Daniel J. Kleitman

Algorithms for Loop Matchings

A simplified (two-base) version of the problem of planar folding of long chains (e.g., RNA and DNA biomolecules) is formulated as a matching problem. The chain is prescribed as a loop or circular sequence of letters A and B, n units long. A matching here means a set of A-B base pairings or matches obeying a planarity condition: no two matches may cross each other if drawn on the interior of the loop. Also, no two adjacent letters may be matched. We present a dynamic programming algorithm requiring

The Structure of Sperner k-families

O( {n^3 } )

Traditional Galleries Require Fewer Watchmen

steps and

Coping with errors in binary search procedures

O( {n^2 } )

Spanning trees with many leaves

storage which computes the size of the maximum for the given A-B base sequence and which also allows reconstructing a particular folded form of the original string which realizes the maximum matching size. The algorithm can be adapted to deal with sequences with larger alphabets and with weighted matchings.An algorithm is also presented for a modified problem closer to the biochemical problem of interest: We demand that every match must be adjacent to another match, forcing ...

The crossing number of K5,n

Abstract If P is a partially ordered set, a k-family of P is a subset which contains no chains of length k + 1. This paper examines the structure of the set of k-families of P. An extension of Dilworths theorem is obtained by relating the maximum size of a k-family to certain partitions of P into chains. A natural lattice ordering on k-families is defined and analyzed, and a number of strong intersection properties are obtained. Finally, the k-families of P are used to define a class of submodular set functions on P, which can be used to generalize a number of results in transversal theory.

On a dual version of the one-dimensional bin packing problem

Chvatal’s watchman theorem shows if the walls of an art gallery form an n-sided polygon then at most

Cost allocation for a spanning tree

[ n /3 ]

Piercing convex sets and the Hadwiger-Debrunner (p, q)-problem☆

watchmen are needed to guard it, and that this number is best possible. In this paper it is shown that if every pair of adjacent sides of the polygon form a right angle then at most

Strong versions of Sperner's theorem☆

[ n / 4 ]