Fenwick trees (Binary Indexed Trees) in Data Structures Theory - Time & Space Complexity

Time Complexity: Fenwick trees (Binary Indexed Trees)

O(log n)

Understanding Time Complexity

Fenwick trees help us quickly update and find sums in a list of numbers. Understanding their time complexity shows how fast these operations grow as the list gets bigger.

We want to know how the time to update or query changes when the input size increases.

Scenario Under Consideration

Analyze the time complexity of the following Fenwick tree update and query operations.

function fenw_update(i, val, fenw, n) {
  while (i <= n) {
    fenw[i] += val;
    i += i & (-i);
  }
}

function fenw_sum(i, fenw) {
  let result = 0;
  while (i > 0) {
    result += fenw[i];
    i -= i & (-i);
  }
  return result;
}
    

This code updates values and calculates prefix sums using a Fenwick tree structure.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The while loops in both update and sum functions.
• How many times: Each loop runs a number of times proportional to the number of bits in the index, roughly the logarithm of the input size.

How Execution Grows With Input

As the input size grows, the number of steps in update or sum grows slowly, increasing by about one step each time the input size doubles.

Input Size (n)	Approx. Operations
10	About 4 steps
100	About 7 steps
1000	About 10 steps

Pattern observation: The steps grow slowly, roughly adding one step for every doubling of input size.

Final Time Complexity

Time Complexity: O(log n)

This means the time to update or query grows slowly and efficiently as the list gets bigger.

Common Mistake

[X] Wrong: "Fenwick tree operations take linear time because they use loops."

[OK] Correct: The loops jump by powers of two, so they run only about as many times as the number of bits in the input size, which is much less than the input length.

Interview Connect

Knowing Fenwick trees and their time complexity shows you can handle efficient data updates and queries, a useful skill for many coding challenges and real-world problems.

Self-Check

"What if we changed the Fenwick tree to store sums of ranges instead of prefix sums? How would the time complexity change?"