3 research outputs found
Faster Sorting Networks for , and Inputs
We present new parallel sorting networks for to inputs. For and inputs these new networks are faster (i.e., they require less
computation steps) than the previously known best networks. Therefore, we
improve upon the known upper bounds for minimal depth sorting networks on and channels. The networks were obtained using a combination of
hand-crafted first layers and a SAT encoding of sorting networks
Sorting Networks: the End Game
This paper studies properties of the back end of a sorting network and
illustrates the utility of these in the search for networks of optimal size or
depth. All previous works focus on properties of the front end of networks and
on how to apply these to break symmetries in the search. The new properties
help shed understanding on how sorting networks sort and speed-up solvers for
both optimal size and depth by an order of magnitude
Methods for Solving Extremal Problems in Practice
During the 20 th century there has been an incredible progress in solving theoretically hard problems in practice. One of the most prominent examples is the DPLL algorithm and its derivatives to solve the Boolean satisfiability problem, which can handle instances with millions of variables and clauses in reasonable time, notwithstanding the theoretical difficulty of solving the problem.
Despite this progress, there are classes of problems that contain especially hard instances, which have remained open for decades despite their relative small size. One such class is the class of extremal problems, which typically involve finding a combinatorial object under some constraints (e.g, the search for Ramsey numbers). In recent years, a number of specialized methods have emerged to tackle extremal problems. Most of these methods are applied to a specific problem, despite the fact there is a great deal in common between different problems.
Following a meticulous examination of these methods, we would like to extend them to handle general extremal problems. Further more, we would like to offer ways to exploit the general structure of extremal problems in order to develop constraints and symmetry breaking techniques which will, hopefully, improve existing tools. The latter point is of immense importance in the context of extremal problems, which often hamper existing tools when there is a great deal of symmetry in the search space, or when not enough is known of the problem structure. For example, if a graph is a solution to a problem instance, in many cases any isomorphic graph will also be a solution. In such cases, existing methods can usually be applied only if the model excludes symmetries