### Lower bounds for linear degeneracy testingLower bounds for linear degeneracy testing

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 Author Ailon, Nir ♦ Chazelle, Bernard Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2005 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Computational geometry ♦ Linear decision trees ♦ Lower bounds Abstract In the late nineties, Erickson proved a remarkable lower bound on the decision tree complexity of one of the central problems of computational geometry: given $\textit{n}$ numbers, do any $\textit{r}$ of them add up to 0? His lower bound of $Ω(n^{⌈r/2⌉}),$ for any fixed $\textit{r},$ is optimal if the polynomials at the nodes are linear and at most $\textit{r}-variate.$ We generalize his bound to $\textit{s}-variate$ polynomials for $\textit{s}$ > $\textit{r}.$ Erickson's bound decays quickly as $\textit{r}$ grows and never reaches above pseudo-polynomial: we provide an exponential improvement. Our arguments are based on three ideas: (i) a geometrization of Erickson's proof technique; (ii) the use of error-correcting codes; and (iii) a tensor product construction for permutation matrices. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2005-03-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 52 Issue Number 2 Page Count 15 Starting Page 157 Ending Page 171

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Source: ACM Digital Library