Sorting arrays by means of swaps

AbstractAs a preparation to the study of a particular sorting machine, the sorting of arrays by sequences of swaps is treated in this paper. In particular it is shown that if a sequence of “miniswaps” sorts the worst possible arrangement of an array, then it sorts every arrangement of that array. The case of permutation arrays was treated before by R.W. Floyd

A Non-Deterministric Parallel Sorting Algorithm

A miniswap Si,1 ≤ i \u3c n, compares two adjacent keys Пi, Пi+1 in the sequence (П1, ... , Пn), and transposes them if they are out of order. A full sweep is any composition of all n - 1 possible miniswaps. We prove that the composition of any n- 1 full sweeps is a sorting function

Multitriangulations, pseudotriangulations and primitive sorting networks

We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial. As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.Comment: 60 pages, 40 figures; minor corrections and improvements of presentatio

Computing pseudotriangulations via branched coverings

We describe an efficient algorithm to compute a pseudotriangulation of a finite planar family of pairwise disjoint convex bodies presented by its chirotope. The design of the algorithm relies on a deepening of the theory of visibility complexes and on the extension of that theory to the setting of branched coverings. The problem of computing a pseudotriangulation that contains a given set of bitangent line segments is also examined.Comment: 66 pages, 39 figure

Sorting by means of swappings

AbstractIf π is a permutation of {1,…,n}, then the effect of the swap Sij on π is that the ith and fth entry are sorted. If [i−j] = l, we call Sij a miniswap (it sorts neighbours). The paper studies a partial order based on the swaps, and establishes some properties of miniswaps