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Topic: Quicksort implementations

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Quicksort

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	Sorting methods
		Quicksort uses the devide and conquer technique to divide the problem into smaller instances of the same problems, then solve the smaller instances recursively, and finally combine the solutions to obtain the solution for the original input.
		Quicksort's strategy is to rearrange the keys to be sorted so that all the small keys precede the large keys.
		Quicksort is a preferred sorting methods in applications that require a sorting algorithm that is usually very fast, but on occasion can have a longer running time.
	www.cse.unr.edu /~chen_q/sorta.html (3201 words)

	Developing for Developers : Efficient selection and partial sorting based on quicksort
		Quicksort is a compelling example of how algorithms with poor worst-case behavior but good average-case or expected behavior can be highly practical.
		Quicksort is fundamentally based on the efficient partition operation, a linear-time operation which divides a list into two parts: the elements less than a value, and the elements greater than or equal to a value.
		In efficient implementations of quicksort, partitioning is usually implemented as an in-place operation that just shuffles elements of the array around as it scans over it.
	blogs.msdn.com /devdev/archive/2006/01/18/514482.aspx (959 words)

	PlanetMath: in-place sorting algorithm
		Quicksort is also considered in-place, although it requires an amount of extra space logarithmic in the size of the array.
		If you try to convert Quicksort to a non-recursive algorithm, you will find that it is necessary to store some intermediatde indexes on a stack, which usually grows up to size
		These are distinctions in complexity that are almost always ignored since they are dwarfed in practical problems by implementation differences.
	planetmath.org /encyclopedia/InPlaceSortingAlgorithm.html (572 words)

	Merge sort - CodeCodex
		Merge sort's most common implementation does not sort in place, meaning memory the size of the input must be allocated for the sorted output to be stored in.
		Unlike some (inefficient) implementations of quicksort, merge sort is a stable sort as long as the merge operation is implemented properly.
		Quicksort, however, is considered by many to be the fastest general-purpose sort algorithm in practice.
	www.codecodex.com /wiki/index.php?title=Merge_sort (1776 words)

	5. Experimental Results
		We have implemented the algorithms in a similar format with linked lists and we have refrained from micro-optimizations such as loop unrolling and manual register allocation.
		Multikey quicksort uses quicksort instead of bucketsort to perform the character sorting required in each stage of the algorithm.
		The speed of quicksort compared to bucketsort depends on both the size of the alphabet and the number of characters to be sorted.
	www.jea.acm.org /TURING/Vol3Nbr7/node5.html (1868 words)

	[No title]
		Sort those smaller lists (using Quicksort, unless they are trivial lists) Create a sorted version of the original list by concatenating the sorted list of smaller elements and the sorted list of larger elements.
		Note: the Quicksort algorithm presented here is an adaptation of a famous algorithm for sorting arrays (vectors in Scheme terminology) to the problem of sorting lists.
		In the array version of Quicksort, there is a very slick way to achieve this result but it does not directly adapt to the list version.
	www.owlnet.rice.edu /~comp210/01spring/Lectures/Lect19.doc (892 words)

	Quicksort (Site not responding. Last check: 2007-11-03)
		Typically, Quicksort is significantly faster in practice than other O(''n'' log ''n'') algorithms, because its inner loop can be efficiently implemented on most architectures, and in most real-world data it is possible to make design choices which minimize the possibility of requiring quadratic time.
		Although it is often believed to work in-place, quicksort uses O(log n) additional stack space on average in recursive implementations and O(''n'') stack space in the worst case.
		One widely used implementation of quicksort, that in the 1997 Microsoft C library, used a cutoff of 8 elements before switching to insertion sort, asserting that testing had shown that to be a good choice.
	quicksort.iqnaut.net (2552 words)

	Question 11)
		Use pseudocode to write the algorithm for the merge subroutine in merge sort OR the partition subroutine in quicksort (if you pick the latter, you may assume that a good pivot has already been chosen and is located at the far right of the array).
		Suppose you were implementing a Queue as a circular array (ring buffer).
		Implement a recursive method palindrome(….) that will check whether a given string is a palindrome or not and write a client call that tests it.
	www.cs.uga.edu /~gtb/1302/Questions-2.htm (1648 words)

	Sorting Algorithms
		Quicksort and merge sort, respectively, are the canonical examples in these categories.
		Insertion sort and selection sort are seen to be instances of merge sort and quicksort, respectively, and sinking sort and bubble sort are in-place versions of insertion sort and selection sort.
		This program started out as a quicksort demonstration applet, and after proposing the idea to Seton Hall's math and computer science department, I was given permission to expand the idea to encompass many of the sorts commonly taught in a data structures/algorithms class.
	www.softpanorama.org /Algorithms/sorting.shtml#Quicksort (3521 words)

