Using a tree array (offline query), calculate the number of elements greater than K in the range L to R

In the field of computer science, we have to deal with large data sets, which include query select and update operations. Performing these operations in real-time with low time complexity is a challenging task for developers.

Using Fenwick trees is an effective way to solve these range-based query problems.

Fenwick Tree is a data structure that can efficiently update elements and calculate the prefix sum of numbers in a table. It is also called a binary index tree.

In this article, we will discuss how to use Fenwick tree to find the number of elements greater than K in the range L to R.

Input and output scenarios

Suppose you have an array with N elements, and you want to find the number of elements in the array that are greater than K in the range L to R.

Input: 
arr = {1, 15, 12, 13, 6, 18, 14, 10, 23, 41, 16, 9}
N = 11, K = 17, L = 1, R = 8
Output: 
2

What is offline query?

Offline query is a query operation performed on a predetermined data set. In other words, these operations are performed only on those datasets that are not further modified while processing the query.

Use Fenwick Tree

Here we will use a Fenwick tree, which has vectors at each node containing all the elements of the array in order. We then use binary search to answer each query and count the number of elements.

• Create two functions, updateTree() and total(), for updating and retrieving the prefix sum in BIT respectively.

• Next, create another function that will count elements in the range L to R that are greater than "K". This function accepts the input values ​​"arr", "N", "L", "R" and "K" ".

• The count is calculated by subtracting the cumulative sum of K from the cumulative sum of the maximum range N.

To exclude out-of-range elements, subtract the cumulative sum of L-1 (if L is greater than 1).

Example

#include <iostream>
#include <vector>
using namespace std;

// Updating the Fenwick Tree
void updateTree(vector<int>& BIT, int index, int value) {
   while (index < BIT.size()) {
      BIT[index] += value;
      index += index & -index;
   }
}

int total(vector<int>& BIT, int index) {
   int result = 0;
   while (index > 0) {
      result += BIT[index];
      index -= index & -index;
   }
   return result;
}

// Counting the number of elements
int elementsGreaterThanK(vector<int>& arr, int N, int K, int L, int R) {
   vector<int> BIT(N+1, 0);

   for (int i = L; i <= R; i++) {
      updateTree(BIT, arr[i], 1);
   }

   int count = total(BIT, N) - total(BIT, K);
   if (L > 1) {
      count -= total(BIT, L-1);
   }
   return count;
}

int main() {
   vector<int> arr = {0, 5, 2, 3, 6, 8, 7, 1, 8, 4, 6, 9};
   int N = 11; // Total range of values
   int K = 4; // Highest value
   int L = 1; // Start index of the range
   int R = 8; // End index of the range

   int count = elementsGreaterThanK(arr, N, K, L, R);
   cout << "Number of elements greater than " << K << " in the range [" << L << ", " << R << "]: " << count << endl;

   return 0;
}

Output

Number of elements greater than 4 in the range [1, 8]: 5

alternative method

Here we will create a separate vector to store the query. Each query consists of two variables, index and val. index is used to store the position in the array, and val is used to store the value of the element at that index. This approach enables developers to perform a variety of query operations. Additionally, we use BIT to count the number of elements greater than K for each query.

Example

#include <iostream>
#include <vector>
using namespace std;
struct Query {
   int index;
   int val;
};

void updateTree(vector<int>& BIT, int index, int value) {
   while (index < BIT.size()) {
      BIT[index] += value;
      index += index & -index;
   }
}

int total(vector<int>& BIT, int index) {
   int result = 0;
   while (index > 0) {
      result += BIT[index];
      index -= index & -index;
   }
   return result;
}

vector<int> elementsGreaterThanK(vector<int>& arr, vector<Query>& queries, int K) {
   int N = arr.size();
   vector<int> BIT(N+1, 0);
   vector<int> result;

   for (int i = 0; i < N; i++) {
      updateTree(BIT, i+1, (arr[i] > K) ? 1 : 0);
   }

   for (auto query : queries) {
      if (query.val > K) {
         result.push_back(total(BIT, query.index));
      }
      else {
         result.push_back(0);
      }
   }

   return result;
}

int main() {
   vector<int> arr = {3, 14, 32, 7, 11, 5, 8, 6, 16, 2, 15};
   vector<Query> queries = {{2, 4}, {5, 3}, {3, 6}, {0, 3}};
   int K = 2;

   vector<int> counts = elementsGreaterThanK(arr, queries, K);

   for (int count : counts) {
      cout << "Number of elements greater than " << K << ": " << count << endl;
   }

   return 0;
}

Output

Number of elements greater than 2: 2
Number of elements greater than 2: 5
Number of elements greater than 2: 3
Number of elements greater than 2: 0

in conclusion

We can simply iterate the array from index L to R and increment by 1 every time the array element is greater than K in order to find the answer to each query. However, in order to reduce the time complexity, we use Fenwick tree to perform such query operations. We can also use Line Segment Tree and Sparse Table instead of Fenwick tree to perform such query operations.

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