Heap Concept Structure and Properties in DSA C# - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

What is the output of this Min-Heap after insertions?

Consider a Min-Heap implemented as an array. Insert the following elements in order: 10, 4, 15, 20, 0, 8. What is the array representation of the heap after all insertions?

DSA C#

int[] heap = new int[10];
int size = 0;

void Insert(int val) {
    heap[size] = val;
    int current = size;
    size++;
    while (current > 0 && heap[current] < heap[(current - 1) / 2]) {
        int temp = heap[current];
        heap[current] = heap[(current - 1) / 2];
        heap[(current - 1) / 2] = temp;
        current = (current - 1) / 2;
    }
}

Insert(10);
Insert(4);
Insert(15);
Insert(20);
Insert(0);
Insert(8);

string result = string.Join(", ", heap.Take(size));
Console.WriteLine(result);

A 0, 4, 8, 20, 10, 15

B 0, 4, 15, 20, 10, 8

C 4, 10, 8, 20, 0, 15

D 4, 0, 8, 20, 10, 15

💡 Hint

Remember that in a Min-Heap, the smallest element is always at the root, and each parent is smaller than its children.

After inserting all elements, the heap property is maintained by bubbling up smaller elements. The smallest element 0 becomes the root, and the array represents the heap in level order.

🧠 Conceptual

intermediate

Which property must a Max-Heap always satisfy?

Select the correct property that a Max-Heap must always satisfy.

A Every parent node is greater than or equal to its children.

B Every parent node is less than or equal to its children.

C All leaf nodes have the same value.

D The heap is always a complete binary tree with all nodes having two children.

💡 Hint

Think about what makes a Max-Heap different from a Min-Heap.

In a Max-Heap, each parent node must be greater than or equal to its children to maintain the heap property.

🔧 Debug

advanced

Identify the error in this heap insertion code snippet

What error will this C# code produce when inserting into a Min-Heap?

DSA C#

int[] heap = new int[5];
int size = 0;

void Insert(int val) {
    heap[size] = val;
    int current = size;
    size++;
    while (current > 0 && heap[current] > heap[(current - 1) / 2]) {
        int temp = heap[current];
        heap[current] = heap[(current - 1) / 2];
        heap[(current - 1) / 2] = temp;
        current = (current - 1) / 2;
    }
}

Insert(3);
Insert(1);
Insert(6);
Insert(5);
Insert(2);
Insert(4);

A The heap property is violated because the comparison operator is incorrect for a Min-Heap.

B No error; the code runs correctly and maintains a Min-Heap.

C IndexOutOfRangeException at Insert(4) because heap array size is exceeded.

D NullReferenceException because heap array is not initialized.

💡 Hint

Check the array size and number of insertions.

The heap array has size 5 but 6 insertions are attempted, causing an IndexOutOfRangeException.

🚀 Application

advanced

How many elements are in a complete binary heap of height 3?

A complete binary heap has height 3 (root at height 0). How many elements does it contain at minimum and maximum?

A Minimum 7, Maximum 16

B Minimum 7, Maximum 15

C Minimum 8, Maximum 16

D Minimum 8, Maximum 15

💡 Hint

A complete binary tree is full except possibly the last level, which is filled from left to right.

Height 3 means levels 0 to 3. Minimum nodes = nodes in full tree up to level 2 plus 1 node at level 3 = 7 + 1 = 8. Maximum nodes = full tree of height 3 = 2^(4)-1 = 15.

🧠 Conceptual

expert

What is the time complexity of building a heap from an unsorted array of n elements?

Choose the correct time complexity for building a heap from an unsorted array of n elements using the optimal heapify method.

A O(n log n)

B O(n)

C O(n^2)

D O(log n)

💡 Hint

Consider the cost of heapifying nodes starting from the bottom.

Building a heap using the bottom-up heapify method takes O(n) time, not O(n log n), because most nodes are near the bottom and require less work.

Practice

1. What is a key property of a max-heap data structure?

easy

A. Every parent node is greater than or equal to its children

B. Every parent node is less than or equal to its children

C. All leaf nodes have the same value

D. The tree is always a complete binary tree with sorted leaves

Solution

1. Step 1: Understand max-heap property

   A max-heap requires that each parent node's value is greater than or equal to the values of its children.

2. Step 2: Compare options with definition

   Every parent node is greater than or equal to its children matches the max-heap property exactly. Every parent node is less than or equal to its children describes a min-heap. Options A and C are incorrect properties.

3. Final Answer:

   Every parent node is greater than or equal to its children -> Option A

4. Quick Check:

   Max-heap = parent ≥ children [OK]

Hint: Max-heap means parent ≥ children values [OK]

Common Mistakes:

• Confusing max-heap with min-heap property
• Thinking leaves must be sorted
• Assuming all nodes have equal values

2. Which of the following is the correct way to declare a min-heap using C#'s PriorityQueue?

easy

A. var minHeap = new PriorityQueue();

B. var minHeap = new PriorityQueue(Comparer<int>.Default);

C. var minHeap = new PriorityQueue(Comparer<int>.Create((x, y) => y - x));

D. var minHeap = new PriorityQueue(Comparer<int>.Create((x, y) => x + y));

Solution

1. Step 1: Recall default behavior of PriorityQueue

   In C#, PriorityQueue by default acts as a min-heap using the natural order of TPriority.

