Fenwick trees (Binary Indexed Trees) in Data Structures Theory - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding Fenwick Tree Structure

Which of the following best describes the main purpose of a Fenwick tree (Binary Indexed Tree)?

A It sorts data in linear time using a tree structure.

B It compresses data to reduce memory usage without losing information.

C It balances binary search trees automatically during insertions.

D It stores prefix sums efficiently and allows updates in logarithmic time.

💡 Hint

Think about what Fenwick trees help calculate quickly.

Fenwick trees are designed to efficiently compute prefix sums and update values in logarithmic time, making them useful for cumulative frequency calculations.

📋 Factual

intermediate

Fenwick Tree Update Operation

In a Fenwick tree, when updating the value at index i, which operation determines the next index to update?

A Subtract the least significant set bit of <code>i</code> from <code>i</code>.

B Add the least significant set bit of <code>i</code> to <code>i</code>.

C Add 1 to <code>i</code>.

D Multiply <code>i</code> by 2.

💡 Hint

Consider how the Fenwick tree moves upward to cover ranges.

The update operation moves to the next index by adding the least significant set bit of the current index, allowing it to cover all relevant ranges.

🔍 Analysis

advanced

Fenwick Tree Query Result

Given a Fenwick tree built over the array [3, 2, -1, 6, 5, 4, -3, 3, 7, 2, 3], what is the result of querying the prefix sum up to index 5 (1-based)?

Data Structures Theory

array = [3, 2, -1, 6, 5, 4, -3, 3, 7, 2, 3]
# Query prefix sum up to index 5

A 15

B 10

C 5

D 20

💡 Hint

Sum the first 5 elements of the array.

The prefix sum up to index 5 is 3 + 2 + (-1) + 6 + 5 = 15.

❓ Comparison

advanced

Fenwick Tree vs Segment Tree

Which of the following statements correctly compares Fenwick trees and segment trees?

A Fenwick trees use less memory and are simpler but support fewer types of queries than segment trees.

B Fenwick trees use more memory and are slower than segment trees for all operations.

C Segment trees cannot handle updates efficiently, unlike Fenwick trees.

D Fenwick trees can only be used for static arrays, while segment trees support dynamic arrays.

💡 Hint

Think about memory usage and query flexibility.

Fenwick trees are simpler and use less memory but mainly support prefix sums and point updates, while segment trees are more flexible but use more memory.

❓ Reasoning

expert

Fenwick Tree Indexing Behavior

Consider a Fenwick tree indexed from 1 to n. If you perform a query for the prefix sum at index 0, what will happen?

A The query will return the value at index 1 only.

B The query will cause an index error because 0 is not a valid Fenwick tree index.

C The query will return 0 because no elements are included.

D The query will return the sum of all elements in the tree.

💡 Hint

Think about how prefix sums are defined starting at index 1.

Querying prefix sum at index 0 means summing zero elements, so the result is 0 without error.

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