TY - JOUR
T1 - Efficient algorithms for maximum regression depth
AU - Van Kreveld, Marc
AU - Mitchell, Joseph S.B.
AU - Rousseeuw, Peter
AU - Sharir, Micha
AU - Snoeyink, Jack
AU - Speckmann, Bettina
N1 - Funding Information:
J.S.B. Mitchell’s research largely conducted while the author was a Fulbright Research Scholar at Tel Aviv University. The author is partially supported by NSF (CCR-9504192, CCR-9732220), Boeing, Bridgeport Machines, Sandia, Seagull Technology, and Sun Microsystems.
Funding Information:
M. van Kreveld partially funded by the Netherlands Organization for Scientific Research (NWO) under FOCUS/BRICKS grant number 642.065.503.
Funding Information:
M. Sharir supported by NSF Grants CCR-97-32101 and CCR-94-24398, by grants from the U.S.–Israeli Binational Science Foundation, the G.I.F., the German–Israeli Foundation for Scientific Research and Development, and the ESPRIT IV LTR project No. 21957 (CGAL), and by the Hermann Minkowski—MINERVA Center for Geometry at Tel Aviv University.
Funding Information:
J. Snoeyink supported in part by grants from NSERC, the Killam Foundation, and CIES while at the University of British Columbia.
PY - 2008/6
Y1 - 2008/6
N2 - We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(n d ) time and O(n d-1) space algorithm computes undirected depth of all points in d dimensions. Properties of undirected depth lead to an O(nlog 2 n) time and O(n) space algorithm for computing a point of maximum depth in two dimensions, which has been improved to an O(nlog n) time algorithm by Langerman and Steiger (Discrete Comput. Geom. 30(2):299-309, [2003]). Furthermore, we describe the structure of depth in the plane and higher dimensions, leading to various other geometric and algorithmic results.
AB - We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(n d ) time and O(n d-1) space algorithm computes undirected depth of all points in d dimensions. Properties of undirected depth lead to an O(nlog 2 n) time and O(n) space algorithm for computing a point of maximum depth in two dimensions, which has been improved to an O(nlog n) time algorithm by Langerman and Steiger (Discrete Comput. Geom. 30(2):299-309, [2003]). Furthermore, we describe the structure of depth in the plane and higher dimensions, leading to various other geometric and algorithmic results.
UR - http://www.scopus.com/inward/record.url?scp=45849127285&partnerID=8YFLogxK
U2 - 10.1007/s00454-007-9046-6
DO - 10.1007/s00454-007-9046-6
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AN - SCOPUS:45849127285
SN - 0179-5376
VL - 39
SP - 656
EP - 677
JO - Discrete and Computational Geometry
JF - Discrete and Computational Geometry
IS - 4
ER -