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

Challenge - 5 Problems

❓ Predict Output

intermediate

What is the output of the heap array after insertions?

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

DSA Java

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

// Insert 10
heap[size++] = 10;
// Insert 4
heap[size++] = 4;
// Insert 15
heap[size++] = 15;
// Insert 20
heap[size++] = 20;
// Insert 0
heap[size++] = 0;

// After each insertion, heapify up to maintain min-heap property
// (Assume heapifyUp is correctly implemented)

// What is the heap array after all insertions?

A [0, 4, 15, 20, 10]

B [0, 10, 4, 15, 20]

C [4, 0, 10, 15, 20]

D [10, 4, 15, 20, 0]

💡 Hint

Remember min-heap means parent is always smaller than children. After inserting 0, it should bubble up to the root.

In a min-heap, the smallest element is at the root (index 0). After inserting 10, 4, 15, 20, and 0, the 0 bubbles up to the root. The final array is [0, 4, 15, 20, 10].

🧠 Conceptual

intermediate

Which property must a max-heap always satisfy?

Choose the correct property that a max-heap must always satisfy.

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

B All leaf nodes are greater than all internal nodes.

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

D The heap is always a complete binary tree but no order property is required.

💡 Hint

Think about what 'max' means in max-heap.

A max-heap requires that every parent node is greater than or equal to its children, ensuring the largest element is at the root.

🔧 Debug

advanced

What error occurs when accessing a heap index out of bounds?

Given a heap array of size 5, what error will this code produce? int[] heap = {10, 20, 30, 40, 50}; int child = heap[10];

A ArrayIndexOutOfBoundsException

B NullPointerException

C No error, returns 0

D ClassCastException

💡 Hint

Accessing an array index that does not exist causes an error.

Accessing an index outside the array bounds throws ArrayIndexOutOfBoundsException in Java.

🚀 Application

advanced

What is the height of a complete binary heap with 31 elements?

A complete binary heap has 31 elements. What is its height?

A 31

B 6

C 5

D 4

💡 Hint

Height is the number of edges on the longest path from root to leaf. For complete binary trees, height = floor(log2(n)).

Height = floor(log2(31)) = 4. This is the number of edges from root to the deepest leaf in a complete binary tree with 31 nodes.

❓ Predict Output

expert

What is the heap array after removing the root from this min-heap?

Given the min-heap array: [2, 5, 3, 10, 7, 8] Remove the root element (2) and re-heapify. What is the resulting array?

DSA Java

int[] heap = {2, 5, 3, 10, 7, 8};
int size = 6;

// Remove root (2), replace with last element (8)
heap[0] = heap[size - 1];
size--;

// Heapify down from root to maintain min-heap
// (Assume heapifyDown is correctly implemented)

// What is the heap array after removal and heapify?

A [3, 5, 7, 10, 8]

B [3, 5, 8, 10, 7]

C [5, 7, 3, 10, 8]

D [8, 5, 3, 10, 7]

💡 Hint

After replacing root with last element, swap down with smaller child until heap property is restored.

After removing 2, replace root with 8: [8, 5, 3, 10, 7, _]. Heapify down swaps 8 with smaller child 3: [3, 5, 8, 10, 7]. Position 2 (now 8) has no children (indices 5 and 6 exceed size 5). Final array: [3, 5, 8, 10, 7].

Practice

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

easy

A. The tree is always a complete binary tree but not necessarily balanced

B. The value of each parent node is less than or equal to its children

C. All leaf nodes have the same value

D. The value of each parent node is greater than or equal to its children

Solution

1. Step 1: Understand max-heap property

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

2. Step 2: Compare options with max-heap definition

   The value of each parent node is greater than or equal to its children matches the max-heap property exactly, while others describe min-heap or unrelated properties.

3. Final Answer:

   The value of each parent node is greater than or equal to its children -> Option D

4. Quick Check:

   Max-heap parent ≥ children = A [OK]

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

Common Mistakes:

• Confusing max-heap with min-heap property
• Thinking leaf nodes must have same value
• Ignoring the complete binary tree requirement

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

easy

A. PriorityQueue<Integer> minHeap = new PriorityQueue<>(Collections.reverseOrder());

B. PriorityQueue<Integer> minHeap = new PriorityQueue<>(Comparator.reverseOrder());

C. PriorityQueue<Integer> minHeap = new PriorityQueue<>();

D. PriorityQueue<Integer> minHeap = new PriorityQueue<>(Collections.naturalOrder());

Solution

1. Step 1: Recall Java PriorityQueue default behavior

   By default, PriorityQueue in Java is a min-heap, so no comparator is needed.

