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

Reviewer: Robert Bruce King

The dominance order on points in the plane is given by the relations p?q if p x ?q x and p y ?q y where p x and p y denote the x and y coordinates of p . A subset C of a set of points P is a chain if any two members p,q of C are comparable, that is, either p?q or q?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 . A subset A of a set P with no two different points comparable is an antichain. This paper describes an algorithm that computes maximum k -chains in time O kn log n and linear space. By the symmetry between chains and antichains in the dominance order, this algorithm for maximum k -chains can also be used to compute maximum k -antichains for planar points in time O kn log n . However, for large k a better algorithm is available for computing maximum k -antichains (and, by symmetry, k -chains) in time O n 2/k log n and linear space. Consequently, a maximum k -chain can be computed in time O n 3/2 log n for arbitrary k . In addition, this method can be extended to weighted point sets, providing an algorithm to compute maximum weighted k -chains with time and space requirements of O 2 kn log 2 kn <__?__Pub Caret> and O 2 kn , respectively.