{"title":"Improved Algebraic Degeneracy Testing","authors":"Jean Cardinal, Micha Sharir","doi":"10.1007/s00454-024-00673-7","DOIUrl":null,"url":null,"abstract":"<p>In the classical linear degeneracy testing problem, we are given <i>n</i> real numbers and a <i>k</i>-variate linear polynomial <i>F</i>, for some constant <i>k</i>, and have to determine whether there exist <i>k</i> numbers <span>\\(a_1,\\ldots ,a_k\\)</span> from the set such that <span>\\(F(a_1,\\ldots ,a_k) = 0\\)</span>. We consider a generalization of this problem in which <i>F</i> is an arbitrary constant-degree polynomial, we are given <i>k</i> sets of <i>n</i> real numbers, and have to determine whether there exists a <i>k</i>-tuple of numbers, one in each set, on which <i>F</i> vanishes. We give the first improvement over the naïve <span>\\(O^*(n^{k-1})\\)</span> algorithm for this problem (where the <span>\\(O^*(\\cdot )\\)</span> notation omits subpolynomial factors). We show that the problem can be solved in time <span>\\(O^*\\left( n^{k - 2 + \\frac{4}{k+2}}\\right) \\)</span> for even <i>k</i> and in time <span>\\(O^*\\left( n^{k - 2 + \\frac{4k-8}{k^2-5}}\\right) \\)</span> for odd <i>k</i> in the real RAM model of computation. We also prove that for <span>\\(k=4\\)</span>, the problem can be solved in time <span>\\(O^*(n^{2.625})\\)</span> in the algebraic decision tree model, and for <span>\\(k=5\\)</span> it can be solved in time <span>\\(O^*(n^{3.56})\\)</span> in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for <i>k</i>-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft’s point-line incidence detection problem in any dimension.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"93 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete & Computational Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00454-024-00673-7","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
In the classical linear degeneracy testing problem, we are given n real numbers and a k-variate linear polynomial F, for some constant k, and have to determine whether there exist k numbers \(a_1,\ldots ,a_k\) from the set such that \(F(a_1,\ldots ,a_k) = 0\). We consider a generalization of this problem in which F is an arbitrary constant-degree polynomial, we are given k sets of n real numbers, and have to determine whether there exists a k-tuple of numbers, one in each set, on which F vanishes. We give the first improvement over the naïve \(O^*(n^{k-1})\) algorithm for this problem (where the \(O^*(\cdot )\) notation omits subpolynomial factors). We show that the problem can be solved in time \(O^*\left( n^{k - 2 + \frac{4}{k+2}}\right) \) for even k and in time \(O^*\left( n^{k - 2 + \frac{4k-8}{k^2-5}}\right) \) for odd k in the real RAM model of computation. We also prove that for \(k=4\), the problem can be solved in time \(O^*(n^{2.625})\) in the algebraic decision tree model, and for \(k=5\) it can be solved in time \(O^*(n^{3.56})\) in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for k-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft’s point-line incidence detection problem in any dimension.
期刊介绍:
Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.