{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article11905","name":"Extending the Centerpoint Theorem to Multiple Points","abstract":"The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set P of n points in R^d, there is a point c, not necessarily from P, such that each halfspace containing c contains at least n\/(d+1) points of P. Such a point c is called a centerpoint, and it can be viewed as a generalization of a median to higher dimensions. In other words, a centerpoint can be interpreted as a good representative for the point set P. But what if we allow more than one representative? For example in one-dimensional data sets, often certain quantiles are chosen as representatives instead of the median.\nWe present a possible extension of the concept of quantiles to higher dimensions. The idea is to find a set Q of (few) points such that every halfspace that contains one point of Q contains a large fraction of the points of P and every halfspace that contains more of Q contains an even larger fraction of P. This setting is comparable to the well-studied concepts of weak epsilon-nets and weak epsilon-approximations, where it is stronger than the former but weaker than the latter. We show that for any point set of size n in R^d and for any positive alpha_1,...,alpha_k where alpha_1 <= alpha_2 <= ... <= alpha_k and for every i,j with i+j <= k+1 we have that (d-1)alpha_k+alpha_i+alpha_j <= 1, we can find Q of size k such that each halfspace containing j points of Q contains least alpha_j n points of P. For two-dimensional point sets we further show that for every alpha and beta with alpha <= beta and alpha+beta <= 2\/3 we can find Q with |Q|=3 such that each halfplane containing one point of Q contains at least alpha n of the points of P and each halfplane containing all of Q contains at least beta n points of P. All these results generalize to the setting where P is any mass distribution. For the case where P is a point set in R^2 and |Q|=2, we provide algorithms to find such points in time O(n log^3 n).","keywords":["centerpoint","point sets","Tukey depth"],"author":[{"@type":"Person","name":"Pilz, Alexander","givenName":"Alexander","familyName":"Pilz","sameAs":"https:\/\/orcid.org\/0000-0002-6059-1821","affiliation":"Institute of Software Technology, Graz University of Technology, Austria","funding":"Supported by a Schr\u00f6dinger fellowship of the Austrian Science Fund (FWF): J-3847-N35."},{"@type":"Person","name":"Schnider, Patrick","givenName":"Patrick","familyName":"Schnider","affiliation":"Department of Computer Science, ETH Zurich, Switzerland"}],"position":53,"pageStart":"53:1","pageEnd":"53:13","dateCreated":"2018-12-06","datePublished":"2018-12-06","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Pilz, Alexander","givenName":"Alexander","familyName":"Pilz","sameAs":"https:\/\/orcid.org\/0000-0002-6059-1821","affiliation":"Institute of Software Technology, Graz University of Technology, Austria","funding":"Supported by a Schr\u00f6dinger fellowship of the Austrian Science Fund (FWF): J-3847-N35."},{"@type":"Person","name":"Schnider, Patrick","givenName":"Patrick","familyName":"Schnider","affiliation":"Department of Computer Science, ETH Zurich, Switzerland"}],"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ISAAC.2018.53","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.1016\/S0167-9473(02)00032-4","http:\/\/dx.doi.org\/10.1016\/j.comgeo.2008.02.005","http:\/\/www.cs.queensu.ca\/cccg\/papers\/cccg13.pdf","http:\/\/dx.doi.org\/10.1007\/978-3-319-44479-6_12","http:\/\/dx.doi.org\/10.1007\/BF02187876","http:\/\/dx.doi.org\/10.1007\/BF02574382","http:\/\/dx.doi.org\/10.1007\/3-540-36494-3_6","http:\/\/dx.doi.org\/10.1007\/BF02187804","http:\/\/dx.doi.org\/10.1007\/BF02574066","http:\/\/dx.doi.org\/10.1007\/BF01303517","https:\/\/hal.archives-ouvertes.fr\/hal-01468664","http:\/\/dx.doi.org\/10.1016\/j.comgeo.2007.10.004","http:\/\/dx.doi.org\/10.1007\/s00373-007-0716-1"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6326","volumeNumber":123,"name":"29th International Symposium on Algorithms and Computation (ISAAC 2018)","dateCreated":"2018-12-06","datePublished":"2018-12-06","editor":[{"@type":"Person","name":"Hsu, Wen-Lian","givenName":"Wen-Lian","familyName":"Hsu"},{"@type":"Person","name":"Lee, Der-Tsai","givenName":"Der-Tsai","familyName":"Lee"},{"@type":"Person","name":"Liao, Chung-Shou","givenName":"Chung-Shou","familyName":"Liao"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article11905","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":"#volume6326"}}}