Abstract
We consider the problem of determining for each pair of vertices of a directed acyclic graph (dag) on n vertices whether or not it has a unique lowest common ancestor, and if so, finding such an ancestor. We show that this problem can be solved in time O(n ωlogn), where ω< 2.376 is the exponent of the fastest known algorithm for multiplication of two n×n matrices.
We show also that the problem of determining a lowest common ancestor for each pair of vertices of an arbitrary dag on n vertices is solvable in time \(\widetilde{O}(n^2p+n^{\omega})\), where p is the minimum number of directed paths covering the vertices of the dag. With the help of random bits, we can solve the latter problem in time \(\widetilde{O}(n^2p)\).
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Kowaluk, M., Lingas, A. (2007). Unique Lowest Common Ancestors in Dags Are Almost as Easy as Matrix Multiplication. In: Arge, L., Hoffmann, M., Welzl, E. (eds) Algorithms – ESA 2007. ESA 2007. Lecture Notes in Computer Science, vol 4698. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75520-3_25
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DOI: https://doi.org/10.1007/978-3-540-75520-3_25
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