STR2: optimized simple tabular reduction for table constraints

Abstract

Table constraints play an important role within constraint programming. Recently, many schemes or algorithms have been proposed to propagate table constraints and/or to compress their representation. In this paper, we describe an optimization of simple tabular reduction (STR), a technique proposed by J. Ullmann to dynamically maintain the tables of supports when generalized arc consistency (GAC) is enforced/maintained. STR2, the new refined GAC algorithm we propose, allows us to limit the number of operations related to validity checking and search of supports. Interestingly enough, this optimization makes simple tabular reduction potentially r times faster where r is the arity of the constraint(s). The results of an extensive experimentation that we have conducted with respect to random and structured instances indicate that STR2 is usually around twice as fast as the original STR, two or three times faster than the approach based on the hidden variable encoding, and can be up to one order of magnitude faster than previously stateof-the-art (generic) GAC algorithms on some series of instances. When comparing STR2 with the more recently developed algorithm based on multi-valued decision diagrams (MDDs), we show that both approaches are rather complementary.

Improving GAC-4 for Table and MDD Constraints

“…It has been successfully used in CP for representing sets or lists [8,7,14,15]. We will use it in GAC-4R and MDD-4R, so we give details of it.…”

“…Some algorithms are lazy (e.g., GAC-Schema [3], STR-2 [14], or STR-3 [15]). They try to reduce the operations executed at each modification (i.e., deletion of value of a domain) at the cost of increasing the complexity of the implementation.…”

An Optimal Filtering Algorithm for Table Constraints

“…The GAC3-Allowed algorithm has been chosen because it is the standard GAC3 algorithm for the table constraints [10]. The two state-of-the-art methods were also reimplemented: The STR2+ algorithm from [11] and the MDD c algorithm from [3]. They are respectively called STR and MDD in the experimental results.…”

“…Experimental results show that, on structured instances, our algorithms improve upon the state-of-the-art STR2+ [11] and MDD c [3]: The speedup is between 1.95 and 3.66 over STR2+ and between 1.83 and 4.57 over MDD c . Our (theoretically) optimal algorithm is not always the fastest in practice.…”

“…We now propose an implementation that only trails one integer, building upon an idea in STR and STR2 [18,11]. The implementation simply keeps invalid elements at the end of the table, with a single variable size representing the boundary between valid (before position size) and invalid elements (after position size).…”