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The master theorem always yields asymptotically tight bounds to recurrences from divide and conquer algorithms that partition an input into smaller subproblems of equal sizes, solve the subproblems recursively, and then combine the subproblem solutions to give a solution to the original problem.
In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
For this recurrence relation, the master theorem for divide-and-conquer recurrences gives the asymptotic bound () = (). It follows that, for sufficiently large n , Karatsuba's algorithm will perform fewer shifts and single-digit additions than longhand multiplication, even though its basic step uses more additions and shifts than the ...
[4] [5] If an algorithm is designed so that it will break a problem into smaller subproblems (divide and conquer), its running time is described by a recurrence relation. A simple example is the time an algorithm takes to find an element in an ordered vector with n {\displaystyle n} elements, in the worst case.
In the most balanced case, a single quicksort call involves O(n) work plus two recursive calls on lists of size n/2, so the recurrence relation is = + (). The master theorem for divide-and-conquer recurrences tells us that T(n) = O(n log n).
Master theorem (analysis of algorithms), analyzing the asymptotic behavior of divide-and-conquer algorithms; Ramanujan's master theorem, providing an analytic expression for the Mellin transform of an analytic function; MacMahon master theorem (MMT), in enumerative combinatorics and linear algebra; Glasser's master theorem in integral calculus
Its primary application is the approximation of the running time of many divide-and-conquer algorithms. For example, in the merge sort , the number of comparisons required in the worst case, which is roughly proportional to its runtime, is given recursively as T ( 1 ) = 0 {\displaystyle T(1)=0} and
The closed form follows from the master theorem for divide-and-conquer recurrences. The number of comparisons made by merge sort in the worst case is given by the sorting numbers . These numbers are equal to or slightly smaller than ( n ⌈ lg n ⌉ − 2 ⌈lg n ⌉ + 1), which is between ( n lg n − n + 1) and ( n lg n + n + O(lg n )). [ 6 ]