Array Representation of Heaps Quiz
40 comprehensive questions exploring array representation of heaps — with 16 code examples covering parent-child indices, index formulas, and memory efficiency in this cpp quiz.
40 Questions
~80 minutes
1
Question 1
What is array representation of heaps?
A
Storing heap elements in a contiguous array where parent-child relationships are maintained through mathematical index calculations, with the root at index 0 and children at predictable positions based on their parent's index
B
Using linked lists for heap storage
C
Storing elements in random order
D
Array representation is not used for heaps
2
Question 2
How is the root stored in array representation?
cpp
Root always at index 0
Children at indices 1 and 2A
The root element is always stored at index 0 in the array, with its left child at index 1 and right child at index 2, establishing the foundation for all subsequent parent-child relationships in the heap structure
B
Root is at the end of the array
C
Root index varies randomly
D
Root is not stored in arrays
3
Question 3
What is the formula for left child index?
A
For a node at index i, the left child is at index (2 * i) + 1, providing a direct mathematical relationship that allows constant-time access to child nodes without storing explicit pointers in the heap structure
B
Left child is at index i + 1
C
Left child is at index 2 * i
D
Left child formula doesn't exist
4
Question 4
How do you calculate right child index?
cpp
Right child: (2 * i) + 2
For parent at index iA
The right child of a node at index i is located at index (2 * i) + 2, following immediately after the left child and maintaining the complete binary tree property in the array representation
B
Right child is at index i + 2
C
Right child is at index 2 * i
D
Right child calculation is different
5
Question 5
What is the parent index formula?
A
For a node at index i, the parent is at index (i - 1) / 2 using integer division, allowing upward traversal in the heap during heapify-up operations and maintaining the tree structure relationships
B
Parent is at index i / 2
C
Parent is at index i - 1
D
Parent formula is not used
6
Question 6
How does array representation achieve memory efficiency?
text
No pointers needed
Contiguous memory layout
Better cache performanceA
Array representation eliminates the need for storing pointers to children, uses contiguous memory blocks for better cache locality, and provides O(1) access to any element through direct index calculations, significantly reducing memory overhead compared to pointer-based tree implementations
B
Arrays use more memory than pointers
C
Memory efficiency is not important
D
Arrays are less efficient than linked structures
7
Question 7
What is the space complexity of array-based heaps?
A
O(n) space complexity where n is the number of elements, with no additional space required for pointers since parent-child relationships are computed mathematically, making array heaps more space-efficient than pointer-based binary trees
B
O(n log n) space
C
O(1) space
D
Array heaps use more space
8
Question 8
How do you check if a node has children in array representation?
cpp
Left child index < heap size
Right child index < heap sizeA
A node at index i has a left child if (2 * i) + 1 < heap_size, and a right child if (2 * i) + 2 < heap_size, allowing boundary checking to prevent access beyond the valid heap elements in the array
B
Check if children exist using pointers
C
Children always exist
D
Array representation can't check children
9
Question 9
What is the advantage of contiguous memory in heap arrays?
A
Contiguous memory layout provides excellent cache locality during heap operations, as parent and child nodes are stored in nearby memory locations, reducing cache misses and improving performance compared to scattered pointer-based implementations
B
Contiguous memory is slower
C
Memory layout doesn't affect performance
D
Contiguous memory causes more cache misses
10
Question 10
How does array representation handle heap resizing?
cpp
Array capacity exceeded
Allocate larger array
Copy elements to new arrayA
When the array capacity is exceeded during insertion, a new larger array is allocated and all existing elements are copied to the new contiguous memory block, maintaining the heap's array-based structure while allowing dynamic growth
B
Arrays don't resize in heaps
C
Resizing breaks the heap property
D
Heap arrays are fixed size
11
Question 11
What is the relationship between heap height and array size?
A
Heap height h relates to array size n by the formula h = floor(log₂(n)), with the maximum number of elements at height h being 2ʰ, determining the heap's logarithmic height growth as elements are added to the array
B
Height is unrelated to array size
C
Array size determines height linearly
D
Height affects array size minimally
12
Question 12
How do index formulas enable O(1) parent access?
cpp
Parent index: (i - 1) / 2
Direct calculation, no traversalA
Index formulas allow direct calculation of parent positions using simple arithmetic, providing O(1) access to parent nodes during heapify-up operations without requiring pointer traversal or tree navigation through the heap structure
B
Parent access requires O(log n) time
C
Index formulas don't help parent access
D
Parent access is O(n)
13
Question 13
What is the memory layout advantage over pointer-based trees?
A
Array representation uses approximately half the memory of pointer-based binary trees by eliminating the storage overhead of two pointers per node, while maintaining all the structural relationships through mathematical calculations instead of explicit links
B
Arrays use more memory than pointers
C
Memory usage is the same
D
Pointer trees are more memory efficient
14
Question 14
How does array representation support heapify operations?
cpp
Direct index access to children
No pointer dereferencing neededA
Array representation enables direct index-based access to child nodes during heapify-down operations, eliminating pointer dereferencing overhead and allowing efficient comparison and swapping operations through simple array indexing arithmetic
B
Array representation slows down heapify
C
Heapify operations don't use arrays
D
Arrays make heapify more complex
15
Question 15
What is the cache performance benefit of array heaps?
