Abstract. We introduce a new approach to compute the common intervals of K permutations based on a very simple and general notion of generators of common intervals. This formalism leads to simple and efficient algorithms to compute the set of all common intervals of K permutations that can contain a quadratic number of intervals, as well as a linear space basis of this set of common intervals. Finally, we show how our results on permutations can be used for computing the modular decomposition of graphs.
Modular decomposition of graphs is a powerful tool with many applications in graph theory and optimization. There are efficient linear-time algorithms that compute the decomposition for undirected graphs. The best previously published time bound for directed graphs is O(n + m log n), where n is the number of vertices and m is the number of edges. We give an O(n + m)-time algorithm.
The first polynomial time algorithm (O(n 4)) for modular decomposition appeared in 1972 [8] and since then there have been incremental improvements, eventually resulting in linear time algorithms [22, 7, 23, 9]. Although an optimal time complexity these algorithms are quite complicated and difficult to implement. In this paper we present an easily implementable linear time algorithm for modular decomposition. This algorithm use the notion of factorizing permutation and a new datastructure, the Ordered Chain Partitions.
Abstract. We introduce a new approach to compute the common intervals of K permutations based on a very simple and general notion of generators of common intervals. This formalism leads to simple and efficient algorithms to compute the set of all common intervals of K permutations that can contain a quadratic number of intervals, as well as a linear space basis of this set of common intervals. Finally, we show how our results on permutations can be used for computing the modular decomposition of graphs.
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