Expected O(ε) error for sequential farthest-point reduction under random permutations

Establish that, when the input point set P is randomly permuted, the simple sequential reduction used to select the farthest point from the segment pq (Algorithm farthest1, which scans points and updates the current farthest using the computed predicate \widehat{frt}(p, q, u, u')) has expected scaled error \mathbb{E}[F_{|P|}] bounded by O(\epsilon). Here F_{|P|} := (d(r, pq) − d(\widehat{r}, pq)) / M, where r is the true farthest point from pq among the |P| points, \widehat{r} is the algorithm’s output, d(·, pq) denotes perpendicular distance to the line through pq, M is the maximum absolute input coordinate used for scaling, and \epsilon is the machine precision.

Background

The paper analyzes the numerical stability of Quickhull by decomposing its building blocks. A key component is selecting the point farthest from a segment pq using reductions over pairwise comparisons of the distance test. Three reduction orders are studied: a simple sequential scan (Algorithm farthest1), blocking, and pairwise recursion, with worst-case error bounds O(nε), O((m + n/m)ε), and O(log n * ε), respectively.

Although the simple sequential method has poor worst-case error growth, the authors hypothesize that in practice—when the input point order is random—the expected error does not accumulate and remains O(ε). They note that a deeper analysis would connect this question to the expected length of subsequences, making it related to variants of Ulam’s Problem, and therefore they provide Monte Carlo evidence but no proof. Establishing the claimed expected bound would justify the tighter forward-error estimate for Quickhull under realistic assumptions.