Simpler O(1) Query Algorithm for Level Ancestors

TL;DR

This paper presents a simplified O(1) query time algorithm for finding level ancestors in trees, improving on previous methods by reducing complexity and preprocessing time, and offering a more straightforward implementation.

Contribution

It introduces a simple, efficient O(1) query algorithm for level ancestors, with optimized preprocessing and potential for parallel implementation, simplifying prior complex algorithms.

Findings

O(1) query time for level ancestors achieved

Preprocessing can be reduced to nearly linear or linear time

Parallel implementation can be built in O(1) time with n processors

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.