cell $(1,1)$ to $e_r$ and $e_r$ to cell $(m,n)$. Thus any event found in subsequent iterations is part of $E_{r-1}$ and of a realizing set with $e_r$.

To determine whether some $E_{k}$ contains a realizing set, we check whether cells $(1,1)$ and $(m,n)$ are connected without “using” the events of $\mathcal{E} \backslash E_{k}$. To do this efficiently, we further modify the Alt and Godau algorithm. We require only a method to prevent events in $\mathcal{E} \backslash E_{k}$ from being used. After $L^R_{i,j}$ is computed, we check whether the event $e$ (if any) that ends at the left boundary of cell $(i,j)$ is part of $\mathcal{E} \backslash E_{k}$ and necessary to obtain $L^R_{i,j}$. If this is the case, we replace $L^R_{i,j}$ with an empty interval. Event $e$ is necessary if and only if $L^R_{i,j}$ is a singleton. To obtain an algorithm that is numerically more stable, we introduce entry points. The *entry point* of the left boundary of cell $(i,j)$ is the maximal $i' < i$ such that $B^R_{i',j}$ is non-empty. These values are easily computed during the decision algorithm. Assume the passage corresponding to event $e$ starts on the left boundary of cell $(i_\text{s}, j)$. Event $e$ is necessary to 