Simpler O(1) Query Algorithm for Level Ancestors
Sanjeev Saxena
TL;DR
The paper tackles constant-time level-ancestor queries in rooted trees. It presents a simple serial re-implementation of Berkman and Vishkin's parallel method and shows how preprocessing can be reduced toward linear via a two-level structure and table lookups, yielding $O(1)$ query time with near-linear preprocessing in the best cases. Central to the approach are Euler-tour representations, the Nearest Smaller problem, and FAR/Near arrays that encode ladder-like ancestry information; a coarse-to-fine refinement using a block-based mod_FAR and Near arrays enables practical, implementable constant-time queries. The work also discusses parallelization implications, showing how table-construction strategies can achieve $O(1)$ preprocessing with $n$ processors, and positions this as a simpler alternative to more complex existing methods like Bender-Farach-Colton, with broad applicability to rooted-tree queries and related LCA problems.
Abstract
This note describes a very simple O(1) query time algorithm for finding level ancestors. This is basically a serial (re)-implementation of the parallel algorithm of Berkman and Vishkin (O.Berkman and U.Vishkin, Finding level-ancestors in trees, JCSS, 48, 214--230, 1994). Although the basic algorithm has preprocessing time of O(n log n), by having additional levels or using table lookup, the preprocessing time can be reduced to almost linear or linear. The table lookup algorithm can be built in O(1) parallel time with $n$ processors and can also be used to simplify the parallel algorithm of Berkman and Vishkin and make it optimal.