Faster Treasure Hunt and Better Strongly Universal Exploration Sequences

TL;DR

This paper presents a faster deterministic algorithm for the treasure hunt problem in networks and introduces improved explicit constructions for strongly universal exploration sequences, advancing the efficiency and applicability of these methods.

Contribution

It provides an $O(n^{c(1+1/\lambda)})$-time algorithm for treasure hunt and a significantly improved explicit construction for strongly universal exploration sequences.

Findings

Achieved a substantial reduction in running time for treasure hunt algorithms.

Developed a more efficient explicit construction for exploration sequences.

Enhanced the theoretical understanding of universal exploration in networks.

Abstract

In this paper, we investigate the explicit deterministic treasure hunt problem in a $n$-vertex network. This problem was firstly introduced by Ta-Shma and Zwick in \cite{TZ07} [SODA'07]. Note also it is a variant of the well known rendezvous problem in which one of the robot (the treasure) is always stationary. In this paper, we propose an $O(n^{c(1+\frac{1}{\lambda})})$-time algorithm for the treasure hunt problem, which significantly improves the currently best known result of running time $O(n^{2c})$ in \cite{TZ07}, where $c$ is a constant induced from the construction of an universal exploration sequence in \cite{R05,TZ07}, and $\lambda \gg 1$ is an arbitrary large, but fixed, integer constant. The treasure hunt problem also motivates the study of strongly universal exploration sequences. In this paper, we also propose a much better explicit construction for strongly universal exploration sequences compared to the one in \cite{TZ07}.