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.