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Folds can be regarded as consistently replacing the structural components of a data structure with functions and values. Lists, for example, are built up in many functional languages from two primitives: any list is either an empty list, commonly called nil ([]), or is constructed by prefixing an element in front of another list, creating what is called a cons node ( Cons(X1,Cons(X2,Cons ...
In computer science, a for-loop or for loop is a control flow statement for specifying iteration. Specifically, a for-loop functions by running a section of code repeatedly until a certain condition has been satisfied. For-loops have two parts: a header and a body. The header defines the iteration and the body is the code executed once per ...
The other two for loops, and the initialization of the output array, each take O(n) time. Therefore, the time for the whole algorithm is the sum of the times for these steps, O(n + k). [1] [2] Because it uses arrays of length k + 1 and n, the total space usage of the algorithm is also O(n + k). [1]
Prefix sums are trivial to compute in sequential models of computation, by using the formula y i = y i − 1 + x i to compute each output value in sequence order. However, despite their ease of computation, prefix sums are a useful primitive in certain algorithms such as counting sort, [1] [2] and they form the basis of the scan higher-order function in functional programming languages.
Thus, each list can be generated in sorted form in time (/). Given the two sorted lists, the algorithm can check if an element of the first array and an element of the second array sum up to T in time (/). To do that, the algorithm passes through the first array in decreasing order (starting at the largest element) and the second array in ...
In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. [1] [2] Each iteration starts with a four digit random number, sorts the digits into descending and ascending order, and calculates the difference between the two new numbers.
This says that an expression is either a number, a product of two expressions, or a sum of two expressions. By recursively referring to expressions in the second and third lines, the grammar permits arbitrarily complicated arithmetic expressions such as (5 * ((3 * 6) + 8)) , with more than one product or sum operation in a single expression.
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.