Kth Smallest Element Using Min Heap in DSA Javascript - Execution Trace

Concept Flow - Kth Smallest Element Using Min Heap

Build Min Heap from array

↓

Repeat k-1 times: Extract Min

↓

Extract Min again

↓

Return extracted value as kth smallest

Build a min heap from the array, then remove the smallest element k times. The last removed element is the kth smallest.

Execution Sample

DSA Javascript

const kthSmallest = (arr, k) => {
  const heap = new MinHeap(arr);
  for (let i = 1; i < k; i++) heap.extractMin();
  return heap.extractMin();
};

This code builds a min heap from the array, extracts the smallest element k-1 times, then returns the kth smallest.

Execution Table

Step	Operation	Nodes in Heap	Pointer Changes	Visual State
1	Build Min Heap from [7, 10, 4, 3, 20, 15]	[3, 7, 4, 10, 20, 15]	Heapify rearranged nodes	3 -> 7 -> 4 -> 10 -> 20 -> 15
2	Extract Min (1st time)	[4, 7, 15, 10, 20]	Root 3 removed, last element 15 moved to root, heapify	4 -> 7 -> 15 -> 10 -> 20
3	Extract Min (2nd time)	[7, 10, 15, 20]	Root 4 removed, last element 20 moved to root, heapify	7 -> 10 -> 15 -> 20
4	Extract Min (3rd time, kth=3)	[10, 20, 15]	Root 7 removed, last element 15 moved to root, heapify	10 -> 20 -> 15

💡 After 3 extractions, the last extracted value 7 is the 3rd smallest element.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4
heap array	[7,10,4,3,20,15]	[3,7,4,10,20,15]	[4,7,15,10,20]	[7,10,15,20]	[10,20,15]
extracted values	[]	[]	[3]	[3,4]	[3,4,7]

Key Moments - 3 Insights

Why do we build the min heap first instead of sorting the array?

Building a min heap allows efficient extraction of the smallest element multiple times without sorting the entire array, as shown in Step 1 and subsequent extractions in the execution_table.

Why do we extract min k times instead of just once?

Each extraction removes the smallest element. Extracting k times ensures the kth smallest is the last extracted, as seen in Steps 2 to 4 in the execution_table.

What happens to the heap after each extraction?

After removing the root, the last element moves to root and heapify restores min heap property, shown by pointer changes and visual state in execution_table rows.

Visual Quiz - 3 Questions

Look at the execution_table, what is the heap array after the 2nd extraction?

A [3, 7, 4, 10, 20, 15]

B [4, 7, 15, 10, 20]

C [7, 10, 15, 20]

D [10, 20, 15]

💡 Hint

Check the 'Nodes in Heap' column at Step 3 in execution_table.

At which step is the kth smallest element extracted?

A Step 1

B Step 4

C Step 2

D Step 3

💡 Hint

Look at the 'Operation' column and note when k=3 extraction happens.

If k was 2 instead of 3, what would be the last extracted value?

A 4

B 7

C 3

D 10

💡 Hint

Refer to 'extracted values' in variable_tracker after Step 3.

Concept Snapshot

Kth Smallest Using Min Heap:
- Build min heap from array
- Extract min element k times
- Last extracted is kth smallest
- Heapify after each extraction
- Efficient for repeated smallest extraction

Full Transcript

To find the kth smallest element using a min heap, first build a min heap from the input array. This organizes the data so the smallest element is at the root. Then, extract the minimum element k times. Each extraction removes the smallest element and re-heapifies the structure to maintain the min heap property. The last extracted element after k removals is the kth smallest element in the array. This method avoids sorting the entire array and is efficient for repeated smallest element retrievals.

Practice

1. What does the Kth Smallest Element Using Min Heap algorithm primarily do?

easy

A. Find the element that is the Kth smallest in a list using a min heap

B. Sort the entire list in ascending order

C. Find the largest element in the list

D. Remove duplicates from the list

Solution

1. Step 1: Understand the min heap role

   A min heap always keeps the smallest element at the root, making it easy to access the smallest values.

2. Step 2: Use min heap to find Kth smallest

   By extracting the root K times, we get the Kth smallest element.

