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When the inputs are linked lists, this algorithm can be implemented to use only a constant amount of working space; the pointers in the lists' nodes can be reused for bookkeeping and for constructing the final merged list. In the merge sort algorithm, this subroutine is typically used to merge two sub-arrays A[lo..mid], A[mid+1..hi] of a single ...
If the running time (number of comparisons) of merge sort for a list of length n is T(n), then the recurrence relation T(n) = 2T(n/2) + n follows from the definition of the algorithm (apply the algorithm to two lists of half the size of the original list, and add the n steps taken to merge the resulting two lists). [5]
Another example is the merging two sorted lists: if their sentinels have data fields set to +∞, the choice of the next output node does not need special handling for empty lists. However, sentinel nodes use up extra space (especially in applications that use many short lists), and they may complicate other operations (such as the creation of ...
Merge sort takes advantage of the ease of merging already sorted lists into a new sorted list. It starts by comparing every two elements (i.e., 1 with 2, then 3 with 4...) and swapping them if the first should come after the second. It then merges each of the resulting lists of two into lists of four, then merges those lists of four, and so on ...
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.
Problems of sufficient simplicity are solved directly. For example, to sort a given list of n natural numbers, split it into two lists of about n/2 numbers each, sort each of them in turn, and interleave both results appropriately to obtain the sorted version of the given list (see the picture). This approach is known as the merge sort algorithm.
balanced k-way merge sort; balanced merge sort; balanced multiway merge; balanced multiway tree; balanced quicksort; balanced tree; balanced two-way merge sort; BANG file; Batcher sort; Baum Welch algorithm; BB α tree; BDD; BD-tree; Bellman–Ford algorithm; Benford's law; best case; best-case cost; best-first search; biconnected component ...
The previous example is a two-pass sort: first sort, then merge. The sort ends with a single k-way merge, rather than a series of two-way merge passes as in a typical in-memory merge sort. This is because each merge pass reads and writes every value from and to disk, so reducing the number of passes more than compensates for the additional cost ...