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

##### MPG-Autoren
http://pubman.mpdl.mpg.de/cone/persons/resource/persons45021

Mehlhorn,  Kurt
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45615

Thiel,  Sven
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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##### Zitation

Mehlhorn, K., & Thiel, S. (2000). Faster Algorithms for Bound-Consistency of the Sortedness and the Alldifferent Constraint. In Principles and practice of constraint programming - CP 2000 (CP-00): 6th international conference, CP 2000 (pp. 306-319). Berlin, Germany: Springer.

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.