{"title":"豪斯多夫距离的复杂性","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":"{\"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}","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}
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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