Data Structures u0026 Algorithms: Segment Trees and Fenwick Trees Quiz

This quiz tests your mastery of complex concepts and competitive programming applications related to Segment Trees, Fenwick Trees (Binary Indexed Trees), range queries, and lazy propagation.

1. Segment Tree Range Sum Efficiency

   Given an array of n integers, which operation does a standard segment tree support in O(log n) time per operation, assuming no lazy propagation?

   1. Range minimum queries and range addition updates
   2. Point sum queries and range minimum updates
   3. Range sum queries and point updates
   4. Range multiplication queries and range sum updates
   5. Point multiplication queries and point minimum updates

2. Fenwick Tree Structure Insight

   What is the significance of the least significant one bit in the index representation when performing range updates in a Fenwick Tree (BIT)?

   1. It determines the size of the current prefix operated upon
   2. It finds the subtree parent in segment trees
   3. It allows lazy propagation of nodes above it
   4. It signals the node is a leaf in the original array
   5. It resets the tree during initialization

3. Lazy Propagation in Range Updates

   How does lazy propagation enhance the efficiency of segment tree range updates, such as adding a value to all elements in a subarray?

   1. It precomputes the answers to all possible queries
   2. It merges queries and updates into a single function call
   3. It directly updates all leaf nodes resulting in O(n) updates
   4. It reduces the tree’s height to make updates faster
   5. It delays updates until the affected segment is required, reducing redundant computations

4. Max Query Application

   Given an array supporting frequent range maximum queries and single element updates, which data structure is typically more space and time efficient than others for large arrays?

   1. Segment Tree
   2. Fenwick Tree
   3. Red-Black Tree
   4. Trie
   5. Treap

5. Prefix Sum with BIT Complexity

   What is the time complexity to compute a prefix sum up to index i using a Fenwick Tree with n elements, assuming standard 1-based indexing?

   1. O(n)
   2. O(log log n)
   3. O(n log n)
   4. O(log n)
   5. O(1)

6. Range Multiplication Challenge

   Suppose you wish to support range multiplication and range sum queries modulo a prime number p using a segment tree. What key technique is necessary to ensure correctness of simultaneous range operations?

   1. Disjoint set union on tree nodes
   2. Heavy-light decomposition
   3. Only point updates and no propagation
   4. Lazy propagation with multiplicative and additive tags
   5. Prefix XOR propagation

7. Building Range Trees: Initial Time

   What is the time complexity to construct a Segment Tree for an array of n elements where each node stores a sum, assuming the array is provided?

   1. O(n^2)
   2. O(n log n)
   3. O(n)
   4. O(log n)
   5. O(1)

8. Competitive Fenwick Tree Use Case

   In competitive programming, which operation can be optimized using Fenwick Trees when solving inversion count problems in an array?

   1. Processing range maximum updates
   2. Counting the number of elements greater than a given prefix value
   3. Reversing a subarray efficiently
   4. Implementing trie-based binary search
   5. Finding the minimum element in a range

9. Dynamic Range Updates Limit

   Which limitation exists for a standard (non-lazy) Fenwick Tree when attempting to perform range updates and range queries concurrently?

   1. It cannot store negative numbers correctly
   2. It cannot perform efficient range queries after range updates
   3. It triples memory space for auxiliary trees
   4. It requires O(n) updates for each operation
   5. It only works with sorted arrays

10. Lowest Common Ancestor Advantage

    Which scenario best illustrates where a segment tree outperforms a Fenwick Tree in handling queries efficiently?

    1. Storing distinct elements in a hash structure
    2. Performing frequent point additions
    3. Calculating the total frequency count of a single value
    4. Finding the minimum element between any two indices in O(log n) time
    5. Maintaining a running prefix sum over an array