{"title":"用矩形覆盖多边形的算法","authors":"D.S. Franzblau, D.J. Kleitman","doi":"10.1016/S0019-9958(84)80012-1","DOIUrl":null,"url":null,"abstract":"<div><p>Decomposing a polygon into simple shapes is a basic problem in computational geometry, with applications in pattern recognition and integrated circuit manufacture. Here we examine the special case of covering a rectilinear polygon (or polyomino) with the minimum number of rectangles, with overlapping allowed. The problem is <em>NP</em>-hard. However, we give here an <em>O</em>(<em>v</em><sup>2</sup>) algorithm for constructing a minimum rectangle cover, when the polygon is vertically convex. (Here <em>v</em> is the number of vertices.) The problem is first reduced to a 1-dimensional interval “basis” problem. In showing our algorithm produces an optimal cover we give a new proof of a minimum basis-maximum independent set duality theorem first proved by E. Györi (<em>J. Combin Theory Ser. B</em> <strong>37</strong>, No. 1, 1–9).</p></div>","PeriodicalId":38164,"journal":{"name":"信息与控制","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"1984-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0019-9958(84)80012-1","citationCount":"51","resultStr":"{\"title\":\"An algorithm for covering polygons with rectangles\",\"authors\":\"D.S. Franzblau, D.J. Kleitman\",\"doi\":\"10.1016/S0019-9958(84)80012-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Decomposing a polygon into simple shapes is a basic problem in computational geometry, with applications in pattern recognition and integrated circuit manufacture. Here we examine the special case of covering a rectilinear polygon (or polyomino) with the minimum number of rectangles, with overlapping allowed. The problem is <em>NP</em>-hard. However, we give here an <em>O</em>(<em>v</em><sup>2</sup>) algorithm for constructing a minimum rectangle cover, when the polygon is vertically convex. (Here <em>v</em> is the number of vertices.) The problem is first reduced to a 1-dimensional interval “basis” problem. In showing our algorithm produces an optimal cover we give a new proof of a minimum basis-maximum independent set duality theorem first proved by E. Györi (<em>J. Combin Theory Ser. B</em> <strong>37</strong>, No. 1, 1–9).</p></div>\",\"PeriodicalId\":38164,\"journal\":{\"name\":\"信息与控制\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1984-12-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/S0019-9958(84)80012-1\",\"citationCount\":\"51\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"信息与控制\",\"FirstCategoryId\":\"1093\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0019995884800121\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"信息与控制","FirstCategoryId":"1093","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019995884800121","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
An algorithm for covering polygons with rectangles
Decomposing a polygon into simple shapes is a basic problem in computational geometry, with applications in pattern recognition and integrated circuit manufacture. Here we examine the special case of covering a rectilinear polygon (or polyomino) with the minimum number of rectangles, with overlapping allowed. The problem is NP-hard. However, we give here an O(v2) algorithm for constructing a minimum rectangle cover, when the polygon is vertically convex. (Here v is the number of vertices.) The problem is first reduced to a 1-dimensional interval “basis” problem. In showing our algorithm produces an optimal cover we give a new proof of a minimum basis-maximum independent set duality theorem first proved by E. Györi (J. Combin Theory Ser. B37, No. 1, 1–9).