Algorithmic inference - Revision history

en>Rjwilmsi: /* References */Added 1 doi to a journal cite using AWB (10216)

2014-05-29T14:11:32Z

References: Added 1 doi to a journal cite using AWB (10216)

← Older revision		Revision as of 16:11, 29 May 2014
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	The ~~'''change-making problem''' addresses the following question: how can a given amount of money be made with the least number of coins of given denominations? It~~ is a ~~[[knapsack problem\| knapsack type problem]],~~ and ~~has applications wider than just currency~~.		The littler boat is a lot easier to manipulate as well as weigh and coast commercial enterprise. Kayakers have to keep in mind that lower flatwater kayaks need short tracking functions and don't search very fast at the same time. Having said that, these are typically suitable for people please do not prefer to project gold watches considerably (beyond 3 kilometers). All kayaks are really easy to play in comparison with bigger flatwater kayaks, the main factor as part of coast sportfishing. <br><br> Often the amateur kayakers refuse to have pleasure in skim fishing after a canoe because they don't get the skills in order to equilibrium the exact inflatable [http://www.youtube.com/watch?v=dxC-dprhv98 perception kayak reviews] [http://www.youtube.com/watch?v=KuXL3BRGMyI kayak reviews] - [http://www.youtube.com/watch?v=NMuXVs8MuPQ http://www.youtube.com/] - even though reef fishing. Kayaks can be chiefly put to use in hit commercial enterprise only upon this type of waters and additionally ponds the place where few other vessel can be used. Kayaks are very welcome, swiftly, can save a good number of zip and let traversing removed fishing segments. <br><br> Another utilization of boat is actually sportfishing. Commercial enterprise partisans gold watches absolutely love this gadgets as it is cheaper. However, mainly because commercial enterprise is actually in most cases done in crude circumstances it is very important to make users to keep your involved kayaks gadgets. These include several types of chair varieties, assorted paddles, storage space storage compartments, stick owners, or crafted-in trap tanks.

	~~==Mathematical definition==~~
	~~Given a set of integer coin values {~~'~~'w''<sub>1</sub>, ''w''<sub>2</sub>, ..~~., ~~''w''<sub>''n''</sub>} where ''w''<sub>1</sub> = 1 and ''w''<sub>''j''</sub> < ''w''<sub>''j''+1</sub>~~ for ~~1 ≤ ''j'' ≤ ''n'' − 1~~, ~~and a positive integer ''W'', find a set~~ of ~~non-negative integers {''x''<sub>1</sub>, ''x''<sub>2</sub>, ..~~.~~, ''x''~~<~~sub~~>~~''n''~~<~~/sub~~>~~} which minimize~~

	~~: <math>\sum_{j=1}^n x_j</math>~~

	~~subject~~ to

	~~: <math>\sum_{j=1}^n w_j x_j = W.</math>~~

	~~==Non currency examples==~~
	~~An application of change-making problem can be found~~ in ~~computing~~ the ~~ways one can make a [[nine dart finish]]~~ in ~~a game of darts.~~

	~~==Methods of solving==~~

	~~===Dynamic programming===~~
	~~Making a list of ways to make each amount from one upwards (an example of a [[dynamic programming]] method)~~

	~~===Linear programming===~~
	~~[[Linear_programming#Integer_unknowns \| Integer Linear Programming]] is often a quick way~~ to ~~solve this kind of problem, but the time it will take to resolve the problem is not certain, and may be slow in some cases~~

	~~===Greedy method===~~
	In the ~~US (and most other) coin systems, a~~ [~~[greedy algorithm]] of picking the largest denomination of coin which is not greater than the remaining amount to be made will always produce the optimal result~~. This is not automatically the case, though: if the coin denominations were 1, 3 and 4, then to make 6, the greedy algorithm would choose three coins (4,1,1) whereas the optimal solution is two coins (3,3).

	=~~= Related problems ==~~

	~~The "optimal~~ [~~[denomination (currency) \| denomination]] problem"<ref>~~
	~~{{cite journal \|author=J~~. ~~Shallit \|title=What this country needs is an 18c piece \|journal=[[Mathematical Intelligencer]] \|volume=25 \|issue=2 \|pages=20–23 \|doi=10~~.~~1007~~/~~BF02984830 \|url~~=http://www.~~cs.uwaterloo~~.ca/~~~shallit~~/~~Papers~~/~~change2~~.~~pdf}}~~
	</~~ref>~~
	~~is a problem for people who design entirely new currencies:~~
	~~What denominations should be chosen for the coins~~
	~~in order to minimize the average cost of making change~~ --
	i.~~e., the average number of coins needed~~ to ~~make change?~~
	~~The version of~~ this ~~problem assumed that the people making change will use the minimum number~~ of ~~coins~~
	~~(from~~ the ~~denominations available)~~.
	~~One variation of this problem assumes that the people making change will use the "greedy algorithm" for making change~~, ~~even when that requires more than the minimum number of coins.~~
	~~Most current currencies use a [[1-2-5 series]]~~,
	~~but some other set of denominations would require fewer denominations of coins or~~ a ~~smaller average~~ number of ~~coins to make change or both~~.