A
Array-based heaps exhibit superior cache performance due to spatial locality, as heapify operations access elements in contiguous memory locations that are likely to reside in the same cache lines, reducing cache misses compared to pointer-based implementations
B
Array heaps have poor cache performance
C
Cache performance is identical
D
Pointer heaps have better cache locality
16
Question 16
How do you determine if an index represents a leaf node?
cpp
If (2 * i) + 1 >= heap_size
Then i is a leaf nodeA
A node at index i is a leaf if (2 * i) + 1 >= heap_size, meaning it has no left child and therefore no children at all, which is crucial for terminating heapify-down operations at the correct boundary
B
Leaf nodes are always at the end
C
Index checking doesn't determine leaves
D
All nodes are leaves in arrays
17
Question 17
What is the trade-off between array and pointer representations?
A
Array representation trades potential wasted space in dynamic arrays for better cache performance and simpler memory management, while pointer-based trees use more memory for pointers but allow flexible node allocation without contiguous memory requirements
B
There are no trade-offs
C
Arrays are always better
D
Pointers are always better
18
Question 18
How does array representation handle incomplete levels?
text
Last level may be incomplete
Array still contiguousA
Array representation naturally handles incomplete levels by storing elements contiguously, with the last level potentially having fewer elements than a complete level while maintaining the complete binary tree property through index calculations
B
Incomplete levels break array representation
C
Arrays require complete levels
D
Incomplete levels cause gaps
19
Question 19
What is the index range for heap elements?
A
Valid heap elements occupy indices from 0 to heap_size - 1, with the array potentially having larger capacity for future insertions, ensuring all heap operations work within the logical heap boundaries
B
Indices start from 1
C
Indices are not consecutive
D
Heap elements use random indices
20
Question 20
How do index formulas support heap traversal?
text
Level-order traversal
Index relationships maintain orderA
Index formulas naturally support level-order traversal since parent and child relationships are encoded in the index values, allowing efficient iteration through heap elements in breadth-first order without additional data structures
B
Index formulas don't support traversal
C
Traversal requires pointers
D
Arrays make traversal difficult
21
Question 21
What is the memory access pattern advantage?
A
Array representation provides predictable memory access patterns that modern CPU prefetchers can anticipate, leading to better performance as heap operations tend to access nearby elements that are likely already in cache
B
Memory access is unpredictable
C
Access patterns are random
D
Arrays have worse access patterns
22
Question 22
How does array size affect index calculations?
cpp
Size determines valid indices
Bounds checking requiredA
The heap size determines which indices are valid for heap operations, requiring bounds checking before accessing parent or child positions to prevent accessing elements outside the current heap's logical boundaries
B
Array size doesn't affect calculations
C
Size makes calculations invalid
D
Index calculations ignore size
23
Question 23
What is the relationship between array index and tree level?
A
Elements at level k in the heap tree occupy indices from 2ᵏ⁻¹ - 1 to 2ᵏ - 2, with level 0 (root) at index 0, level 1 at indices 1-2, and each subsequent level doubling the index range, maintaining the complete binary tree structure
B
Levels don't correspond to indices
C
Indices are unrelated to levels
D
Levels determine random indices
24
Question 24
How does array representation enable fast heap construction?
cpp
Bottom-up construction
Direct index access to subtreesA
Array representation enables efficient bottom-up heap construction by allowing direct index-based access to subtree roots, facilitating the heapify process on each subtree without pointer traversal or recursive navigation through the tree structure
B
Array representation slows construction
C
Construction requires pointers
D
Arrays make construction complex
25
Question 25
What is the space overhead comparison?
A
Array-based heaps have lower space overhead compared to pointer-based trees, typically using about 50% less memory per element since they don't store left and right child pointers, though they may waste some space in dynamic array capacity
B
Arrays have higher overhead
C
Space overhead is identical
D
Pointer trees have lower overhead
26
Question 26
How do index formulas handle the root case?
cpp
Root at index 0
No parent: (0-1)/2 = -0.5 -> 0 in integer divisionA
For the root at index 0, the parent formula (i-1)/2 yields 0 in integer division, correctly indicating that the root has no valid parent, which serves as the termination condition for heapify-up operations reaching the tree root
B
Root case causes errors
C
Index formulas fail at root
D
Root has a parent
27
Question 27
What is the cache line utilization advantage?
