Thorsten Ehlers

New Bounds on Optimal Sorting Networks

We present new parallel sorting networks for \(17\) to \(20\) inputs. For \(17, 19,\) and \(20\) inputs these new networks are faster (i.e., they require fewer computation steps) than the previously known best networks. Therefore, we improve upon the known upper bounds for minimal depth sorting networks on \(17, 19,\) and \(20\) channels. Furthermore, we show that our sorting network for \(17\) inputs is optimal in the sense that no sorting network using less layers exists. This solves the main open problem of [D. Bundala & J. Zavodný. Optimal sorting networks, Proc. LATA 2014].

k-Abelian pattern matching

Two words are called k-abelian equivalent, if they share the same multiplicities for all factors of length at most k. We present an optimal linear time algorithm for identifying all occurrences of factors in a text that are k-abelian equivalent to some pattern P. Moreover, an optimal algorithm for finding the largest k for which two words are k-abelian equivalent is given. Solutions for online versions of the k-abelian pattern matching problem are also proposed.

Sorting networks: To the end and back again

Abstract New properties of the front and back ends of sorting networks are studied, illustrating their utility when searching for bounds on optimal networks. Search focuses first on the “out-sides” of the network and then on the inner part. Previous works focused on properties of the front end to break symmetries in the search. The new, out-side-in, properties shed understanding on how sorting networks sort, and facilitate the computation of new bounds on optimality. We present new, faster, parallel sorting networks for 17–20 inputs. For 17 inputs, we show that no sorting network using less layers exists.

Communication in Massively-Parallel SAT Solving

The exchange of learnt clauses is a key feature in parallel SAT solving. We present an approach based on a communication graph. Each solver thread corresponds to a node in this graph. Communication between two solvers is allowed if the respective nodes are connected by an edge. This yields another dimension in controlling the amount of communication. We show results for this approach, gaining significant speedups for up to 256 parallel solvers.

A Generalization of the Directed Graph Layering Problem

The Directed Layering Problem (DLP) solves a step of the widely used layer-based approach to automatically draw directed acyclic graphs. To cater for cyclic graphs, usually a preprocessing step is used that solves the Feedback Arc Set Problem (FASP) to make the graph acyclic before a layering is determined.

k -Abelian Pattern Matching

Two words are called k-abelian equivalent, if they share the same multiplicities for all factors of length at most k. We present an optimal linear time algorithm for identifying all occurrences of factors in a text that are k-abelian equivalent to some pattern P. Moreover, an optimal algorithm for finding the largest k for which two words are k-abelian equivalent is given. Solutions for various online versions of the k-abelian pattern matching problem are also proposed.

Generalized Layerings for Arbitrary and Fixed Drawing Areas

The Directed Layering Problem (DLP) solves a step of the widely used layer-based approach to automatically draw directed acyclic graphs. To cater for cyclic graphs, usually a preprocessing step is used that solves the Feedback Arc Set Problem (FASP) to make the graph acyclic before a layering is determined. Here we present the Generalized Layering Problem (GLP), which solves the combination of DLP and FASP simultaneously, allowing general graphs as input. We present an integer programming model and a heuristic to solve the NP-complete GLP and perform thorough evaluations on different sets of graphs and with different implementations for the steps of the layer-based approach. We observe that GLP reduces the number of dummy nodes significantly, can produce more compact drawings, and improves on graphs where DLP yields poor aspect ratios. The drawings resulting from GLP also turn out to be more suitable for making the best possible use of a given drawing area. However, we show that a specialized variant of GLP can yield considerable improvements w. r. t. this particular optimization goal.

Parallelizing Constraint Programming with Learning

Parallel Constraint Programming (CP) solvers typically split the search space in disjoint subspaces, and run solvers independently on these. This may induce significant overhead when solving optimization problems. Parallel Boolean Satisfiability (SAT) solvers typically run a portfolio of solvers, all solving the same problem but sharing some limited learnt clause information. In this paper we consider parallelizing a lazy clause generation (LCG) constraint programming solver, which is a constraint programming solver with learning. Since it is both a kind of CP solver and a kind of SAT solver it is not clear which approach to parallelization is likely to be most effective. We give examples of very different kinds of optimization problems we wish to parallelize and show that a hybrid approach to parallelization can provide a robust and high performing parallel LCG solver.

Breaking Symmetries with Lex Implications

Breaking symmetries is crucial when solving hard combinatorial problems. A common way to eliminate symmetries in CP/SAT is to add symmetry breaking constraints. Ideally, symmetry breaking constraints should be complete and compact. The aim of this paper is to find compact and complete symmetry breaks applicable when solving hard combinatorial problems using CP/SAT approach. In particular: graph search problems and matrix model problems where symmetry breaks are often specified in terms of lex constraints. We show that sets of lex constraints can be expressed with only a small portion of their inner lex implications which are a particular form of Horn clauses. We exploit this fact and compute a compact encoding of the row-wise LexLeader and state of the art partial symmetry breaking constraints. We illustrate the approach for graph search problems and matrix model problems.

Tuning Parallel SAT Solvers

In this paper we present new implementation details and benchmarking results for our parallel portfolio solver TopoSAT2. In particular, we discuss ideas and implementation details for the exchange of learned clauses in a massively-parallel SAT solver which is designed to run more that 1, 000 solver threads in parallel. Furthermore, we go back to the roots of portfolio SAT solving, and discuss the impact of diversifying the solver by using different restart, branchingand clause database management heuristics. We show that these techniques can be used to tune the solver towards different problems. However, in a case study on formulas derived from Bounded Model Checking problems we see the best performance when using a rather simple clause exchange strategy. We show details of these tests and discuss possible explanations for this phenomenon. As computing times on massively-parallel clusters are expensive, we consider it especially interesting to share these kind of experimental results.