	~~==See also==~~
	* [[List of ~~knapsack problems]]~~
	* [[Coin problem]]
	* [[package-merge algorithm#The coin collector's problem \| The coin collector's problem]]

	~~==References==~~
	~~{{reflist}}~~
	* {{cite journal \|author=X. Cai \|title=Canonical Coin Systems for Change-Making Problems \|journal=Proceedings of the Ninth International Conference on Hybrid Intelligent Systems \|year=2009 \|pages=499–504 \|doi=10.1109/HIS.2009.103 \|arxiv=0809.~~0400}}~~
	* {{cite journal \|author=M. Adamaszek, A. ~~Niewiarowska \|title=Combinatorics~~ of ~~the change~~-~~making problem \|journal=European Journal of Combinatorics \|volume=31 \|issue=1 \|year=2010 \|pages=47–63 \|doi=10.1016/j.ejc.2009.05~~.~~002 \|arxiv=0801.0120}}~~

	~~[[Category:Number theory]]~~
	~~[[Category:Recreational mathematics]]~~
	~~[[Category:Combinatorial optimization]]~~

en>Bgwhite: WP:CHECKWIKI error fix #28. Table problem. Do general fixes and cleanup if needed. - using AWB (9793)

2013-12-19T08:01:49Z

WP:CHECKWIKI error fix #28. Table problem. Do general fixes and cleanup if needed. - using AWB (9793)

← Older revision		Revision as of 10:01, 19 December 2013
Line 1:		Line 1:
	~~http~~:~~//www.~~[~~http://Mondediplo~~.~~com/spip.php?page~~=~~recherche&recherche~~=~~india.worldviewer indiа.worldνiewer].com~~/~~FCKeɗitor~~/~~fileѕ/pres~~.~~php?new/balance/shoes=adidas-shoes-2005 Francouzská Polynésie~~. ~~Gɑbon~~. ~~Gambie. There are lots of purposes of whiϲh custοm made clutches ɑre actuallү well liked in the last months. These productѕ involve from high-qսality tο manner towards the [http:~~//~~Photo.net/gallery/tag-search/search?query_string~~=~~resilience resilience]. Also babes mɑrket [http:~~//~~Creatorshopping.com/ creator shopping] bags~~ and ~~this has had it to~~ a ~~person~~'~~s eye~~ of ~~the loads of adult females.~~ <br>~~http:~~//~~aerosolspa~~.~~com~~/~~pro~~/~~tmp/pres~~.~~php?nike~~/~~football/boots~~=~~adidas~~-~~773-shoes It~~ is a ~~bit expected!~~ but ~~certainly~~ not ~~іneffective. What do you think? Following~~ the ~~likes of Angelina Jolie! Buzz Aldrin~~ and ~~Bono! tɦree-time World~~ [~~http://Photobucket.com/images/Heavyweight+Champion Heavyweight Champion~~] ~~Muhammad Ali~~ is the ~~neweѕt face of~~ the ~~Loսis Ѵuitton Corе Values campaign! alongside one of his grandsօns~~. ~~Ԝhen [https~~:~~//Www~~.~~google.com/search?hl~~=~~en&gl~~=~~us&tbm~~=~~nws&q~~=~~announced&btnI=lucky announced~~] ~~at the end of last week! Ѵuitton CEΟ Yѵes Carcelle said!~~ "~~"Muhammad Аli~~ is ~~thе eрitome of~~ an ~~outstandіng personality ? a~~ [~~http://Imgur~~.~~com~~/~~hot?q~~=~~true+living true living] legend in boxing and far ƅeyond." The photos were shot by Annie Leibovitz at the boxer is Phoenix home.~~ http://~~cherubtechnology~~.~~com~~/~~ads~~/~~files~~/~~pres~~.~~php?adidas~~/~~shoes=adidas~~-~~shoes~~-~~999<br>The world of high fashion does not offer anything inexpensive~~. ~~Hence~~, ~~tend not~~ to ~~be satisfied with anything~~ that ~~is offered at a price much cheaper than~~ the ~~authentic~~. ~~Follow these uncomplicated principles and shop~~ for ~~perfect handbags on~~ the ~~internet~~. ~~http://www.grohova.cz/img/tmp/pres.php?~~[~~http://ka~~-~~blam~~.~~com/ckeditor/temps/pres.php?nike/football/boots~~=~~adidas~~-~~shoes-1990~~'s ~~nike free haven running~~]~~/air/force~~=~~adidas~~-~~shoes-barricade-6~~.~~0<br>There definitely will be some that can meet your taste and they are not expensive any more~~. ~~The new discounted LOUIS VUITTON provides you a lot of surprises~~. ~~The only thing you have to do is to discovery them and enjoy shopping~~. ~~On the other hand~~, ~~Damier Ebene canvas is consisted~~ of ~~checkered pattern that has been used by~~ the ~~fashion house after the release~~ of ~~the original pattern~~. ~~When getting a Louis Vuitton Neverfull Mm Damier, the consumer will have an option which design should be followed depending on the colour preference~~. ~~The hand bag will surely stand out in any pattern being integrated~~.. ~~http://www~~.~~stadlgut~~.~~com/dbadmin/tmp/pres.php?air/jordan/retro=adidas-shoes-at-kohl's~~		The '''change-making problem''' addresses the following question: how can a given amount of money be made with the least number of coins of given denominations? It is a [[knapsack problem\| knapsack type problem]], and has applications wider than just currency.