A
Array representation maximizes cache line utilization during heap operations, as multiple heap elements often fit within a single cache line, allowing efficient bulk memory access and reducing the number of cache misses during heapify operations
B
Cache lines are underutilized
C
Utilization is the same as pointers
D
Arrays waste cache lines
28
Question 28
How does array representation support parallel heap operations?
cpp
Independent subtrees
Index ranges allow parallel processingA
Array representation facilitates parallel heap construction and operations by allowing independent processing of disjoint subtrees through their index ranges, enabling concurrent heapify operations on different parts of the heap without data dependencies
B
Arrays don't support parallelism
C
Parallel operations are slower
D
Pointers are better for parallelism
29
Question 29
What is the memory allocation strategy for array heaps?
A
Array heaps typically use dynamic arrays with capacity doubling strategy, allocating contiguous memory blocks that can accommodate future growth while maintaining the O(1) access properties and cache-friendly memory layout of the heap structure
B
Fixed-size arrays only
C
Memory allocation is complex
D
Arrays don't support dynamic allocation
30
Question 30
How do index formulas enable heap sorting?
cpp
In-place sorting
Index relationships maintainedA
Index formulas enable in-place heap sort by maintaining parent-child relationships through mathematical calculations, allowing the sorting process to work within the original array without requiring additional storage or pointer manipulation
B
Index formulas don't help sorting
C
Sorting requires pointer changes
D
Arrays make sorting impossible
31
Question 31
What is the boundary condition for heap indices?
A
Valid heap indices range from 0 to heap_size - 1, with any access beyond this range considered invalid, requiring bounds checking in all index calculations to prevent array access violations during heap operations
B
No boundary conditions exist
C
Boundaries are flexible
D
Indices can be any value
32
Question 32
How does array representation affect heap stability?
text
Stability depends on implementation
Array storage doesn't affect orderingA
Array representation itself doesn't affect heap stability, as stability depends on the comparison logic and insertion order, with the array storage providing the underlying structure that maintains heap properties regardless of stability requirements
B
Arrays make heaps unstable
C
Arrays guarantee stability
D
Stability is impossible in arrays
33
Question 33
What is the index calculation efficiency?
A
Index calculations using simple arithmetic operations (multiplication, addition, division) are extremely efficient, typically executing in constant time on modern processors and providing the foundation for O(1) parent-child access in heap operations
B
Index calculations are slow
C
Calculations require complex operations
D
Efficiency is not important
34
Question 34
How does array representation support heap merging?
cpp
Concatenate arrays
Rebuild heap structureA
Array representation supports heap merging by allowing concatenation of heap arrays followed by heap reconstruction, leveraging the contiguous storage to efficiently combine multiple heaps into a single valid heap structure through rebuild operations
B
Arrays don't support merging
C
Merging requires pointers
D
Arrays make merging complex
35
Question 35
What is the memory fragmentation impact?
A
Array-based heaps avoid memory fragmentation issues common in pointer-based trees, as they allocate contiguous memory blocks that don't create scattered small allocations, improving memory management efficiency and reducing overhead from memory fragmentation
B
Arrays cause more fragmentation
C
Fragmentation is the same
D
Pointer trees avoid fragmentation
36
Question 36
How do index formulas handle negative indices?
cpp
Integer division handles negatives
Root case: (-1)/2 = 0 in integer mathA
Index formulas handle the root case gracefully through integer division, where (0-1)/2 = -0.5 which truncates to 0 in integer arithmetic, correctly identifying the root as having no valid parent index and preventing invalid array access
B
Negative indices cause crashes
C
Formulas produce invalid results
D
Negative indices are impossible
37
Question 37
What is the practical performance difference in large heaps?
A
In large heaps, array representation significantly outperforms pointer-based implementations due to superior cache performance and reduced memory overhead, with the performance gap widening as heap size increases and cache misses become more costly
B
Performance difference decreases
C
Arrays are slower for large heaps
D
Performance is identical
38
Question 38
How does array representation enable heap serialization?
cpp
Direct array copy
No pointer serialization neededA
Array representation enables straightforward heap serialization by allowing direct copying of the contiguous memory block, eliminating the need for complex pointer serialization and graph traversal required in pointer-based tree implementations
B
Arrays make serialization difficult
C
Serialization requires pointers
D
Arrays can't be serialized
39
Question 39
What is the index formula generalization for d-ary heaps?
A
For d-ary heaps, the k-th child of node i is at index (d * i) + k, and the parent of node i is at index (i - 1) / d, generalizing the binary heap formulas to support heaps with arbitrary branching factors while maintaining array-based storage
B
Formulas don't generalize
C
d-ary heaps require pointers
D
Generalization is complex
40
Question 40
Considering array representation of heaps and its fundamental advantages, which property makes contiguous memory storage essential for achieving optimal heap performance despite the mathematical complexity of index calculations and boundary checking requirements?
A
Contiguous memory storage provides the spatial locality and cache efficiency that enables fast heap operations, while index formulas deliver the mathematical framework for parent-child relationships, creating a powerful combination that eliminates pointer overhead and supports logarithmic-time heap maintenance through direct array access and arithmetic calculations
B
Memory storage is not important
C
Cache performance is irrelevant
D
Index calculations are too complex