{"title":"The Complexity of the Hausdorff Distance","authors":"Paul Jungeblut, Linda Kleist, Tillmann Miltzow","doi":"10.1007/s00454-023-00562-5","DOIUrl":null,"url":null,"abstract":"Abstract We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class $${ \\forall \\exists _{<}\\mathbb {R}} $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>∀</mml:mo> <mml:msub> <mml:mo>∃</mml:mo> <mml:mo><</mml:mo> </mml:msub> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> . This implies that the problem is -, -, $$\\exists \\mathbb {R} $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>∃</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> -, and $$\\forall \\mathbb {R} $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>∀</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> -hard.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"101 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete and Computational Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00454-023-00562-5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class $${ \forall \exists _{<}\mathbb {R}} $$ ∀∃<R . This implies that the problem is -, -, $$\exists \mathbb {R} $$ ∃R -, and $$\forall \mathbb {R} $$ ∀R -hard.
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