Abstract:
We present two new methods for finding a lowest common ancestor (LCA) for each pair of vertices of a directed acyclic graph (dag) on $n$ vertices and $m$ edges. The first method is surprisingly natural and solves the all-pairs LCA problem for the input dag on $n$ vertices and $m$ edges in time $O(nm)$. The second method relies on a novel reduction of the all-pairs LCA problem to the problem of finding maximum witnesses for Boolean matrix product. We solve the latter problem and hence also the all-pairs LCA problem in time $O(n^{2+\lambda})$, where $\lambda $ satisfies the equation $\omega(1, \lambda, 1) = 1 + 2 \, \lambda $ and $\omega(1, \lambda, 1)$ is the exponent of the multiplication of an $n \times n^{\lambda}$ matrix by an $n^{\lambda} \times n$ matrix. By the currently best bounds on $\omega(1,\lambda,1)$, the running time of our algorithm is $O(n^{2.575})$. Our algorithm improves the previously known $O(n^{2.688})$ time-bound for the general all-pairs LCA problem in dags by Bender \emph{et al.} Our additional contribution is a faster algorithm for solving the all-pairs lowest common ancestor problem in dags of small depth, where the depth of a dag is defined as the length of the longest path in the dag. For all dags of depth at most $h \le n^{\alpha}$, where $\alpha \approx 0.294$, our algorithm runs in time asymptotically the same as that of multiplying two $n \times n$ matrices, that is, $O(n^{\omega})$; we also prove that this running time is optimal even for dags of depth $1$. For dags with depth $h > n^{\alpha}$, the running time of our algorithm is at most $O(n^{\omega} \cdot h^{0.468})$. This algorithm is faster than our algorithm for arbitrary dags for all values of $h \le n^{0.42}$.

Abstract:
We present two new methods for finding a lowest common ancestor (LCA) for each pair of vertices of a directed acyclic graph (dag) on $n$ vertices and $m$ edges. The first method is surprisingly natural and solves the all-pairs LCA problem for the input dag on $n$ vertices and $m$ edges in time $O(nm)$. The second method relies on a novel reduction of the all-pairs LCA problem to the problem of finding maximum witnesses for Boolean matrix product. We solve the latter problem and hence also the all-pairs LCA problem in time $O(n^{2+\lambda})$, where $\lambda $ satisfies the equation $\omega(1, \lambda, 1) = 1 + 2 \, \lambda $ and $\omega(1, \lambda, 1)$ is the exponent of the multiplication of an $n \times n^{\lambda}$ matrix by an $n^{\lambda} \times n$ matrix. By the currently best bounds on $\omega(1,\lambda,1)$, the running time of our algorithm is $O(n^{2.575})$. Our algorithm improves the previously known $O(n^{2.688})$ time-bound for the general all-pairs LCA problem in dags by Bender \emph{et al.} Our additional contribution is a faster algorithm for solving the all-pairs lowest common ancestor problem in dags of small depth, where the depth of a dag is defined as the length of the longest path in the dag. For all dags of depth at most $h \le n^{\alpha}$, where $\alpha \approx 0.294$, our algorithm runs in time asymptotically the same as that of multiplying two $n \times n$ matrices, that is, $O(n^{\omega})$; we also prove that this running time is optimal even for dags of depth $1$. For dags with depth $h > n^{\alpha}$, the running time of our algorithm is at most $O(n^{\omega} \cdot h^{0.468})$. This algorithm is faster than our algorithm for arbitrary dags for all values of $h \le n^{0.42}$.

Abstract:
We present two new methods for finding a lowest common ancestor (LCA) for each pair of vertices of a directed acyclic graph (dag) on n vertices and m edges. The first method is a natural approach that solves the all-pairs LCA problem for the input dag in time O(nm). The second method relies on a novel reduction of the all-pairs LCA problem to the problem of finding maximum witnesses for Boolean matrix product. We solve the latter problem and hence also the all-pairs LCA problem in time O(n^{2.575}). Our additional contribution is a faster algorithm for solving the all-pairs LCA problem in graphs of small depth.