In this paper we study the complexity of the problems: given a loop, described by linear constraints over a finite set of variables, is there a linear or lexicographical-linear ranking function for this loop? While existence of such functions implies termination, these problems are not equivalent to termination. When the variables range over the rationals (or reals), it is known that both probl...

Fredricksen, Kessler and Maiorana discovered a simple but elegant construction of a universal cycle for binary strings of length n: Concatenate the aperiodic prefixes of length n binary necklaces in lexicographic order. We generalize their construction to binary strings of length n whose weights are in the range c, c + 1, . . . , n by simply omitting the necklaces with weight less than c. We al...

We can break symmetry by eliminating solutions within each symmetry class. For instance, the Lex-Leader method eliminates all but the smallest solution in the lexicographical ordering. Unfortunately, the Lex-Leader method is intractable in general. We prove that, under modest assumptions, we cannot reduce the worst case complexity of breaking symmetry by using other orderings on solutions. We a...

In this paper parallel algorithms are presented for enumeration and unranking of t–ary trees with n internal nodes. Generation algorithms are designed in the associative computing model ASC that belongs to a broad category of SIMD models. Tree sequences are generated in lexicographical order, with O(1) time per object, in a new representation, as combinations with repetitions with restricted gr...

The extension of mathematical morphology to colour, and more generally to multivariate image data, continues to be an open problem. As its underlying theory is defined in terms of complete lattices, the main challenge lies in introducing a complete lattice structure on the image intensity range, hence vectorial extrema computation methods are necessary. In this paper, we circumvent the need for...