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| | The author is called Irwin Wunder but it's not the most masucline name out there. The factor she adores most is body building and now she is trying to earn cash with it. I am a meter reader but I strategy on changing it. Years in the past we moved to Puerto Rico and my family members loves it.<br><br>my weblog: [http://premium.asslikethat.com/blog/9567 std testing at home] |
| {{Refimprove|date=October 2007}}
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| {{expert-subject|Mathematics|date=December 2008}}
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| In [[mathematics]] a '''combinatorial explosion''' describes the effect of [[Function (mathematics)|functions]] that grow very rapidly as a result of [[combinatorial]] considerations.<ref name="Krippendorff">{{cite web|last=Krippendorff|first=Klaus|title=Combinatorial Explosion|url=http://pespmc1.vub.ac.be/ASC/Combin_explo.html|work=Web Dictionary of Cybernetics and Systems|publisher=PRINCIPIA CYBERNETICA WEB|accessdate=29 November 2010}}</ref>
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| Examples of such functions include the [[factorial]] function and related functions. Pathological examples of combinatorial explosion include functions such as the [[Ackermann function]].
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| == Examples ==
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| === Computing ===
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| Combinatorial explosion can occur in computing environments in a way analogous to communications and [[multi-dimensional space]]. Imagine a simple system with only one variable, a [[boolean]] called A. The system has two possible states, ''A'' = true or ''A'' = false. Adding another boolean variable ''B'' will give the system four possible states, ''A'' = true and ''B'' = true, ''A'' = true and ''B'' = false, ''A'' = false and ''B'' = true, ''A'' = false and ''B'' = false. A system with ''n'' booleans has 2<sup>''n''</sup> possible states, while a system of ''n'' variables each with Z allowed values (rather than just the 2 (true and false) of booleans) will have ''Z''<sup>''n''</sup> possible states.
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| The possible states can be thought of as the leaf nodes of a tree of height ''n'', where each node has ''Z'' children. This rapid increase of leaf nodes can be useful in areas like [[searching]], since many results can be accessed without having to descend very far. It can also be a hindrance when manipulating such structures.
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| Consider a [[class hierarchy]] in an [[object-oriented language]]. The hierarchy can be thought of as a tree, with different types of object inheriting from their parents. If different classes need to be combined, such as in a comparison (like ''A'' < ''B'') then the number of possible combinations which may occur explodes. If each type of comparison needs to be programmed then this soon becomes intractable for even small numbers of classes. [[Multiple inheritance]] can solve this, by allowing subclasses to have multiple parents, and thus a few parent classes can be considered rather than every child, without disrupting any existing hierarchy.
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| For example, imagine a hierarchy where different vegetables inherit from their ancestor species. Attempting to compare the tastiness of each vegetable with the others becomes intractable since the hierarchy only contains information about genetics and makes no mention of tastiness. However, instead of having to write comparisons for carrot/carrot, carrot/potato, carrot/sprout, potato/potato, potato/sprout, sprout/sprout, they can all inherit from a separate class of tasty whilst keeping their current ancestor-based hierarchy, then all of the above can be implemented with only a tasty/tasty comparison.
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| ===Arithmetics===
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| Suppose we take the [[factorial]] for ''n'':
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| <math>n! = (n)(n-1)...(2)(1)</math> | |
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| Then 1! = 1, 2! = 2, 3! = 6, and 4! = 24. However, we quickly get to extremely large numbers, even for relatively small ''n''. For example, 100! = 9.33262154 × 10<sup>157</sup>, a number so large that it cannot be displayed on most calculators, and vastly larger than the estimated number of fundamental particles in the Universe.<ref>http://www.physicsoftheuniverse.com/numbers.html</ref>
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| {{other uses|Combinatorial explosion}}
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| {{unreferenced|date=August 2012}}
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| {| align="right"
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| |[[Image:4x2.svg|thumb|125px|Using separate lines of communication, four organizations require six channels]]
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| |[[Image:4xn.svg|thumb|125px|Using an intermediary, only one channel per organization is required]]
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| |}
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| ===Communication===
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| In administration and [[computing]], a ''combinatorial explosion'' is the rapidly accelerating increase in communication lines as organizations are added in a process. (Casually described as "exponential" it is actually strictly only [[polynomial]])
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| If two organizations need to communicate about a particular topic, it may be easiest to communicate directly in an ad hoc manner—only one [[communication channel|channel of communication]] is required. However, if a third organization is added, three separate channels are required. Adding a fourth organization requires six channels; five, ten; six, fifteen; etc.<!-- This is obviously polynomial rather than exponential growth; that's why I deleted the reference to exponential growth. [[User:Michael Hardy]] -->
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| In general, going on like that, it will take
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| <math>
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| l=\frac{n(n-1)}{2} = {n \choose 2}
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| </math>
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| communication lines for ''n'' organizations, which is just the number of 2-[[combination]]s of ''n'' elements, see also [[binomial coefficient]] .
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| The alternative approach is to realize when this communication will not be a one-off requirement, and produce a generic or intermediate way of passing information. The drawback is that this requires more work for the first pair, since each must convert its internal approach to the common one, rather than the superficially easier approach of just understanding the other.
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| ==See also==
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| * [[Birthday paradox]]
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| * [[Metcalfe's law]]
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| * [[Curse of dimensionality]]
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| * [[Intractability (complexity)]]
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| * [[Second half of the chessboard]]
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| ==References==
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| <references/>
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| [[Category:Combinatorics]]
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| {{combin-stub}}
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The author is called Irwin Wunder but it's not the most masucline name out there. The factor she adores most is body building and now she is trying to earn cash with it. I am a meter reader but I strategy on changing it. Years in the past we moved to Puerto Rico and my family members loves it.
my weblog: std testing at home