A scalable approach to computing representative lowest common ancestor in directed acyclic graphs

Abstract

LCA computation for vertex pairs in trees can be achieved in constant time after linear-time preprocessing. However, extension of these techniques to compute LCA for vertex-pairs in DAGs has been not possible due to the non-tree edges in a DAG. In this paper, we present an algorithm for computing the LCA for vertex pairs in a DAG which treats the DAG??s spanning tree and its non-tree edges separately. Our approach enables us to tap the efficiency of existing LCA algorithms for trees. Furthermore, our algorithm decomposes the DAG into a set of component trees called clusters which significantly reduces the preprocessing necessary to incorporate non-tree edges in the LCA computation. Our algorithm seamlessly interpolates the performance graph between the best reported algorithms for trees and the best reported algorithms for DAGs depending on the incidence of non-tree edges in the DAG. Using the proposed techniques, it is possible to achieve near-linear preprocessing and constant query time for sparse DAG

An efficient, large-scale, non-lattice-detection algorithm for exhaustive structural auditing of biomedical ontologies

“…Many attempts have been made on improving the efficiency of algorithms for the all-pairs all-LCA problem [13, 14], i.e., finding all LCAs associated with each pair of nodes. More recently, Dash et al [15] presented an approach that combines the efficiency of existing LCA algorithms on trees with range-interval labeling scheme and an efficient matrix multiplication. This approach achieves near-linear time for tree-like, rooted DAGs, but query results are limited to a single representative LCA per each pair of nodes.…”

“…Our all LCA requirement is motivated by real-world application needs for implementing lattice-based approach to ontology quality assurance. Compared with the fastest all pairs representative LCA algorithm known to date with an O (| V | · | E |) time complexity [15], ANT-LCA provides a rough speed-up of three orders of magnitude for SNOMED CT. However, the worst time-complexity for our algorithm, | V | 2 · | E |, is attained when virtually all nodes are pairable with all other nodes.…”

On MatBase’s algorithm for preventing cycles in binary Cartesian function products

“…Detecting cycles in both undirected and directed graphs were intensively studied (e.g., [5,6,7,9,10,11,16,17,18,19,20,21,22]). Cartesian function products were studied as well, but not intensively (e.g., [23,24]).…”

All-Pairs LCA in DAGs: Breaking through the O(n2.5) barrier

“…For other results on All-Pairs LCA, see [27] on finding unique LCA. Other work in the area include [16].…”