prob006: Golomb rulers

proposed by Peter van Beek
vanbeek@cs.ualberta.ca

Specification

These problems are said to have many practical applications including sensor placements for x-ray crystallography and radio astronomy. A Golomb ruler may be defined as a set of m integers 0 = a_1 < a_2 < ... < a_m such that the m(m-1)/2 differences a_j - a_i, 1 <= i < j <= m are distinct. Such a ruler is said to contain m marks and is of length a_m. The objective is to find optimal (minimum length) or near optimal rulers.

Note that a symmetry can be removed by adding the constraint that a_2 - a_1 < a_m - a_{m-1}, the first difference is less than the last.

There exist several interesting generalizations of the problem which have received attention like modular Golomb rulers (differences are all distinct mod a given base), disjoint Golomb rulers, Golomb rectangles (the 2-dimensional generalization of Golomb rulers), and difference triangle sets (sets of rulers with no common difference).