2. Step 2: Analyze options

   var minHeap = new PriorityQueue(); uses default constructor, creating a min-heap. var minHeap = new PriorityQueue(Comparer<int>.Default); incorrectly tries to pass a comparer to constructor which does not exist. var minHeap = new PriorityQueue(Comparer<int>.Create((x, y) => y - x)); reverses order for max-heap. var minHeap = new PriorityQueue(Comparer<int>.Create((x, y) => x + y)); uses invalid comparer logic.

3. Final Answer:

   var minHeap = new PriorityQueue<int, int>(); -> Option A

4. Quick Check:

   Default PriorityQueue is min-heap [OK]

Hint: Default PriorityQueue is min-heap without comparer [OK]

Common Mistakes:

• Trying to pass comparer to constructor (not supported)
• Using wrong comparer logic for min-heap
• Confusing min-heap and max-heap syntax

3. Consider the following C# code using a max-heap implemented as a binary heap array:

int[] heap = { 50, 30, 40, 10, 20 };
// Insert 60 and then heapify up
Insert(heap, 60);

What will be the heap array after insertion?

medium

A. [60, 50, 40, 10, 20, 60]

B. [60, 30, 50, 10, 20, 40]

C. [60, 50, 40, 30, 20, 10]

D. [50, 30, 40, 10, 20, 60]

Solution

1. Step 1: Insert 60 at the end of heap array

   Initial heap: [50, 30, 40, 10, 20]. Insert 60 at index 5: [50, 30, 40, 10, 20, 60]

2. Step 2: Heapify up to maintain max-heap

   Compare 60 with parent at index 2 (40). 60 > 40, swap. New array: [50, 30, 60, 10, 20, 40]. Compare 60 with parent at index 0 (50). 60 > 50, swap. Final array: [60, 30, 50, 10, 20, 40]

3. Final Answer:

   [60, 30, 50, 10, 20, 40] -> Option B

4. Quick Check:

   Insert + heapify up = [60,30,50,10,20,40] [OK]

Hint: Insert at end, swap up while parent smaller [OK]

Common Mistakes:

• Not swapping with parent correctly
• Placing new element at wrong index
• Confusing heapify up with heapify down

4. The following C# code attempts to remove the root from a max-heap stored in an array but has a bug:

void RemoveRoot(int[] heap, int size) {
    heap[0] = heap[size - 1];
    size--;
    HeapifyDown(heap, 0, size);
}

What is the main issue causing incorrect heap behavior?

medium

A. HeapifyDown is called with wrong starting index

B. The last element is not removed before assignment

C. The size variable is decremented locally but not updated outside

D. The root is not replaced with the maximum child

Solution

1. Step 1: Analyze size decrement scope

   The size variable is decremented inside the method but since it's passed by value, the caller's size remains unchanged.

2. Step 2: Effect on heap operations

   Because size outside is not updated, heap operations may access invalid elements or not reflect the reduced heap size, causing incorrect behavior.

3. Final Answer:

   Size variable is decremented locally but not updated outside -> Option C

4. Quick Check:

   Size must update outside method [OK]

Hint: Pass size by ref or return new size [OK]

Common Mistakes:

• Assuming size changes persist without ref
• Thinking heapify down index is wrong
• Ignoring size update importance

5. You have a min-heap represented as an array: [5, 10, 15, 20, 25, 30]. You want to replace the root with 22 and restore the min-heap property. Which sequence of array states correctly shows the heap after replacing and heapifying down?

hard

A. [22, 10, 15, 20, 25, 30] -> [10, 22, 15, 20, 25, 30]

B. [22, 10, 15, 20, 25, 30] -> [10, 15, 22, 20, 25, 30]

C. [22, 10, 15, 20, 25, 30] -> [15, 10, 22, 20, 25, 30]

D. [22, 10, 15, 20, 25, 30] -> [10, 20, 15, 22, 25, 30]

Solution

1. Step 1: Replace root with 22

   Initial heap: [5, 10, 15, 20, 25, 30]. Replace root 5 with 22: [22, 10, 15, 20, 25, 30]

2. Step 2: Heapify down to restore min-heap

   Compare 22 with children 10 and 15. Smallest child is 10. Swap 22 and 10: [10, 22, 15, 20, 25, 30]. Now 22 at index 1 has children at indices 3 (20) and 4 (25). Smallest is 20 < 22, swap 22 and 20: [10, 20, 15, 22, 25, 30]. 22 now at index 3 has no children, stop.

3. Final Answer:

   [22, 10, 15, 20, 25, 30] -> [10, 20, 15, 22, 25, 30] -> Option D

4. Quick Check:

   Replace root + heapify down = [22,10,15,20,25,30] -> [10,20,15,22,25,30] [OK]

Hint: Swap root down with smallest child until heap restored [OK]

Common Mistakes:

• Swapping with wrong child
• Stopping heapify too early
• Confusing min-heap with max-heap rules