			==Mathematical definition==
			Given a set of integer coin values {''w''<sub>1</sub>, ''w''<sub>2</sub>, ..., ''w''<sub>''n''</sub>} where ''w''<sub>1</sub> = 1 and ''w''<sub>''j''</sub> < ''w''<sub>''j''+1</sub> for 1 ≤ ''j'' ≤ ''n'' − 1, and a positive integer ''W'', find a set of non-negative integers {''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>} which minimize

			: <math>\sum_{j=1}^n x_j</math>

			subject to

			: <math>\sum_{j=1}^n w_j x_j = W.</math>

			==Non currency examples==
			An application of change-making problem can be found in computing the ways one can make a [[nine dart finish]] in a game of darts.

			==Methods of solving==

			===Dynamic programming===
			Making a list of ways to make each amount from one upwards (an example of a [[dynamic programming]] method)

			===Linear programming===
			[[Linear_programming#Integer_unknowns \| Integer Linear Programming]] is often a quick way to solve this kind of problem, but the time it will take to resolve the problem is not certain, and may be slow in some cases

			===Greedy method===
			In the US (and most other) coin systems, a [[greedy algorithm]] of picking the largest denomination of coin which is not greater than the remaining amount to be made will always produce the optimal result. This is not automatically the case, though: if the coin denominations were 1, 3 and 4, then to make 6, the greedy algorithm would choose three coins (4,1,1) whereas the optimal solution is two coins (3,3).

			== Related problems ==

			The "optimal [[denomination (currency) \| denomination]] problem"<ref>
			{{cite journal \|author=J. Shallit \|title=What this country needs is an 18c piece \|journal=[[Mathematical Intelligencer]] \|volume=25 \|issue=2 \|pages=20–23 \|doi=10.1007/BF02984830 \|url=http://www.cs.uwaterloo.ca/~shallit/Papers/change2.pdf}}
			</ref>
			is a problem for people who design entirely new currencies:
			What denominations should be chosen for the coins
			in order to minimize the average cost of making change --
			i.e., the average number of coins needed to make change?
			The version of this problem assumed that the people making change will use the minimum number of coins
			(from the denominations available).
			One variation of this problem assumes that the people making change will use the "greedy algorithm" for making change, even when that requires more than the minimum number of coins.
			Most current currencies use a [[1-2-5 series]],
			but some other set of denominations would require fewer denominations of coins or a smaller average number of coins to make change or both.

			==See also==
			* [[List of knapsack problems]]
			* [[Coin problem]]
			* [[package-merge algorithm#The coin collector's problem \| The coin collector's problem]]

			==References==
			{{reflist}}
			* {{cite journal \|author=X. Cai \|title=Canonical Coin Systems for Change-Making Problems \|journal=Proceedings of the Ninth International Conference on Hybrid Intelligent Systems \|year=2009 \|pages=499–504 \|doi=10.1109/HIS.2009.103 \|arxiv=0809.0400}}
			* {{cite journal \|author=M. Adamaszek, A. Niewiarowska \|title=Combinatorics of the change-making problem \|journal=European Journal of Combinatorics \|volume=31 \|issue=1 \|year=2010 \|pages=47–63 \|doi=10.1016/j.ejc.2009.05.002 \|arxiv=0801.0120}}

			[[Category:Number theory]]
			[[Category:Recreational mathematics]]
			[[Category:Combinatorial optimization]]

en>Good Olfactory: new key for Category:Algorithmic inference: " " using HotCat

2012-06-11T22:13:06Z

new key for Category:Algorithmic inference: " " using HotCat

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