Simple Grid Polygon Online Exploration Revisited

TL;DR

The paper revisits online exploration of unknown simple grid polygons, where an agent only observes the four-neighborhood and must visit all free cells before returning to the start. It identifies a fundamental flaw in the recent $7/6$-competitive upper bound claimed by Wei et al. and presents a corrected lower-bound construction that yields an asymptotic bound of $13/11$ via a concatenated five-block adversary. The results reinforce that the best known upper bound remains $5/4$, while the new lower bound narrows the gap between online and offline performance, clarifying model nuances such as the impact of narrow passages. The work thus advances the theoretical understanding of online grid exploration and highlights directions for closing the remaining gap in competitive ratios, with the potential for future refinements or tighter bounds.

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.

Figures (8)

• Figure 1: (i) The local view of the agent in the beginning, only the status (free/boundary) of the direct 4 neighbouring cells can be achieved. (ii) The currently known cell map after one step from $c$ to the neighbouring cell. (iii) The model can also be directly interpreted as a (special) grid graph model where cells are represented by vertices located in the center of the cell. Note that by visiting a vertex$c_w$ and detecting the edge $(c_w,c")$ the edge $(c_s,c")$ has to exist for a one-to-one correspondence. In the given model it is also already known at position $c$ that a connected vertex $c_n$ cannot exist, this can be different in the notion of solid grid graphs. (iv) The aforementioned local situation embedded in a simple grid cell environment also denoted as a simple grid (cell) polygon by interpretation of the boundary path as a simple polygon (in red). Here the boundary cell $B$ is not a hole, it is not fully surrounded. The grey cells indicate the five independent narrow passages of the environment. By (online) DFS movements keeping along the boundary these passages will always be passed optimally (exemplified by some blue paths). We do not count turnings, so the blue (sub-)path in the left/lower narrow passage starting and ending at $c'$ consists of 6 steps.
• Figure 2: (i) The strategy of Wei et. al wei tries to avoid single locally isolated cells $c$. To this end a tangent line is computed. If it separates the currently active start cell $s$ from the cell $c$, the cell $c$ is visited first. (ii) The idea is to meander in remaining corridors of width three optimally by applying this rule successively, first at time step $t$ and then in the whole remaining corridor. (iii) The worst-case example presented in Figure 8 of Wei et. al wei applies this rule twice, see $l_1$ and $l_2$ for time stamps $t_1$ and $t_2$ and the corresponding isolated cells $c_1$ and $c_2$, respectively. The strategy locally changes the preference and then moves on with left-hand rule. 28 steps are required in total which gives a ratio of $\frac{7}{6}$ against the optimal offline solution which requires 24 steps only as presented in (iv).
• Figure 3: The situation can be strengthened such that the ratio of the length of the path of the given strategy (i) and the optimal path length shown in (ii) attains a ratio of $\frac{26}{22}=\frac{13}{11}>\frac{7}{6}$. The narrow passages $N_1$ and $N_2$ both cause 2 extra revisits by running in the passage and also for reentering the remaining part again, respectively. In the optimal solution (ii) narrow passages are resolved optimally along the boundary.
• Figure 4: The general scheme of the adversary strategy. The strategy starts in a (potentially already known) rectangle of height 2 and width 1. The adversary strategy starts when this rectangle is left to the right. For any strategy the adversary finally presents one of the blocks (b), (d), (f), (h) or (i), depending on the motion of the agent. In these final blocks each blue arrow indicates a different option for the movement starting from cell $c$, for different blue arrow options the same block will be presented. The finalization of (g) in block (h) or (i) depends on the first visit (blue arrow options) of a cell $c_0$, $c_1$ or $c_2$ after $c$.
• Figure 5: All possible starting situations when the agent leaves a starting rectangle of height 2 and width 1 to the right. After that the adversary makes use of the scheme presented in Figure \ref{['fig-lb-total']}. For $(a_2)$ and $(a_2')$ the scheme of Figure \ref{['fig-lb-total']} simply has to be mirrored horizontally.

Theorems & Definitions (1)

• Theorem 1