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?"

Practice

1. What is the primary purpose of a Fenwick tree (Binary Indexed Tree)?

easy

A. To find the maximum element in a list

B. To quickly update elements and calculate prefix sums in a list

C. To sort elements in ascending order

D. To store elements in a balanced binary search tree

Solution

1. Step 1: Understand Fenwick tree functionality

   Fenwick trees are designed to efficiently update values and compute prefix sums in a list.

2. Step 2: Compare with other data structures

   Unlike sorting or maximum finding, Fenwick trees focus on cumulative sums and updates.

3. Final Answer:

   To quickly update elements and calculate prefix sums in a list -> Option B

4. Quick Check:

   Fenwick tree = fast updates + prefix sums [OK]

Hint: Fenwick trees = fast prefix sums and updates [OK]

Common Mistakes:

• Confusing Fenwick trees with sorting algorithms
• Thinking Fenwick trees find max/min values
• Assuming Fenwick trees store elements in sorted order

2. Which of the following is the correct way to update a Fenwick tree at index i with value val in pseudocode?

easy

A. for i in range(i, n): tree[i] += val

B. for i in range(n): tree[i] += val

C. while i <= n: tree[i] += val; i += i & (-i)

D. while i <= n: tree[i] = val; i *= 2

Solution

1. Step 1: Recall Fenwick tree update logic

   Updating involves adding val to tree[i] and moving to the next index using i += i & (-i).

2. Step 2: Check each option

   while i <= n: tree[i] += val; i += i & (-i) matches the correct loop and update pattern; others either use wrong loops or incorrect index updates.

3. Final Answer:

   while i <= n: tree[i] += val; i += i & (-i) -> Option C

4. Quick Check:

   Update uses i += i & (-i) [OK]

Hint: Update index by adding last set bit (i & -i) [OK]

Common Mistakes:

• Using a for loop instead of while
• Assigning value instead of adding
• Incorrect index increment like i *= 2

3. Given a Fenwick tree built for array [1, 2, 3, 4, 5], what is the prefix sum returned by querying index 3?

medium

A. 6

B. 9

C. 10

D. 5

Solution

1. Step 1: Calculate prefix sum up to index 3

   Sum of elements at indices 1, 2, and 3 is 1 + 2 + 3 = 6.

2. Step 2: Confirm Fenwick tree query result

   Fenwick tree query returns the sum of elements from start to index 3, which is 6.

3. Final Answer:

   6 -> Option A

4. Quick Check:

   Sum(1 to 3) = 6 [OK]

Hint: Sum up to index 3 = 1+2+3 = 6 [OK]

Common Mistakes:

• Adding extra elements beyond index 3
• Confusing prefix sum with total sum
• Using zero-based indexing incorrectly

4. Identify the error in this Fenwick tree update pseudocode snippet:

function update(i, val):
  while i < n:
    tree[i] += val
    i += i & (-i)

medium

A. The tree array should be zero-indexed

B. The update should subtract val, not add

C. The index increment should be i -= i & (-i)

D. The loop condition should be while i <= n, not < n

Solution

1. Step 1: Check loop boundary condition

   The Fenwick tree size is n, so updates must include index n, requiring <= instead of <.

2. Step 2: Verify update logic

   Adding val and incrementing index by i & (-i) is correct; no subtraction or decrement needed.

3. Final Answer:

   The loop condition should be while i <= n, not < n -> Option D

4. Quick Check:

   Update loop uses i <= n [OK]

Hint: Use <= n in update loop to include last index [OK]

Common Mistakes:

• Using < instead of <= in loop condition
• Decrementing index instead of incrementing
• Confusing add with subtract in update

5. You have a Fenwick tree for an array of size 8. After updating index 4 by +3, which indices in the Fenwick tree are guaranteed to be updated?

hard

A. 4, 8

B. 4, 5, 6, 7, 8

C. 1, 2, 4

D. Only index 4

Solution

1. Step 1: Understand Fenwick tree update jumps

   Starting at index 4, the next index updated is 4 + (4 & -4) = 4 + 4 = 8.

2. Step 2: List updated indices

   Indices 4 and 8 are updated; no other indices between 4 and 8 are updated in this step.

3. Final Answer:

   4, 8 -> Option A

4. Quick Check:

   Update jumps: i, i + (i & -i) [OK]

Hint: Update indices jump by last set bit (4 to 8) [OK]

Common Mistakes:

• Assuming all indices after 4 update
• Confusing update with query indices
• Forgetting Fenwick tree uses bitwise jumps