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Else, recursively merge the first ⌊k/2⌋ lists and the final ⌈k/2⌉ lists, then binary merge these. When the input lists to this algorithm are ordered by length, shortest first, it requires fewer than n ⌈log k ⌉ comparisons, i.e., less than half the number used by the heap-based algorithm; in practice, it may be about as fast or slow ...
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]
Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered ...
It then merges each of the resulting lists of two into lists of four, then merges those lists of four, and so on; until at last two lists are merged into the final sorted list. [24] Of the algorithms described here, this is the first that scales well to very large lists, because its worst-case running time is O( n log n ).
Recursively sort the ⌊ / ⌋ larger elements from each pair, creating a sorted sequence of ⌊ / ⌋ of the input elements, in ascending order, using the merge-insertion sort. Insert at the start of S {\displaystyle S} the element that was paired with the first and smallest element of S {\displaystyle S} .
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.
A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively. The term "infinite tuple" is occasionally used for "infinite sequences". Tuples are usually written by listing the elements within parentheses "( )" and separated by commas; for example, (2, 7, 4, 1, 7) denotes a 5-tuple. Other types of brackets are ...
The input to the + sorting problem consists of two finite collections of numbers and , of the same length.The problem's output is the collection of all pairs of a number from and a number from , arranged into sorted order by the sum of each pair. [1]