Maximum k-Chains in Planar Point Sets: Combinatorial Structure and Algorithms

13 years 4 months ago

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A chain of a set P of n points in the plane is a chain of the dominance order on P. A k-chain is a subset C of P that can be covered by k chains. A k-chain C is a maximum k-chain if no other k-chain contains more elements than C. This paper deals with the problem of nding a maximum k-chain of P in the cardinality and in the weighted case. Using the skeleton S(P) of a point set P introduced by Viennot we describe a fairly simple algorithm that computes maximum k-chains in time O(kn logn) and linear space. The basic idea is that the canonical chain partition of a maximum(k?1)-chain in the skeleton S(P) provides k regions in the plane, such that a maximum k-chain for P can be obtained as the union of a maximal chain from each of these regions. By the symmetry between chains and antichains in the dominance order we may use the algorithmfor maximumk-chains to compute maximumk-antichains for planar points in time O(kn logn). However, for large k one can do better. We describe an algorithm c...

Stefan Felsner, Lorenz Wernisch

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