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Topic: Recursive algorithm

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	IMEJ Article - Animating Recursive Algorithms
		Two recursive calls are then made, one to the part of the array with the smaller elements, and another to the part of the array with the larger elements.
		For these algorithms, where data items within an existing linear data structure are manipulated without the data structure changing its size or shape, we represent each recursive call by a horizontal line under the sequence of data items referred to in the recursive call (Figure 1).
		Iterative insertion algorithms perform splitting and rotation as the tree is traversed from root to the leaf, where the data is eventually inserted [10], and can trivially be recoded as a tail recursive implementation, with the operations performed just before the recursive call.
	imej.wfu.edu /articles/2002/2/02/index.asp (3618 words)

	Recursive Algorithm
		is an algorithm which calls itself with "smaller (or simpler)" input values, and which obtains the result for the current input by applying simple operations to the returned value for the smaller (or simpler) input.
		In general, recursive computer programs require more memory and computation compared with iterative algorithms, but they are simpler and for many cases a natural way of thinking about the problem.
		Recursive algorithms can also be used to test objects for membership in a set.
	www.cs.odu.edu /~toida/nerzic/content/recursive_alg/rec_alg.html (605 words)

	Recursion
		Recursion appears both as recursive algorithms and as recursive data structures.
		Recursive data structures are arguably the most important data structure in computer science as they are able to represent arbitrarily complex data.
		One of the key aspects of a recursive algorithm is that in the inductive case, the inductive method makes the recursive call to another object's method.
	www.owlnet.rice.edu /~tchjava/2005/lectures/recursion (1509 words)

	Instructor Notes for Recursion Labs
		To avoid overwhelming students with too many new ideas at once, we recommend introducing recursion as an algorithm design idea first, and later introducing induction, but induction should come as soon as students are comfortable enough with the basic concept of recursion to learn a new way of approaching it.
		Their algorithms will use recursion to move a robot to a wall, but will use a parameter to accumulate a count of red tiles, or will accumulate a count in a global variable.
		However, an important reason for including this exercise in the lab is to give students a chance to practice writing a recursive algorithm that calculates and returns (in the technical sense of the word) a result, and instructors should insist that students’ algorithms do so.
	www.charlesriver.com /algorithms/note-reclabs.html (1305 words)

	Recursive Surface Algorithm
		The algorithm starts by replacing the initial cube and all that result until all the cubes have some minimum required depth.
		The algorithm continues by subdividing when a cube has not reached the maximum allowed depth and, the eight numbers that result when you evaluated the function at the vertices of the cube do not have the same sign.
		The algorithm was inspired after working on implementing a brute force method for finding curves in the plane.
	www.geom.uiuc.edu /~streed/pisces/docs/algorithms/rsurf.html (407 words)

	TH algorithm
		In Algorithms no. 1 and 2, fr points to the tower from which a disk is being moved, and to to the tower where it is being transferred.
		I have compared this algorithm with the loop less iterative algorithm LLHanoi of M. Er [2], which seems to be the fastest previously known algorithm for the Tower of Hanoi problem.
		Algorithm no. 4 is implemented in a variation of my JavaScript Tower of Hanoi puzzle that is intended for automatic demonstration of the solution only.
	hanoitower.mkolar.org /algo.html (2116 words)

	Homework 3.3 (Site not responding. Last check: 2007-10-29)
		Write a recursive algorithm that takes in a natural number n and determines if n is prime.
		Write a recursive algorithm that takes two natural numbers a and b, and finds the summation of all natural numbers from a to b.
		Write a recursive algorithm that takes in a natural number n and, and prints out all of the factors of n.
	www.omegamath.com /Discrete/cp3.3.html (234 words)

	Mastering recursive programming
		Recursion is a tool not often used by imperative language developers, because it is thought to be slow and to waste space, but as the author demonstrates, there are several techniques that can be used to minimize or eliminate these problems.
		With recursive programs that recurse instead of change state, the number of occasions of state change is small and the program variables maintain self-consistency by setting all of the recursion variables at once.
		Recursion is a great art, enabling programs for which it is easy to verify correctness without sacrificing performance, but it requires the programmer to look at programming in a new light.
	www-128.ibm.com /developerworks/linux/library/l-recurs.html (5938 words)

