Just for the analysis: suppose we renumber the vertices $w_1, w_2, \dots, w_n$ in order of their distance from the edge $e = (u,v)$.

The crucial definitions (that avoid pain later) are the following:

- At some time instant in this procedure, one (or both) of $u$ or $v$ gets assigned to some $w_i$. Say that $w_i$ settles the edge $(u,v)$.
- At the moment the edge is settled, if only one endpoint of this edge gets assigned, then we say that $w_i$ cuts the edge $(u,v)$.

Consider $w_j$ and let its distance $d(w_j,u) = a_j$ and $d(w_j, v) = b_j$. Assume $a_j < b_j$, the other case is identical. If $w_j$ cuts $(u,v)$ when the random values are $X$ and $\pi$, the following two properties must hold:

- The random variable $X$ must lie in the interval $[a_j, b_j]$ (else either none or both of $(u,v)$ would get marked).
- The node $w_j$ must come before $w_1, ..., w_{j-1}$ in the permutation $\pi$.

Suppose not, and one of them came before $w_j$ in the permutation. Since all these vertices are closer to the edge than $w_j$ is, then for the current value of $X$, they would have settle the edge (either capture one or both of the endpoints) at some previous time point, and hence $w_j$ would not settle---and hence not cut---the edge $(u,v)$.

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