K-way merge with heaps in Data Structures Theory - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to describe the main data structure used in a k-way merge with heaps.

Data Structures Theory

The primary data structure used in a k-way merge is a [1].

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Aqueue

Bstack

Cheap

Dlinked list

Attempts:

3 left

💡 Hint

Common Mistakes

Confusing heap with stack or queue.

Thinking linked list is used for efficient minimum retrieval.

The k-way merge algorithm uses a heap to efficiently find the smallest element among k sorted lists.

2fill in blank

medium

Complete the sentence to explain the initial step in k-way merge using heaps.

Data Structures Theory

Initially, the heap is built by inserting the [1] element from each sorted list.

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Alast

Bfirst

Cmiddle

Dlargest

Attempts:

3 left

💡 Hint

Common Mistakes

Choosing last or largest element instead of first.

Assuming middle element is used initially.

The algorithm starts by inserting the first element of each sorted list into the heap to begin merging.

3fill in blank

hard

Fix the error in the explanation of the heap operation during k-way merge.

Data Structures Theory

After extracting the smallest element from the heap, we [1] the next element from the same list into the heap.

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Ainsert

Bignore

Cremove

Dswap

Attempts:

3 left

💡 Hint

Common Mistakes

Thinking we remove or ignore the next element instead of inserting it.

Confusing swap with insert operation.

After removing the smallest element, the next element from that list is inserted into the heap to continue the merge.

4fill in blank

hard

Fill both blanks to describe the heap size and complexity in k-way merge.

Data Structures Theory

The heap size is [1] to the number of lists, and each insertion or extraction takes [2] time.

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Aequal

Blogarithmic

Clinear

Dconstant

Attempts:

3 left

💡 Hint

Common Mistakes

Assuming heap size grows with total elements instead of number of lists.

Thinking heap operations take linear or constant time.

The heap size equals the number of lists (k), and each heap operation takes logarithmic time relative to k.

5fill in blank

hard

Fill all three blanks to complete the dictionary comprehension representing the k-way merge condition.

Data Structures Theory

merged = [1]: [2] for [3] in range(len(lists)) if lists[[3]]

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Ai

Blists[i].pop(0)

Dj

Attempts:

3 left

💡 Hint

Common Mistakes

Using wrong variable names for iteration.

Not checking if the list is non-empty before popping.

This comprehension extracts the first element from each non-empty list indexed by i to merge them.

Practice

1. What is the main purpose of using a min-heap in a K-way merge algorithm?

easy

A. To store all elements in a single large array

B. To sort each individual list before merging

C. To reverse the order of elements in each list

D. To efficiently find and extract the smallest element among multiple sorted lists

Solution

1. Step 1: Understand the role of a min-heap in merging

   A min-heap helps quickly find the smallest element among all current candidates from each list.

2. Step 2: Connect min-heap usage to K-way merge

   By always extracting the smallest element, the algorithm efficiently merges sorted lists without scanning all elements repeatedly.

3. Final Answer:

   To efficiently find and extract the smallest element among multiple sorted lists -> Option D

4. Quick Check:

   Min-heap = smallest element extraction [OK]

Hint: Min-heap always gives smallest element fast in merging [OK]

Common Mistakes:

• Thinking min-heap sorts individual lists
• Assuming min-heap stores all elements at once
• Confusing min-heap with max-heap

2. Which of the following is the correct initial step when implementing a K-way merge using a min-heap?

easy

A. Remove the largest element from each list

B. Insert all elements from all lists into the min-heap at once

C. Insert the first element of each sorted list into the min-heap

D. Sort each list again before merging

Solution

1. Step 1: Identify the initial heap population

   The algorithm starts by inserting the first element of each sorted list into the min-heap to track the smallest candidates.

2. Step 2: Explain why not all elements are inserted initially

   Inserting all elements at once would be inefficient and defeat the purpose of incremental merging.

3. Final Answer:

   Insert the first element of each sorted list into the min-heap -> Option C

4. Quick Check:

   Start heap with first elements only [OK]

Hint: Heap starts with first elements, not all at once [OK]

Common Mistakes:

• Inserting all elements at once causing inefficiency
• Sorting lists again unnecessarily
• Removing largest elements instead of smallest

3. Consider three sorted lists: [1, 4, 7], [2, 5, 8], and [3, 6, 9]. Using a K-way merge with a min-heap, what is the first element extracted from the heap?

medium

A. 1

B. 2

C. 3

D. 4

Solution

1. Step 1: Insert first elements of each list into the min-heap

   The heap initially contains 1, 2, and 3 from the three lists.

2. Step 2: Extract the smallest element from the heap

   The smallest among 1, 2, and 3 is 1, so it is extracted first.

3. Final Answer:

   1 -> Option A

4. Quick Check:

   Smallest first element = 1 [OK]

Hint: First extracted is smallest first element from all lists [OK]

Common Mistakes:

• Choosing second or third smallest element first
• Confusing heap order with list order
• Ignoring initial heap contents

4. You implemented a K-way merge with heaps but the merged output is not sorted. What is the most likely mistake?

medium

A. Using a max-heap instead of a min-heap

B. Not inserting the next element from a list after extracting its smallest element

C. Sorting each list before merging

D. Inserting duplicate elements into the heap

Solution

1. Step 1: Understand the heap update process in K-way merge

   After extracting the smallest element from a list, the next element from that list must be inserted into the heap to maintain correct order.

2. Step 2: Identify the effect of missing this step

   If the next element is not inserted, the heap misses candidates, causing the merged output to be unsorted or incomplete.

3. Final Answer:

   Not inserting the next element from a list after extracting its smallest element -> Option B

4. Quick Check:

   Insert next element after extraction [OK]

Hint: Always add next element after extraction to keep order [OK]

Common Mistakes:

• Using max-heap which reverses order
• Sorting lists again unnecessarily
• Thinking duplicates cause sorting errors

5. You have 4 sorted lists of different lengths: [1, 3, 5], [2, 4], [0, 6, 7, 8], and [9]. Using a K-way merge with a min-heap, what is the correct order of elements extracted?

hard

A. [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

B. [1, 2, 3, 4, 5, 6, 7, 8, 9, 0]

C. [9, 8, 7, 6, 5, 4, 3, 2, 1, 0]

D. [0, 2, 1, 3, 4, 5, 6, 7, 8, 9]

Solution

1. Step 1: Insert first elements of all lists into the min-heap

   Heap starts with 1, 2, 0, and 9.

2. Step 2: Extract elements in ascending order, inserting next from each list

   Extract 0, insert 6; extract 1, insert 3; extract 2, insert 4; extract 3, insert 5; extract 4, no next; extract 5, no next; extract 6, insert 7; extract 7, insert 8; extract 8, no next; extract 9, no next.

3. Final Answer:

   [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] -> Option A

4. Quick Check:

   Sorted merge order = ascending combined list [OK]

Hint: Extract smallest, insert next from same list until all done [OK]

Common Mistakes:

• Ignoring list lengths causing wrong order
• Extracting elements out of heap order
• Confusing max-heap with min-heap results