	Lecture 11 Sequential Algorithms
		A recursive algorithm is one that uses the method within itself to solve a simpler problem.
		A tail-end recursive algorithm is one that calls upon itself at the "tail end" of the algorithm.
		A truly recursive algorithm is one in which the recursive call upon itself somewhere inside the method (other than just at the end.) A recursive algorithm may call upon itself more than once.
	www.tcnj.edu /~scott/Lectures/imm120lecture11recursion.htm (1210 words)

	Fibonacci Numbers
		In general, any recursive algorithm such as this one gives us a recurrence relation: the time for any routine is the time within the routine itself, plus the time for the recursive calls.
		This is an iterative algorithm (one that uses loops instead of recursion) so we analyze it a little differently than we would a recursive algorithm.
		Algorithm 2 and algorithm 3 are both O(n).
	www.ics.uci.edu /~eppstein/161/960109.html (3061 words)

	Applying recursion to serial and parallel QR factorization leads to better performance
		Hence, RGEQRF is a modified version of Algorithm 4.2 of Bischof and Van Loan, in [3], and RGEQR3 is a recursive Level 3 counterpart of their Algorithm 4.1.
		It is related to the LU factorization algorithm described in [6] and the dynamic load-balancing versions of LU and Cholesky factorization in [7].
		The algorithm is based on a dynamic load-balancing strategy, implemented using the pool-of-tasks principle in which each processor enters a critical section to assign itself more work as soon as it has completed its last task.
	www.research.ibm.com /journal/rd/444/elmroth.html (7822 words)

	Recursion - Wikipedia, the free encyclopedia
		Recursion, in mathematics and computer science, is a method of defining functions in which the function being defined may be used within its own definition.
		So a recursive meal might be potato skins, baby greens salad, chicken parmesan, and for dessert, a four course meal, consisting of crab cakes, Caesar salad, for an entrée, a four course meal, and chocolate cake for dessert, so on until each of the meals within the meals is completed.
		The function calls itself recursively on a smaller version of the input (n - 1) and multiplies the result of the recursive call by n, until reaching the base case, analogously to the mathematical definition of factorial.
	en.wikipedia.org /wiki/Recursive_algorithm (1828 words)

	The Recursive Ray Tracing Algorithm
		Recursion is a means of obtaining a result in which a given process repeats itself an arbitrary number of times.
		It then recurses (or repeats itself) by shooting more rays from the point of intersection to see what objects are reflected at that point, what objects may be seen through the object at that point, which light sources are directly visible from that point, and so on.
		Because the standard ray tracing algorithm actually follows the path of light from the eye of the viewer backward to the light source, it is, in that sense, often referred to as backward ray tracing.
	www.geocities.com /jamisbuck/raytracing.html (4225 words)

	Fabulous Adventures In Coding : How Not To Teach Recursion
		When you introduce recursion by using it to solve a problem which iteration solves much, much better, students come away thinking that recursion is dumb -- that it is both harder to understand and produces worse solutions.
		Recursive factorial, fibonacci and tower of hanoi are easy to understand and useful to explain the concept.
		The recursive solution for the tower of hanoi is far from optimal, but elegant, and a very simple solution for a problem that a novice could believe it's very complex.
	blogs.msdn.com /ericlippert/archive/2004/05/19/135392.aspx (3517 words)

	CMSC 435 Algorithm Analysis & Design
		This non-recursive algorithm is clearly linear in n (T(n) = Q(n)) as the for-loop executes a constant number of statements n times.
		The performance of this algorithm is exponential, as the algorithm suffers from acute senility.
		The algorithm for retrieving the solution set proceeds as follows -- {we must redefine the matrix M so that it is not local to a single function}
	www.academic.marist.edu /jte/algorithms/DynamicProgramming1.htm (1024 words)

