Faster Algorithms for Bound-Consistency of the Sortedness and the Alldifferent Constraint

We present narrowing algorithms for the sortedness and the alldifferent constraint which achieve bound-consistency. The algorithm for the sortedness constraint takes as input $2n$ intervals $X_1, \dots, X_n$, $Y_1, \dots, Y_n$ from a linearly ordered set $D$. Let $\mathcal{S}$ denote the set of all tuples $t \in X_1 \times \cdots \times X_n \times Y_1 \times \cdots \times Y_n$ such that the last $n$ components of $t$ are obtained by sorting the first $n$ components. Our algorithm determines whether $\mathcal{S}$ is non-empty and if so reduces the intervals to bound-consistency. The running time of the algorithm is asymptotically the same as for sorting the interval endpoints. In problems where this is faster than $O(n \log n)$, this improves upon previous results. The algorithm for the alldifferent constraint takes as input $n$ integer intervals $Z_1, \dots, Z_n$. Let $\mathcal{T}$ denote all tuples $t \in Z_1 \times \cdots \times Z_n$ where all components are pairwise different. The algorithm checks whether $\mathcal{T}$ is non-empty and if so reduces the ranges to bound-consistency. The running time is also asymptotically the same as for sorting the interval endpoints. When the constraint is for example a permutation constraint, i.e. $Z_i \subseteq \range{1}{n}$ for all $i$, the running time is linear. This also improves upon previous results