{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article10258","name":"Finding Axis-Parallel Rectangles of Fixed Perimeter or Area Containing the Largest Number of Points","abstract":"Let P be a set of n points in the plane in general position, and consider the problem of finding an axis-parallel rectangle with a given perimeter, or area, or diagonal, that encloses the maximum number of points of P. We present an exact algorithm that finds such a rectangle in O(n^{5\/2} log n) time, and, for the case of a fixed perimeter or diagonal, we also obtain (i) an improved exact algorithm that runs in O(nk^{3\/2} log k) time, and (ii) an approximation algorithm that finds, in O(n+(n\/(k epsilon^5))*log^{5\/2}(n\/k)log((1\/epsilon) log(n\/k))) time, a rectangle of the given perimeter or diagonal that contains at least (1-epsilon)k points of P, where k is the optimum value.\r\n\r\nWe then show how to turn this algorithm into one that finds, for a given k, an axis-parallel rectangle of smallest perimeter (or area, or diagonal) that contains k points of P. We obtain the first subcubic algorithms for these problems, significantly improving the current state of the art.","keywords":["Computational geometry","geometric optimization","rectangles","perimeter","area"],"author":[{"@type":"Person","name":"Kaplan, Haim","givenName":"Haim","familyName":"Kaplan"},{"@type":"Person","name":"Roy, Sasanka","givenName":"Sasanka","familyName":"Roy"},{"@type":"Person","name":"Sharir, Micha","givenName":"Micha","familyName":"Sharir"}],"position":52,"pageStart":"52:1","pageEnd":"52:13","dateCreated":"2017-09-01","datePublished":"2017-09-01","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kaplan, Haim","givenName":"Haim","familyName":"Kaplan"},{"@type":"Person","name":"Roy, Sasanka","givenName":"Sasanka","familyName":"Roy"},{"@type":"Person","name":"Sharir, Micha","givenName":"Micha","familyName":"Sharir"}],"copyrightYear":"2017","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ESA.2017.52","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6290","volumeNumber":87,"name":"25th Annual European Symposium on Algorithms (ESA 2017)","dateCreated":"2017-09-01","datePublished":"2017-09-01","editor":[{"@type":"Person","name":"Pruhs, Kirk","givenName":"Kirk","familyName":"Pruhs"},{"@type":"Person","name":"Sohler, Christian","givenName":"Christian","familyName":"Sohler"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article10258","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6290"}}}