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In computer science, K-Way Merge Algorithms or Multiway Merges are a specific type of sequence merge algorithms that specialize in taking in multiple sorted lists and merging them into a single sorted list. These merge algorithms generally refer to merge algorithms that take in a number of sorted lists greater than two. 2-Way Merges are also referred to as binary merges.
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2-Way Merge
A 2-Way Merge, or a binary merge, has been studied extensively due to its key role in merge sort. An example of such is the classic merge that appears frequently in merge sort examples. The classic merge outputs the data item with the lowest key at each step; given some sorted lists, it produces a sorted list containing all the elements in any of the input lists, and it does so in time proportional to the sum of the lengths of the input lists.
Denote by A[1..p] and B[1..q] two arrays sorted in increasing order. Further, denote by C[1..n] the output array. The canonical 2-Way Merge algorithm stores indices i, j, and k into A, B, and C respectively. Initially, these indices refer to the first element, i.e., are 1. If A[i] < B[j], then the algorithm copies A[i] into C[k] and increases i and k. Otherwise, the algorithm copies B[j] into C[k] and increases j and k. A special case arises if either i or j have reached the end of A or B. In this case the algorithm copies the remaining elements of B or A into C and terminates.
K-Way Merge
The K-way merge problem consists of merging k sorted arrays to produce a single sorted array with the same elements. Denote by n the total number of elements. n is equal to the size of the output array and the sum of the sizes of the k input arrays. For simplicity, we assume that none of the input arrays is empty. As a consequence k < n, which simplifies the reported running times. The problem can be solved in O(n log k) running time with O(n) space. Several algorithms that achieve this running time exist.
Iterative 2-Way Merge
The problem can be merged by iteratively merging two of the k arrays using a 2-Way Merge until only a single array is left. If the arrays are merged in arbitrary order, then the resulting running time is only O(kn). This is suboptimal.
The running time can be improved by iteratively merging the first with the second, the third with the fourth, and so on. As the number of arrays is halved in each iteration, there are only Θ(log k) iteration. In each iteration every element is moved exactly once. The running time per iteration is therefore in Θ(n) as n is the number of elements. The total running time is therefore in Θ(n log k).
We can further improve upon this algorithm, by iteratively merging the two shortest arrays. It is clear that this minimizes the running time and can therefore not be worse than the stratedy described in the previous paragraph. The running time is therefore in O(n log k). Fortunately, in border cases the running time can be better. Consider for example the degenerate case, where all but one array contain only one element. The strategy explained in the previous paragraph needs Θ(n log k) running time, while the improved one only needs Θ(n) running time.
Direct K-Way Merge
The basic idea of a direct K-Way Merge consists of efficiently computing the minimum element of all k arrays and then transferring it to the output array. A straightforward implementation would scan all k arrays to determine the minimum. This straightforward implementation results in a running time of Θ(kn).
We can improve upon this by computing the smallest element faster. By using either heaps or tournament trees the smallest element can be determined in O(log k) time. The resulting running times are therefore in O(n log k).
Heap
The heap algorithm allocates a min-heap of pointers into the input arrays. Initially these pointers point to the smallest elements of the input array. The pointers are sorted by the value that they point to. In an O(k) preprocessing step the heap is created using the standard heapify procedure. Afterwards, the algorithm iteratively transfers the element that the root pointer points to, increases this pointer and executes the standard increase key procedure upon the root element. The running time of the increase key procedure is bounded by O(log k). As there are n elements, the total running time is O(n log k).
Tournament Tree
The tournament tree algorithms constructs a balanced binary tree with k leafs. Every leaf stores a pointer into one of the input arrays. Further, for every node in the tree, a value and an index is stored. For the k-th leaf the value is the value that its pointer points to and the index is k. For every internal leaf the value is minimum of the values of the node's children. The index of an internal leaf indicates which input array the value comes from. The root of the tournament tree contains the smallest element and the index of the corresponding input array. The algorithm iteratively transfers this element and advances the corresponding pointer. It then updates the values of the nodes on the path from the corresponding leaf to the root. As the tree is balanced, this path contains only Θ(log k) element. As there n elements that need to be extracted, the resulting total running time is Θ(n log k). the sorted items in a heap, then the running time becomes Θ(n log k).
Example
Let's say we have 4 sorted arrays:
{5, 10, 15, 20}
{10, 13, 16, 19}
{2, 19, 26, 40}
{18, 22, 23, 24}
We start with the heads of each array and then build a binary tree from there.
The nodes from each array are compared to each other, before the value 2 is found as the lowest list head element. That value is then popped off, and its leaf is refilled with the next value in the list.
The value 2 is repopulated by the next value in the sorted list, 19. The comparisons end with 5 being the smallest value, and thus the next value to be popped off. This continues until all of the sorted lists are empty.
Lower Running Time Bound
One can show that no comparison-based K-Way Merge algorithm exists with a running time in O(n f(k)) where f grows asymptotically slower than a logarithm. The proof is a straightforward reduction from comparison-based sorting. Suppose that such an algorithm existed, then we could construct a comparison-based sorting algorithm with running time O(n f(n)) as follows: Chop the input array into n arrays of size 1. Merge these n arrays with the K-Way Merge algorithm. The resulting array is sorted and the algorithm has a running time in O(n f(n)). This is a contradiction to the well-known result that no comparison-based sorting algorithm with a worst case running time below O(n log n) exists.