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A basic example of short-circuiting is given in depth-first search (DFS) of a binary tree; see binary trees section for standard recursive discussion. The standard recursive algorithm for a DFS is: base case: If current node is Null, return false; recursive step: otherwise, check value of current node, return true if match, otherwise recurse on ...
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 ...
The most important basic example of a datatype that can be defined by mutual recursion is a tree, which can be defined mutually recursively in terms of a forest (a list of trees). Symbolically: f: [t[1], ..., t[k]] t: v f
Another example is a similar singly linked type in Java: class List < E > { E value ; List < E > next ; } This indicates that non-empty list of type E contains a data member of type E, and a reference to another List object for the rest of the list (or a null reference to indicate that this is the end of the list).
An example of a primitive recursive programming language is one that contains basic arithmetic operators (e.g. + and −, or ADD and SUBTRACT), conditionals and comparison (IF-THEN, EQUALS, LESS-THAN), and bounded loops, such as the basic for loop, where there is a known or calculable upper bound to all loops (FOR i FROM 1 TO n, with neither i ...
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 ...
The use of Turing machines here is not necessary; there are many other models of computation that have the same computing power as Turing machines; for example the μ-recursive functions obtained from primitive recursion and the μ operator. The terminology for computable functions and sets is not completely standardized.
The first consists of many recursive calls that repeatedly perform the same division process until the subsequences are trivially sorted (containing one or no element). An intuitive approach is the parallelization of those recursive calls. [19] Following pseudocode describes the merge sort with parallel recursion using the fork and join keywords: