in the tutorial

  cout << "Putting It All Together!" << endl;

​

  // Core Fenwick Tree implementation logic

  // ... Replace this comment with relevant code ...

​

  return 0;

}

What is the time complexity of updating a value in a Fenwick Tree?