Simple Grid Polygon Online Exploration Revisited

TL;DR

This paper revisits the online exploration problem for grid environments, clarifies contradictions in previous research, and provides improved bounds on the competitive ratio of exploration strategies.

Contribution

It identifies flaws in prior optimal strategies and introduces a new, tighter lower bound on the competitive ratio for grid exploration algorithms.

Findings

Previous optimal ratio claim of 7/6 is flawed

Lower bound on competitive ratio improved to 13/11

Clarifies contradictions in earlier research results

Abstract

Due to some significantly contradicting research results, we reconsider the problem of the online exploration of a simple grid cell environment. In this model an agent attains local information about the direct four-neigbourship of a current grid cell and can also successively build a map of all detected cells. Beginning from a starting cell at the boundary of the environment, the agent has to visit any cell of the grid environment and finally has to return to its starting position. The performance of an online strategy is given by competitive analysis. We compare the number of overall cell visits (number of steps) of an online strategy to the number of such visits in the optimal offline solution under full information of the environment in advance. The corresponding worst-case ratio gives the competitive ratio. The aforementioned contradiction among two publications turns out to be as follows: There is a journal publication that claims to present an optimal competitive strategy with ratio 7/6 and a former conference paper that presents a lower bound of 20/17. In this note we extract the flaw in the upper bound and also present a new slightly improved and (as we think) simplified general lower bound of 13/11.