• Explanation:
  • The Binary Indexed Tree (BIT), also known as a Fenwick Tree, is a data structure that efficiently supports dynamic cumulative frequency tables.
  • It allows you to compute prefix sums and update elements in O(log n) time. It is particularly useful for problems that require frequent range sum queries and updates.

Steps:

• The tree is implemented using an array, where each node represents a cumulative sum of elements.
• Update Operation: Updates an element in the array, and its impact propagates upward.
• Query Operation: Computes the sum of elements from the start to a given index by summing the values of the relevant nodes.

Time Complexity:

• Update: O(log n)
• Query: O(log n)

class FenwickTree:
    def __init__(self, n):
        self.n = n
        self.tree = [0] * (n + 1)
 
    def update(self, index, delta):
        while index <= self.n:
            self.tree[index] += delta
            index += index & -index  # Move to the parent node
 
    def query(self, index):
        sum = 0
        while index > 0:
            sum += self.tree[index]
            index -= index & -index  # Move to the parent node
        return sum
 
# Example usage
fenwick = FenwickTree(5)
fenwick.update(1, 3)
fenwick.update(3, 5)
 
print(f"Sum of first 3 elements: {fenwick.query(3)}")  # Output: 8
print(f"Sum of first 5 elements: {fenwick.query(5)}")  # Output: 8