	Dr. Dobb's \| Generic<Programming>: Efficient Generic Sorting and Searching in C++ (II) - Sorting through Sorts of ...
		So if your only intuition of Quicksort is that it might be an improved version of Slowsort, you may want to refer to [1] first.
		The result is a concatenation of three sequences: Quicksort on those elements in the sequence's tail that are less than or equal to the pivot; the pivot itself; and finally, Quicksort on the greater elements.
		What many modern Quicksort implementations do is to select the median of the first, last, and mid element as the pivot.
	www.ddj.com /dept/cpp/184403848 (2991 words)

	Dr. Dobb's \| Testing Sort Functions \| April 15, 2003
		Two-way merge sort is a solid (stodgy but guaranteed) O(NLgN) algorithm, which means that its timing performance is proportional to N times the binary logarithm of N. (So, for that matter, is Heapsort [5, 3].) Two-way merge sorting requires a companion storage area the size of the original sort target.
		This is clearly the poorest performing Quicksort of the lot — at least on this test suite — and should be avoided.
		The timing for Borland Quicksort is an estimate based on runs of three populations — 5,000, 10,000, and 15,000.
	www.ddj.com /dept/cpp/184403049?pgno=10 (2637 words)

	EE108B - Programming Assignment #1
		This assignment is intended to familiarize you with the SPIM environment for later assignments; to test your understanding of pointer implementations and procedure call conventions.
		Implement quicksort using the MIPS instruction set outlined in the SPIM documentation and in lecture.
		Your job is to add the required assembly code to the skeleton file, quicksort.s, such that the assembly code has the same functionality as the C code.
	www.stanford.edu /class/ee108b/pas/PA1/pa1.html (539 words)

	Merge sort - Wikipedia, the free encyclopedia
		It is easy to implement merge sort such that it is stable, meaning it preserves the input order of equal elements in the sorted output.
		Merge sort's most common implementation does not sort in place; therefore, the memory size of the input must be allocated for the sorted output to be stored in.
		Unlike some (efficient) implementations of quicksort, merge sort is a stable sort as long as the merge operation is implemented properly.
	en.wikipedia.org /wiki/Mergesort (1388 words)

	Quicksort (Site not responding. Last check: 2007-11-03)
		Recall that Quicksort selects a pivot for the elements in the array.
		The Quicksort algorithm is then recursively applied on the higher and lower arrays.
		Selection of the pivot is not too important for the sequential algorithm however it is important for the parallel algorithm in order to keep the tree of processes reasonably balanced.
	www.cs.mu.oz.au /677/notes/node79.html (93 words)

	Reference.com/Encyclopedia/Quicksort
		Quicksort is a well-known sorting algorithm developed by C.
		Typically, quicksort is significantly faster in practice than other Θ(n log n) algorithms, because its inner loop can be efficiently implemented on most architectures, and in most real-world data it is possible to make design choices which minimize the possibility of requiring quadratic time.
		Lecture 5 - quicksort CSE 373/548 - Analysis of Algorithms.
	www.reference.com /browse/wiki/Quicksort (4154 words)

	CPS 100, Sortall, Fall 1999
		This assignment is primarily an empirical study of different sorting algorithms whose implementations are given to you.
		To "fix" this problem, you're going to write a new version of quicksort that splits the array into 3 sections: one less than the pivot, one equal to the pivot, and one greater than the pivot.
		When quicksort is applied to small vectors (size 20 or less) it calls insertionsort, so your pivot code may not be tested unless you change the call to the constructor to look like this
	www.cs.duke.edu /~dr/104.99f/asgn5.html (2235 words)

	flipcode - Quick Sort On Linked List (Site not responding. Last check: 2007-11-03)
		Quicksort averages O(n), and in the worst case scenario you want to switch to heap sort, which is O(n log n), because quicksort gets nastier than that (O(n^2)).
		Also many many quicksort implementations pick the first element in the list as pivot, which is really stupid.
		Quicksort is often preferred over merge-sort with arrays, because its constant factor is smaller.
	www.flipcode.com /cgi-bin/fcarticles.cgi?show=4&id=64000 (2189 words)

	UnrealWiki: QuickSort
		QuickSort is a useful, very fast, algorithm that is used for sorting arrays.
		QuickSort calls itself twice recursivly to sort the upper/lower parts of the array.
		Wormbo: Function calls are slow, so it might be a good idea to include the code for swapping the array elements in the main function.
	wiki.beyondunreal.com /wiki/QuickSort (312 words)

	Combsort (Site not responding. Last check: 2007-11-03)
		A naive quicksort like this one is very slow in the presence of an already sorted (or almost sorted) list.
		In any case, because this deficiency of quicksort is so easy to fix, and because it won't change the performance of quicksort much, we won't worry about already sorted lists in this comparison.
		I didn't do a benchark of a C++ quicksort that returns a copy of the array (rather than modifying it in place) because it seems too hard to bother with.
	yagni.com /combsort/index.php (1396 words)