3. Final Answer:

   Find the element that is the Kth smallest in a list using a min heap -> Option A

4. Quick Check:

   Kth smallest element = find using min heap [OK]

Hint: Extract min K times from min heap to get Kth smallest [OK]

Common Mistakes:

• Confusing min heap with max heap
• Thinking it sorts the entire list
• Assuming it finds the largest element

2. Which of the following is the correct way to insert an element into a min heap in JavaScript?

easy

A. heap.pop(element); heapifyDown(0);

B. heap.push(element); heapifyUp(heap.length - 1);

C. heap.unshift(element); heapifyDown(0);

D. heap.shift(element); heapifyUp(heap.length - 1);

Solution

1. Step 1: Insert element at the end

   Inserting into a heap means adding the element at the end of the array.

2. Step 2: Restore heap property by heapify up

   After insertion, heapify up is called from the inserted index to maintain min heap order.

3. Final Answer:

   heap.push(element); heapifyUp(heap.length - 1); -> Option B

4. Quick Check:

   Insert + heapifyUp = heap.push(element); heapifyUp(heap.length - 1); [OK]

Hint: Insert at end, then heapify up to fix min heap [OK]

Common Mistakes:

• Using pop or shift to insert elements
• Calling heapifyDown after insertion
• Inserting at the front of the array

3. What is the output of the following code snippet?

const heap = [1, 3, 6, 5, 9, 8];
function extractMin(heap) {
  const min = heap[0];
  heap[0] = heap.pop();
  heapifyDown(heap, 0);
  return min;
}
function heapifyDown(heap, i) {
  const left = 2 * i + 1;
  const right = 2 * i + 2;
  let smallest = i;
  if (left < heap.length && heap[left] < heap[smallest]) smallest = left;
  if (right < heap.length && heap[right] < heap[smallest]) smallest = right;
  if (smallest !== i) {
    [heap[i], heap[smallest]] = [heap[smallest], heap[i]];
    heapifyDown(heap, smallest);
  }
}
console.log(extractMin(heap));

medium

A. 6

B. 3

C. 5

D. 1

Solution

1. Step 1: Extract min element

   The min element is at heap[0], which is 1.

2. Step 2: Replace root and heapify down

   Replace root with last element 8, then heapify down to restore min heap.

3. Final Answer:

   1 -> Option D

4. Quick Check:

   extractMin returns root = 1 [OK]

Hint: extractMin returns root element of min heap [OK]

Common Mistakes:

• Returning last element instead of root
• Not heapifying down after extraction
• Confusing min heap root with max

4. Identify the bug in this code snippet for finding the Kth smallest element using a min heap:

function kthSmallest(arr, k) {
  const heap = [];
  for (const num of arr) {
    heap.push(num);
  }
  for (let i = Math.floor(heap.length / 2); i >= 0; i--) {
    heapifyDown(heap, i);
  }
  let result;
  for (let i = 0; i < k; i++) {
    result = heap.shift();
    heapifyDown(heap, 0);
  }
  return result;
}

medium

A. Using shift() removes first element but does not maintain heap structure

B. Heap is not built correctly from array

C. Loop for heapifyDown should start from 0 to end

D. Variable result is not initialized

Solution

1. Step 1: Check element removal method

   Using shift() removes first element but shifts all others, breaking heap structure.

2. Step 2: Correct removal method

   Should remove root by swapping with last element, popping last, then heapify down.

3. Final Answer:

   Using shift() removes first element but does not maintain heap structure -> Option A

4. Quick Check:

   shift() breaks heap property [OK]

Hint: Remove root by swap and pop, not shift() [OK]

Common Mistakes:

• Using shift() instead of swap and pop
• Incorrect heap building loop range
• Ignoring heapify after removal

5. Given an array [7, 10, 4, 3, 20, 15] and k = 3, what is the Kth smallest element using a min heap approach?

hard

A. 4

B. 10

C. 7

D. 3

Solution

1. Step 1: Build min heap from array

   Min heap root is smallest element: 3.

2. Step 2: Extract min 3 times

   Extract 1st min: 3, 2nd min: 4, 3rd min: 7.

3. Final Answer:

   7 -> Option C

4. Quick Check:

   3rd smallest element = 7 [OK]

Hint: Extract min K times; last extracted is Kth smallest [OK]

Common Mistakes:

• Counting index instead of extraction times
• Confusing Kth smallest with Kth largest
• Not building min heap properly