2. Step 2: Analyze options for min-heap declaration

   PriorityQueue<Integer> minHeap = new PriorityQueue<>(); uses default constructor, which creates a min-heap. A and B use reverseOrder creating max-heap, C uses non-existent Collections.naturalOrder().

3. Final Answer:

   PriorityQueue<Integer> minHeap = new PriorityQueue<>(); -> Option C

4. Quick Check:

   Default PriorityQueue = min-heap = D [OK]

Hint: Default PriorityQueue in Java is min-heap [OK]

Common Mistakes:

• Using reverseOrder() creates max-heap, not min-heap
• Confusing naturalOrder() with reverseOrder()
• Adding unnecessary comparator for default min-heap

3. What will be the output of the following Java code snippet?

import java.util.PriorityQueue;
public class TestHeap {
  public static void main(String[] args) {
    PriorityQueue<Integer> heap = new PriorityQueue<>();
    heap.add(30);
    heap.add(10);
    heap.add(20);
    System.out.print(heap.poll() + ", " + heap.poll() + ", " + heap.poll());
  }
}

medium

A. 20, 10, 30

B. 10, 20, 30

C. 30, 20, 10

D. 10, 30, 20

Solution

1. Step 1: Understand PriorityQueue behavior

   Java's PriorityQueue is a min-heap by default, so poll() returns the smallest element.

2. Step 2: Trace poll() calls

   Elements added: 30, 10, 20. First poll() returns 10, second 20, third 30.

3. Final Answer:

   10, 20, 30 -> Option B

4. Quick Check:

   poll() returns min element each time = 10,20,30 [OK]

Hint: poll() returns smallest element in min-heap [OK]

Common Mistakes:

• Assuming poll() returns elements in insertion order
• Confusing min-heap with max-heap behavior
• Not realizing poll() removes elements

4. Identify the error in the following code that attempts to create a max-heap using Java's PriorityQueue:

import java.util.PriorityQueue;
import java.util.Collections;
public class MaxHeapExample {
  public static void main(String[] args) {
    PriorityQueue<Integer> maxHeap = new PriorityQueue<>();
    maxHeap.add(5);
    maxHeap.add(15);
    maxHeap.add(10);
    System.out.println(maxHeap.poll());
  }
}

medium

A. PriorityQueue is not initialized with a comparator for max-heap

B. PriorityQueue cannot store Integer objects

C. poll() method does not exist for PriorityQueue

D. add() method cannot be used to insert elements

Solution

1. Step 1: Check PriorityQueue initialization

   The code initializes PriorityQueue without a comparator, so it defaults to min-heap.

2. Step 2: Identify max-heap creation mistake

   To create max-heap, PriorityQueue must be initialized with Collections.reverseOrder() comparator.

3. Final Answer:

   PriorityQueue is not initialized with a comparator for max-heap -> Option A

4. Quick Check:

   Missing comparator = B [OK]

Hint: Max-heap needs reverseOrder() comparator in PriorityQueue [OK]

Common Mistakes:

• Assuming default PriorityQueue is max-heap
• Thinking poll() method is invalid
• Believing add() cannot insert elements

5. Given a max-heap represented as an array: [50, 30, 40, 10, 20], which of the following arrays represents the heap after inserting the element 35 and re-heapifying?

hard

A. [50, 30, 40, 10, 20, 35]

B. [50, 35, 40, 10, 20, 30]

C. [50, 40, 30, 10, 20, 35]

D. [50, 40, 35, 10, 20, 30]

Solution

1. Step 1: Insert 35 at the end of the array

   Array becomes [50, 30, 40, 10, 20, 35].

2. Step 2: Re-heapify by comparing 35 with its parent

   Parent of index 5 is index 2 (40). Since 35 <= 40, no swap needed. Heap property is maintained.

3. Step 3: Check if bubbling up is needed

   Since 35 < 40, no bubble up occurs, so the array remains [50, 30, 40, 10, 20, 35]. However, this matches [50, 30, 40, 10, 20, 35], but [50, 30, 40, 10, 20, 35] is not the correct max-heap after insertion because 35 should bubble up if possible.

4. Step 4: Correct bubbling up

   Actually, 35 is less than 40, so no swap with parent. The heap remains [50, 30, 40, 10, 20, 35].

5. Final Answer:

   [50, 30, 40, 10, 20, 35] -> Option A

6. Quick Check:

   Insert 35 at end, parent(5)=40 >= 35, no bubble up = D [OK]

Hint: Insert at end, then bubble up to fix max-heap [OK]

Common Mistakes:

• Not bubbling up after insertion
• Swapping with wrong parent index
• Assuming insertion at root