On some distance problems in fixed orientations
SIAM Journal on Computing
On Pareto optima, the Fermat-Weber problem, and polyhedral gauges
Mathematical Programming: Series A and B
On the two-dimensional davenport-schinzel problem
Journal of Symbolic Computation
Journal of Algebraic Combinatorics: An International Journal
A rounding algorithm for approximating minimum manhattan networks
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
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Given a vector objective function f = (f1,...,fn) defined on a set X, a point y∈X is dominated by a point x∈ X if fi(x) i(y) forall i∈(1,...,n) and there exists an index j∈(1,...,n) such that fj(x) j(y). The non-dominated pointsof X are called the Pareto optima of f. H. Kuhn(1973) applied the concept of Pareto optimality to distancefunctions and characterized the convex hull conv (T) of any set T=(t1,...,tn) of Rm as the set of all Paretooptima of the vector function d2(x)=(d2(x,t1),...,d2(x,tn)), where d2(x,y)is the Euclidean distance between x,y∈ Rm. Motivatedby this result, given a set T=(t1,...,tn) of points of ametric space (X,d), we call the set Pd(T) of all Paretooptima of the function d(x)=(d(x,t1),...,d(x,tn)) the Pareto envelope of T. In this paper, we investigate the Pareto envelopes in Rm endowed with l1- or l∞-distances. We characterize PI(T) in all dimensions and PM(T) in R3. Usingthese results, we design efficient algorithms for constructing theseenvelopes in R3, in particular, an optimal O(n logn)-time algorithm for PM(T) and an O(n log2n)-time algorithmfor PI(T).