Tabulation hashing: Difference between revisions

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en>David Eppstein
Undid revision 593724376 by Kbulgakov (talk) no, it's a two-dimensional array
en>Thomasda
The table is two dimensional. This is very important.
 
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'''Numerical 3-dimensional matching''' is an [[NP-complete]] decision problem. It is given by three [[multisets]] of [[integer]]s <math>X</math>, <math>Y</math> and <math>Z</math>, each containing <math>k</math> elements, and a bound <math>b</math>. The goal is to select a subset <math>M</math> of <math>X\times Y\times Z</math> such that every integer in <math>X</math>, <math>Y</math> and <math>Z</math> occurs exactly once and that for every triple <math>(x,y,z)</math> in the subset <math>x+y+z=b</math> holds.
This problem is labeled as [SP16] in.<ref name=garey>Garey, Michael R. and David S. Johnson (1979), Computers and Intractability; A Guide to the Theory of NP-Completeness. ISBN 0-7167-1045-5</ref>


== Example ==
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Take <math>X=\{3,4,4\}</math>, <math>Y=\{1,4,6\}</math> and <math>Z=\{1,2,5\}</math>, and <math>b=10</math>. This instance has a solution, namely <math>\{(3,6,1), (4,4,2), (4,1,5)\}</math>. Note that each triple sums to <math>b=10</math>. The set <math>\{(3,6,1), (3,4,2), (4,1,5)\}</math> is not a solution for several reasons: not every number is used (a <math>4\in X</math> is missing), a number is used too often (the <math>3\in X</math>) and not every triple sums to <math>b</math> (since <math>3+4+2=9\neq b=10</math>). However, there is at least one solution to this problem, which is the property we are interested in with decision problems.
If we would take <math>b=11</math> for the same <math>X</math>, <math>Y</math> and <math>Z</math>, this problem would have no solution (all numbers sum to <math>30</math>, which is not equal to <math>k\cdot b=33</math> in this case).
 
== Related problems ==
Every instance of the Numerical 3-dimensional matching problem is an instance of both the [[3-partition problem]], and the 3-dimensional matching problem.
 
== Proof of NP-completeness ==
NP-completeness of the 3-partition problem is stated by Garey and Johnson in "Computers and Intractability; A Guide to the Theory of NP-Completeness".<ref name=garey/> It is done by a reduction from 3-dimensional matching via 4-partition.
To prove NP-completeness of the numerical 3-dimensional matching, the proof is similar, but a reduction from 3-dimensional matching via the numerical 4-dimensional matching problem should be used.
 
== References ==
{{reflist}}
 
[[Category:Strongly NP-complete problems]]

Latest revision as of 22:06, 4 June 2014


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