Introduction to K-way Merge

About the Pattern

The K-way merge pattern is a fundamental algorithmic strategy used to merge K sorted data structures—such as arrays and linked lists—into a single sorted structure. It extends the traditional merge sort algorithm, which combines two sorted structures into one.

Note: For simplicity, the term lists will refer to both arrays and linked lists in this discussion.

The core idea behind the K-way merge algorithm is simple: repeatedly select the smallest (or largest, for descending order) element from the first elements of the K input lists and append it to a new output list. This process continues until all elements are merged into the output list, maintaining the sorted order.

How the Algorithm Works

The K-way merge algorithm has two main approaches for achieving its goal, each leading to the same result. Let's explore these approaches in detail.

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Approach 1: Using a Min Heap

1. Initialize the Heap
   Insert the first element of each list into a min heap. This heap helps efficiently track the smallest current element among the lists.

2. Build the Output List
   Remove the smallest element from the heap (always at the top) and add it to the output list.

3. Track Element Origins
   Keep track of which list each element in the heap originated from to determine the next element to add.

4. Replace Elements in the Heap
   After removing the smallest element, replace it with the next element from the same list.

5. Repeat Steps
   Repeat steps 2–4 until all elements are merged into the output list.

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Approach 2: Grouping and Merging Pairs of Lists

1. Divide into Pairs
   Divide the K sorted lists into pairs, grouping them for two-way merges.

2. Perform Two-Way Merges
   For each pair, perform a standard two-way merge to create a single sorted list for that pair.

3. Handle Odd Lists
   If there is an odd number of lists, leave one list unmerged for the current round.

4. Repeat Pairing and Merging
   Continue pairing and merging the resulting lists until only one sorted list remains.

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Examples of Problems Solved with K-way Merge

1. Median of K Sorted Arrays
   Find the median of K sorted arrays.

2. K-th Smallest Element Across Arrays
   Find the K-th smallest element among multiple sorted arrays.

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Identifying Problems That Match This Pattern

A problem likely fits this pattern if:

• Merging Sorted Structures: It involves merging multiple sorted arrays or matrices.
• Finding K-th Smallest/Largest: It requires identifying the K-th smallest or largest element across sorted collections.

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Real-World Applications

1. Patient Records Aggregation
   Combine sorted streams of patient data (lab results, notes, reports) into a unified, chronologically ordered record for comprehensive healthcare management.

2. Merging Financial Transaction Streams
   Merge sorted transactions (trades, payments, transfers) into a single stream for real-time analysis and fraud detection.

3. Log File Analysis
   Merge server logs into a single, time-ordered stream for performance monitoring, user behavior analysis, or error diagnosis.

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Merge Sorted Array solution:

The following example demonstrates merging two sorted arrays in place:

1. Initialize two pointers: p1 (last element of nums1) and p2 (last element of nums2).
2. Initialize a pointer p pointing to the last element of nums1.
3. Compare elements:
   • If nums1[p1] > nums2[p2], set nums1[p] = nums1[p1] and decrement p1 and p.
   • Otherwise, set nums1[p] = nums2[p2] and decrement p2 and p.
4. Repeat until all elements of nums2 are merged into nums1.

The algorithm processes the arrays in reverse order, starting from the largest elements and filling nums1 from the end. This approach prevents overwriting existing data in nums1 as it is merged in place. The merge only considers the first m elements in nums1 and the first n elements in nums2. Any extra space in nums1 beyond m + n is irrelevant for the merging process.