Partially ordered ring: Difference between revisions

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In [[computational complexity theory]]  there is an open problem whether some information about the '''sum of radicals''' may be computed in [[polynomial time]] depending on the input size, i.e., in the number of bits necessary to represent this sum. It is of importance for many problems in [[computational geometry]], since the computation of the [[Euclidean distance]] between two points in general case involves the computation of a [[square root]], and therefore the [[perimeter]] of a [[polygon]] or the length of a polygonal chain takes the form of a sum of radicals.<ref name=mulzer>Wolfgang Mulzer, Günter Rote, "Minimum-Weight Triangulation Is NP-Hard", In: Proceedings of the 22nd ''[[Annual Symposium on Computational Geometry]]'', Sedona, June 5&ndash;7, 2006, ''[[Journal of the ACM]]'', Vol. 55, No. 2, 2008.</ref>  
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The sum of radicals is defined as a finite [[linear combination]] of radicals:
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:<math>
\Sigma_{i=1}^n k_i \sqrt[r_i]{x_i},
</math>
where <math>n, r_i</math> are [[natural number]]s and <math>k_i, x_i</math> are [[real number]]s.
 
Most theoretical research in [[computational geometry]] of combinatorial character assumes the [[computational model]] of infinite precision real RAM, i.e., an [[abstract computer]] in which real numbers and operations with them are performed with infinite precision and the input size of a real number and the cost of an elementary operation are constants.<ref>{{cite book
|author = [[Franco P. Preparata]] and [[Michael Ian Shamos]] | title = Computational Geometry - An Introduction | publisher = [[Springer-Verlag]]| year = 1985 | id = 1st edition:  2nd printing, corrected and expanded, 1988: ISBN 3-540-96131-3; Russian translation, 1989: ISBN 5-03-001041-6
|isbn = 0-387-96131-3}}</ref> However there is  research in computational complexity, especially in [[computer algebra]], where the input size of a number is the number of bits necessary for its representation.<ref>''Computer Algebra Handbook'', 2003, ISBN 3-540-65466-6</ref>  
 
In particular, of interest in computational geometry is the problem of determining the '''[[sign function|sign]] of the sum of radicals'''. For instance, the length of a polygonal path in which all vertices have integer coordinates may be expressed using the [[Pythagorean theorem]] as a sum of integer square roots, so in order to determine whether one path is longer or shorter than another in a [[Euclidean shortest path]] problem, it is necessary to determine the sign of an expression in which the first path's length is subtracted from the second; this expression is a sum of radicals.
 
In a similar way, the sum of radicals problem is inherent in the problem of [[minimum-weight triangulation]] in the [[Euclidean metric]].  
 
In 1991, Blömer proposed a polynomial time [[Monte Carlo algorithm]] for determining '''whether a sum of radicals is zero''', or more generally whether it represents a rational number.<ref name=blomer>{{Cite journal|first=Johannes|last=Blömer|title=Computing sums of radicals in polynomial time|doi=10.1109/SFCS.1991.185434|year=1991|pages=670|postscript=.}}.</ref>
While Blömer's result does not resolve the computational complexity of finding the sign of the sum of radicals, it does imply that if the latter problem is in [[class NP]], then it is also in [[co-NP]].<ref name=blomer/>
 
==See also==
*[[Nested radicals]]
*[[Abel–Ruffini theorem]]
 
==References==
{{reflist}}
 
[[Category:Computational problems]]
[[Category:Computer algebra]]
[[Category:Computational geometry]]

Revision as of 16:18, 10 February 2014

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