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where a represents the number of recursive calls at each level of recursion, b represents by what factor smaller the input is for the next level of recursion (i.e. the number of pieces you divide the problem into), and f(n) represents the work that the function does independently of any recursion (e.g. partitioning, recombining) at each level ...
Notable examples of systems employing polymorphic recursion include Dussart, Henglein and Mossin's binding-time analysis [2] and the Tofte–Talpin region-based memory management system. [3] As these systems assume the expressions have already been typed in an underlying type system (not necessary employing polymorphic recursion), inference can ...
In the mathematical subfield of numerical analysis, de Boor's algorithm [1] is a polynomial-time and numerically stable algorithm for evaluating spline curves in B-spline form. It is a generalization of de Casteljau's algorithm for Bézier curves.
A recursive step — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ancestor. One's ancestor is either: One's parent (base case), or; One's parent's ancestor (recursive step). The Fibonacci sequence is another classic example of recursion: Fib(0) = 0 as ...
Mutual recursion is very common in functional programming, and is often used for programs written in LISP, Scheme, ML, and similar programming languages. For example, Abelson and Sussman describe how a meta-circular evaluator can be used to implement LISP with an eval-apply cycle. [7] In languages such as Prolog, mutual recursion is almost ...
Another notable example is the algorithm invented by Anatolii A. Karatsuba in 1960 [8] that could multiply two n-digit numbers in () operations (in Big O notation). This algorithm disproved Andrey Kolmogorov 's 1956 conjecture that Ω ( n 2 ) {\displaystyle \Omega (n^{2})} operations would be required for that task.
The leaves of the tree are the base cases of the recursion, the subproblems (of size less than k) that do not recurse. The above example would have a child nodes at each non-leaf node. Each node does an amount of work that corresponds to the size of the subproblem n passed to that instance of the recursive call and given by (). The total amount ...
In the worst case, i = 1 or i = n − 2 at each recursive invocation yields a running time of O(n 2). In the best case, i = n / 2 or i = n ± 1 / 2 at each recursive invocation yields a running time of O(n log n). Using (fully or semi-) dynamic convex hull data structures, the simplification performed by the algorithm can be ...