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1"""Pareto frontier (maximise y, minimise x) over labelled points. Pure Python, deterministic.23A point is *Pareto-efficient* when no other point has both a lower-or-equal x and a higher-or-equal y with at least one strict inequality.4Exact ties (same x and same y) are kept together on the frontier — we never break a tie by choosing one model over another."""5from __future__ import annotations67from typing import Any8910def pareto_frontier(points: list[dict[str, Any]], *, x: str = "x", y: str = "y", key: str = "id", maximize_y: bool = True) -> list[Any]:11 """Return the `key`s of the Pareto-efficient points, ordered by increasing x. Points with a missing x or y are ignored."""12 valid = [p for p in points if p.get(x) is not None and p.get(y) is not None]13 if not valid:14 return []15 sign = 1.0 if maximize_y else -1.016 # sort by x ascending, then by y descending (best first) so a sweep with the running best y finds the frontier17 ordered = sorted(valid, key=lambda p: (float(p[x]), -sign * float(p[y])))18 frontier: list[Any] = []19 best_y: float | None = None20 best_x: float | None = None21 for p in ordered:22 px, py = float(p[x]), sign * float(p[y])23 if best_y is None or py > best_y:24 frontier.append(p[key])25 best_y, best_x = py, px26 elif py == best_y and best_x is not None and px == best_x:27 frontier.append(p[key]) # exact tie: keep both28 return frontier293031def is_dominated(p: dict[str, Any], others: list[dict[str, Any]], *, x: str = "x", y: str = "y", maximize_y: bool = True) -> bool:32 """True when some other point is at least as good on both axes and strictly better on one."""33 sign = 1.0 if maximize_y else -1.034 px, py = float(p[x]), sign * float(p[y])35 for o in others:36 if o is p or o.get(x) is None or o.get(y) is None:37 continue38 ox, oy = float(o[x]), sign * float(o[y])39 if ox <= px and oy >= py and (ox < px or oy > py):40 return True41 return False424344__all__ = ["is_dominated", "pareto_frontier"]45