DSA Fenwick Tree & Range Queries - Practice Questions 2026
Course Overview
About This Free Course
Mastering complex learn data structures algorithms interview preparation practice like the Fenwick Tree (Binary Indexed Tree) and Range Queries is a significant milestone for any software engineer or competitive programmer. This course is meticulously designed to provide you with the rigorous practice needed to transition from understanding the theory to implementing efficient solutions in high-pressure environments.
Why Serious Learners Choose These Practice Exams
Aspiring top-tier developers choose these practice exams because they bridge the gap between "knowing" a data structure and "applying" it. Fenwick Trees are notoriously tricky to visualize and implement without errors. Our questions are crafted to challenge your understanding of index manipulation, bitwise operations, and the nuances of range sum updates. By working through these problems, you develop the muscle memory required for technical interviews at companies like Google, Meta, and Amazon, as well as competitive programming platforms like Codeforces and LeetCode.
Course Structure
Our curriculum is divided into logical tiers to ensure a smooth learning curve:
Basics / Foundations: Focuses on the prerequisite knowledge of bit manipulation and the structural logic of a Binary Indexed Tree. You will answer questions regarding why we use $i \ \& \ (-i)$ and how the tree structure is mapped over a standard array.
Core Concepts: Deep dives into the fundamental operations: Point Update and Prefix Sum. These tests ensure you understand the $O(\log n)$ complexity and the traversal paths from a leaf to the root.
Intermediate Concepts: Moves beyond simple sums to explore Range Updates and Point Queries. This section challenges you to think about how to manipulate the Fenwick Tree to handle more complex query types.
Advanced Concepts: Covers sophisticated applications such as 2D Fenwick Trees and Range Updates combined with Range Queries. This is where you learn to optimize space and time for multi-dimensional data.
Real-world Scenarios: Situates Fenwick Trees within larger problems. You will encounter questions that ask you to choose the best data structure for a specific scenario, comparing Fenwick Trees against Segment Trees or Square Root Decomposition.
Mixed Revision / Final Test: A comprehensive, timed mock exam featuring a random selection of questions from all previous tiers to test your retention and speed.
Question 1
In a standard 1-indexed Fenwick Tree (BIT), which formula is used to find the parent index while performing a prefix sum query for index $i$?
Option 1: $i = i + (i \ \& \ -i)$
Option 2: $i = i - (i \ \& \ -i)$
Option 3: $i = i \ | \ (i + 1)$
Option 4: $i = i \ \& \ (i - 1)$
Option 5: $i = 2 \times i$
Correct Answer: Option 2
Correct Answer Explanation: To calculate the prefix sum up to index $i$ in a Fenwick Tree, we sum the value stored at tree[i] and then move to the next responsible range by stripping the last set bit. The operation $i \ \& \ -i$ isolates the least significant bit (LSB). Subtracting this from $i$ effectively moves the pointer to the preceding cumulative frequency in the structure.
Wrong Answers Explanation:
Option 1: This formula is used for the Update operation, not the query operation. It moves the index "up" to all ranges that include the current index.
Option 3: This is a bitwise trick often used in different data structures but does not correctly identify the parent in a Fenwick Tree query.
Option 4: While $i \ \& \ (i - 1)$ also removes the lowest set bit, the standard free gen ai readiness implementation and transformation course of a BIT query specifically utilizes the two’s complement identity $i \ \& \ -i$ for clarity and consistency.
Option 5: This is the logic for finding a child in a Binary Heap or Segment Tree, not a Fenwick Tree.
Question 2
What is the space complexity of a 2D Fenwick Tree designed to handle range sum queries on an $M \times N$ matrix?
Option 1: $O(M + N)$
Option 2: $O(\log M \times \log N)$
Option 3: $O(M \times N)$
Option 4: $O(M^2)$
Option 5: $O(N \log M)$
Correct Answer: Option 3
Correct Answer Explanation: A Fenwick Tree, whether 1D or 2D, requires an auxiliary array of the same dimensions as the input data to store the cumulative frequencies. Therefore, for an $M \times N$ matrix, the tree requires $O(M \times N)$ space.
Wrong Answers Explanation:
Option 1: This is linear space relative to dimensions, which is insufficient to store the cumulative data of a 2D grid.
Option 2: This is the time complexity for a single update or query in a 2D BIT, not the space complexity.
Option 4: This would only be true if $N = M$, but it generally overestimates space if the matrix is rectangular.
Option 5: This does not account for the full storage requirements of the 2D structure.
Question 3
When comparing a Fenwick Tree to a Segment Tree for simple Range Sum Queries and Point Updates, which statement is true?
Option 1: Segment Trees are easier to implement than Fenwick Trees.
Option 2: Fenwick Trees use significantly more memory than Segment Trees.
Option 3: Fenwick Trees are generally faster and use less memory than Segment Trees.
Option 4: Fenwick Trees can solve any problem a Segment Tree can solve.
Option 5: Fenwick Trees have $O(1)$ query time.
Correct Answer: Option 3
Correct Answer Explanation: Fenwick Trees are preferred for simple prefix sums because they are iterative and require exactly $N$ space (the same as the input array). Segment Trees, while more versatile, usually require $4N$ space and involve recursive calls, which adds overhead.
Wrong Answers Explanation:
Option 1: Fenwick Trees are widely considered much shorter and easier to code (often just a few lines).
Option 2: This is the opposite of the truth; Fenwick Trees are more memory-efficient.
Option 4: Fenwick Trees are primarily for functions that have an inverse (like addition/subtraction). Segment Trees can handle non-invertible functions like Range Minimum Query (RMQ) much more easily.
Option 5: Both structures have $O(\log N)$ query time.
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