	Longest Common Subsequences
		The top down (memoizing) method is closer to the original recursive algorithm, so easier to understand, but the bottom up method is usually a little more efficient.
		The current (theoretically) fastest algorithm for longest common subsequences (due to myself and co-authors) runs in time O(n log s + c log log min(c,mn/c)) where c is the number of these corners, and s is the number of characters appearing in the two strings.
		One disadvantage of the dynamic programming methods we've described, compared to the original recursion, is that they use a lot of space: O(mn) for the array L (the recursion only uses O(n+m)).
	www.ics.uci.edu /~eppstein/161/960229.html (2866 words)

	recursive (Site not responding. Last check: 2007-10-29)
		For instance, a tree is composed of smaller trees (subtrees) and leaf nodes, and a list may have other lists as elements.
		Note: Infinite data structures may be represented by having a tree include (point back to) itself recursively, a list include itself, etc. Recursive data structures are often best handled with a recursive algorithm, or an algorithm using recursion.
		Go to the Dictionary of Algorithms and Data Structures home page.
	www.nist.gov /dads/HTML/recursive.html (154 words)

	Algorithm - Wikipedia, the free encyclopedia
		In mathematics, computing, linguistics and related disciplines, an algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state.
		Algorithms can be composed to create more complex algorithms.
		The word algorithm comes from the name of the 9th century Persian mathematician Abu Abdullah Muhammad ibn Musa al-Khwarizmi whose works introduced Arabic numerals and algebraic concepts.
	en.wikipedia.org /wiki/Algorithm (6140 words)

	A FAST RECURSIVE GIS ALGORITHM FOR COMPUTING STRAHLER STREAM ORDER IN BRAIDED AND NONBRAIDED NETWORKS1 Journal of the ... (Site not responding. Last check: 2007-10-29)
		This paper describes an algorithmic framework that was developed to create the fastest possible procedure for vector stream network ordering using the Strahler method.
		Hence, a formal and thorough algorithmic definition of the recursive stream order computation framework is of interest.
		Network structure in GIS could be viewed as a graph, where nodes are also graph 'nodes' and arcs are graph 'edges.' For stream networks in particular, statistical graph theory was used (e.g., Scheidegger, 1968, to describe Horton's law of stream numbers).
	www.findarticles.com /p/articles/mi_qa4038/is_200408/ai_n9447596 (952 words)

	CodeGuru Forums - a recursive algorithm problem
		Now you can call this recursively with the second, third etc. element and you will get all subsets.
		The algorithm for writing such a sequence in a recursive way is like this (pseudo-code):
		If interested, tell me. Generally prolog is the best language for recursive algorithms and with list operations.
	www.codeguru.com /forum/showthread.php?t=297149 (839 words)

	Franklin & Marshall - Polygon triangulation: recursive algorithm
		The recursive algorithm for triangulating the polygon begins by searching for two vertices on opposite sides ot the polygon, and connecting them with a diagonal.
		This procedure is then continued recursively until all smaller polygons are triangles.
		This algorithm is used in the "art gallery problem," found elsewhere.
	www.fandm.edu /x7741.xml (199 words)

	The Tree of Life & Epistemology of The Recursive Evolutionary Algorithm
		The Tree of Life and Epistemology of The Recursive Evolutionary Algorithm
		’ (by now algorithmic [and much more on this in the presentation on 'Evolution']) and consistent with all the personal, subjective yet meaningful understanding of the same that the Darhana of Yoga has to offer from an inner point of view.
		The categories and subcategories here all refer to the process of Yoga as such and are not to be ideologically transferred to any other domain, they will rather become infused into life and living depending on application and circumstance.
	hem.bredband.net /columbia/epistemology.htm (2144 words)

	Example-Fibonacci Numbers
		The running time of this algorithm is clearly O(n) as shown by the analysis in Table
		What happens if instead of using the iterative algorithm, we use the definition of Fibonacci numbers to implement directly a recursive algorithm ?
		shows the actual running times of both the non-recursive and recursive algorithms for computing Fibonacci numbers.
	www.brpreiss.com /books/opus4/html/page73.html (455 words)

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