Understanding K-way Merge

K-way Merge is a common algorithmic pattern that merges multiple sorted lists or arrays into a single sorted list. This pattern is particularly useful in problems involving external sorting, priority queues, and other applications where multiple sorted sequences must be combined efficiently.

Process

1. Initialization: Define the problem space and initialize the necessary data structures. This typically involves creating a min-heap to keep track of the smallest elements from each K-sorted list.

2. Heap Construction: Insert the first element from each K-sorted list into the min-heap. Each element in the heap should also store a reference to its originating list to facilitate further processing.

3. Heap Operations:

   • Extraction: Remove the smallest element (the root of the min-heap) and add it to the result list.

   • Insertion: Insert the next element from the list from which the extracted element came into the min-heap. If the list is exhausted, do not insert any element.

4. Termination: The process terminates when the heap is empty, which means all elements from all K lists have been processed and added to the result list.

When to Use

1. External Sorting: To merge multiple sorted chunks of data when sorting large datasets that do not fit into memory.

2. Priority Queues: To merge multiple priority queues into a single priority queue.

3. Database Query Processing: To merge results from multiple sorted indexes.

How Does It Reduce Time Complexity?

1. Efficient Merging: Using a min-heap allows for efficient extraction of the smallest element and insertion of the next element from the same list, with a time complexity of O(log K) for each operation. This results in an overall time complexity of O(N log K), where N is the total number of elements across all K lists.

2. Reduced Comparisons: By using a heap, the number of comparisons required to merge the lists is significantly reduced compared to a naive approach that involves comparing elements from all lists.

Example problem for better understanding

Merge K Sorted Lists

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