Finger Indexed Sets: New Approaches

TL;DR

This paper introduces new, simpler algorithms for finger searching in sets that achieve optimal bounds in amortized and expected cases, improving practicality for insertions and deletions at the tail or anywhere in the set.

Contribution

The paper presents novel, simplified algorithms for finger searching that match optimal bounds, including for tail-specific and general cases, emphasizing practical efficiency.

Findings

Achieves $O(\sqrt{ ext{log}d/ ext{loglog}d})$ bounds for tail insertions/deletions.

Introduces a randomized algorithm with expected optimal bounds for general insertions/deletions.

Simplifies previous algorithms while maintaining optimal theoretical performance.

Abstract

In the particular case we have insertions/deletions at the tail of a given set S of $n$ one-dimensional elements, we present a simpler and more concrete algorithm than that presented in [Anderson, 2007] achieving the same (but also amortized) upper bound of $O(\sqrt{logd/loglogd})$ for finger searching queries, where $d$ is the number of sorted keys between the finger element and the target element we are looking for. Furthermore, in general case we have insertions/deletions anywhere we present a new randomized algorithm achieving the same expected time bounds. Even the new solutions achieve the optimal bounds in amortized or expected case, the advantage of simplicity is of great importance due to practical merits we gain.