to cell $(m,n)$. This requires only two changes: $B^R_{1,2}$ should be initialized with $B^F_{1,2}$ and $L^R_{2,1}$ with $L^F_{2,1}$; the answer should be “yes” if and only if $B^R_{m,n}$ or $L^R_{m,n}$ is non-empty.

![Locally correct matching produced by Algorithm \[alg:findmatching\]. Free space diagram drawn at $\varepsilon = \delta_\text{F}(P,Q)$.[]{data-label="fig:examplematching"}](examplematching)

#### Realizing set

By computing the [Fréchet distance]{} using the modified Alt and Godau algorithm, we obtain an ordered, potentially non-minimal realizing set $\mathcal{E} = \{ e_1, \ldots, e_l \}$. The algorithm must find an event that is contained in a realizing set. Let $E_{k}$ denote the first $k$ events of $\mathcal{E}$. For now we assume that the events in $\mathcal{E}$ end at different cell boundaries. We use a binary search on $\mathcal{E}$ to find the $r$ such that $E_r$ contains a realizing set, but $E_{r-1}$ does not. This implies that event $e_r$ is contained in a realizing set and can be used to split the curves. Note that $r$ is unique due to monotonicity. For correctness, the order of events in $\mathcal{E}$ must be consistent in different iterations, for example, by using a lexicographic order. Set $E_r$ contains only realizing sets that use $e_r$. Hence, $E_{r-1}$ contains a realizing set to connect 