	Quick Sort
		Quicksort works by "partitioning" the array to be sorted, then recursively sorting each partition.
		Alternatively, stop early and do a final pass of InsertionSort at the end; this imposes less overhead in cycles than doing lots of separate little insertion sorts, but may interact less well with cache (for moderate datasets) or VM (for immoderate ones).
		As with quicksort (but even more so), you want to stop "early" and switch to a simpler, lower-overhead algorithm once the pieces are small.
	c2.com /cgi/wiki?QuickSort (560 words)

	Quicksort
		This file contains only blank implementations ("stubs") because this is a Programming Project for my students.
		The main method illustrates the use of a quicksort to sort a small array.
		Sort an array of integers from smallest to largest, using a quicksort algorithm.
	www.cs.colorado.edu /~main/docs/Quicksort.html (125 words)

	quicksort
		Note: Quicksort has running time Θ(n²) in the worst case, but it is typically O(n log n).
		In practical situations, a finely tuned implementation of quicksort beats most sort algorithms, including sort algorithms whose theoretical complexity is O(n log n) in the worst case.
		Robert Sedgewick's talk showing that with Bentley-McIlroy 3-way partitioning Quicksort Is Optimal (C) (pdf format) for random files possibly with duplicate keys; includes discussion and proof.
	www.nist.gov /dads/HTML/quicksort.html (249 words)

	How to demonstrate bigO cost of algorithms? - Python
		since the big O for quicksort is nlogn for all but the worst case.
		implement a __cmp__ method that increments a global whenever called.
		> since the big O for quicksort is nlogn for all but the worst case.
	www.thescripts.com /forum/thread31926.html (1758 words)

	Computer Science Logo Style vol 3 ch 3: Algorithms and Data Structures
		Quicksort includes many improvements over this algorithm, not primarily in reducing the number of comparisons but in decreasing the overhead time by, for example, exchanging pairs of input items in their original memory locations instead of making copies of sublists.
		Nevertheless, quicksort is more widely used than mergesort, because the very best implementations of quicksort seem to require less overhead time, for the average input, than the best implementations of mergesort.
		But the implementation is a bit trickier than looking through a sequential list, because each invocation gives rise to several recursive invocations (one per child), not merely one recursive invocation as usual.
	www.cs.berkeley.edu /~bh/v3ch3/v3ch3.html (13467 words)

	eqntott (SPECint92)
		Both merge sort and quicksort are on average O(N log N) algorithms; however, in the real world, constant factors and not just asymptotic algorithmic complexity matters.
		It is generally acknowledged that implementations of some of the quicksort variants have the lowest constant factors of well-known comparison-based sorting routines; this is borne out by some basic experimentation --- each row of Table
		which shows that the merge sort implementation tends to perform fewer comparisons --- and the number of comparisons, when each comparison can be quite expensive, will dominate the running time more than the efficiency of the inner loops of the sorting algorithm.
	www.ai.mit.edu /projects/reinventing_computing/papers/theses/pshuang/meng/node20.html (1543 words)

	Joe Maller
		Seeing how Quicksort compares to other methods of sorting reinforces why it’s one of the most important algorithms in current use.
		This page has examples of Quicksort implementations in a bunch of languages.
		Here’s another QuickSort animation which shows exactly what is happening during the sort.
	joemaller.com /2004/09/15 (150 words)

	quicksort : Java Glossary
		QuickSort can be pathologically slow if the data are already ordered.
		Typical QuickSort implementations are unstable since they scramble keys to avoid pathological pre-orderings.
		The second line reads: “to sort a list whose first element is x and the rest of which is called xs, sort the elements of xs that are less than x, sort the elements of xs that are greater than or equal to x, and concatenate (++) the results with x sandwiched in the middle.”
	mindprod.com /jgloss/quicksort.html (289 words)

	A library of internal sorting routines
		Certain current implementations of sorting in C++ standard libraries like STL (the HP/SGI Standard Template Library which is now a part of the GNU free software collection) are comparable in speed to
		Like quicksort, and other methods which pick elements at random positions in order to possibly exchange their positions, it is not 'stable'; elements of the same magnitude, might switch their relative order.
		This is a relatively optimized version of heap-sort which minimizes neighboring elements exchanges in the innermost loop, using an insertion strategy.
	www.yendor.com /programming/sort (845 words)

	Functional Programming
		A stray thought: These implementations of QuickSort have in common that they use the first element as a pivot.
		This undesirable property is not a forced consequence of functional programming, but because functional languages tend to prefer list-like data structures, the simple implementations of QuickSort tend to have that drawback.
		For that reason, the second Haskell implementation is a better example here, since it shows off a more normal HigherOrderFunction than the first implementation, which is a little more of a trick.
	c2.com /cgi/wiki?FunctionalProgramming (2383 words)

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