Merging $k$ Sorted Lists with a Thin Heap: A $\mathcal{O}(n \lg k)$ Algorithm

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In summary, the conversation discusses the task of merging k sorted lists into one sorted list using a thin heap. The hint suggests using a thin heap for a k-way merging, and there is confusion about what a thin heap is and how it can be used for this task. The conversation also considers using a min heap with k positions and a procedure for merging the lists using heapify and selecting the minimum element.
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evinda
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Hello! (Wave)

I am asked to write a $Ο (n \lg k)$ - time algorithm that merges $k$ sorted lists into one sorted list, where $n$ is the the total number of elements in all the input lists.
Hint: Use a thin heap for a $k$ -way merging.

Do you have an idea what could be meant with [m] thin heap [/m] ? (Worried)

Also, how could we merge $k$ sorted lists into one using a heap? :confused:
 
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Finally, a min heap is meant...
So do we have to have a heap with $k$ positions, put the elements of the first positions of the $k$ lists in the heap, heapify and delete the root, which will be the smallest element, and put it into the new list, then place at the root the second element from the list from which the minimum was, then heapify and continue the same procedure? (Thinking)
 

FAQ: Merging $k$ Sorted Lists with a Thin Heap: A $\mathcal{O}(n \lg k)$ Algorithm

What is a thin heap?

A thin heap is a data structure that is used to efficiently merge multiple sorted lists. It is a type of priority queue that supports operations such as insertion, deletion, and merging in logarithmic time.

How does the $\mathcal{O}(n \lg k)$ algorithm work?

The algorithm works by first creating a thin heap with the first element from each of the k sorted lists. It then repeatedly takes the minimum element from the heap and adds the next element from the corresponding list to the heap. This process continues until all the elements from the lists have been merged into a single sorted list.

What is the time complexity of merging k sorted lists using this algorithm?

The time complexity of this algorithm is $\mathcal{O}(n \lg k)$, where n is the total number of elements in the input lists and k is the number of lists. This is because the algorithm involves merging the lists in a divide-and-conquer fashion, reducing the number of elements to be merged in each step.

Can this algorithm handle lists of different lengths?

Yes, this algorithm can handle lists of different lengths. Because it uses a thin heap, the length of the lists does not affect its time complexity. The only factor that affects the time complexity is the total number of elements in the lists.

What are the advantages of using this algorithm over other methods of merging sorted lists?

One advantage is its time complexity of $\mathcal{O}(n \lg k)$, which is better than most other methods. Additionally, this algorithm does not require the input lists to be sorted beforehand, making it more efficient for dynamically changing lists. It also has a lower space complexity compared to other methods, as it only uses a thin heap instead of creating a new list for the merged elements.

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