Constructing arrangements of lines and hyperplanes with applications
SIAM Journal on Computing
On k-hulls and related problems
SIAM Journal on Computing
On orthogonal linear approximation
Numerische Mathematik
Topologically sweeping an arrangement
Topologically sweeping an arrangement
Linear Programming in Linear Time When the Dimension Is Fixed
Journal of the ACM (JACM)
Lower bounds for algebraic computation trees
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Walking on an arrangement topologically
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Monotone paths in line arrangement
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Monotone paths in line arrangements
Computational Geometry: Theory and Applications
Subgradient and sampling algorithms for l1 regression
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
How to get close to the median shape
Proceedings of the twenty-second annual symposium on Computational geometry
How to get close to the median shape
Computational Geometry: Theory and Applications - Special issue on the 21st European workshop on computational geometry (EWCG 2005)
Weighted rectilinear approximation of points in the plane
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Overlap properties of geometric expanders: extended abstract
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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This paper presents algorithms for approximating a set of n points by a linear function, or a line, that minimizes the L1 norm of vertical and orthogonal distances. The algorithms find exact solutions based upon geometric properties of the problems as opposed to approximate solutions based upon existing numerical techniques. The algorithmic complexity of these problems appears not to have been investigated before our work in [9], although &Ogr;(n3) naive algorithms can be easily obtained based on some simple characteristics of optimal L1 solutions. In this paper, an &Ogr;(n) optimal time algorithm for the weighted vertical L1 problem is presented. The algorithm is based upon a modified multi-dimensional search technique which extends the applicability of the basic technique to a wider class of problems. An &Ogr;(n1.5 log2 n) algorithm is presented for the unweighted orthogonal problem, and an &Ogr;(n2) algorithm is presented for the weighted problem. An &OHgr;(n log n) lower bound for the orthogonal L1 problem is shown under a certain model of computation. Also, the complexity of solving the orthogonal L1 problem is related to the construction of the k-belt of an arrangement of lines.