"""Difference bases: minimize |B|^2 / v (EinsteinArena difference-bases, MIT).

v = largest integer such that 1..v are all differences of elements of B (0 is added to B if missing).
Ported from the upstream verifier. Changes: elements must be JSON integers in [0, inst["max_value"]]
(upstream cast with int(), so floats/strings were silently truncated) and distinct (upstream deduplicated);
at most inst["max_size"] elements after adding 0."""
import numpy as np

from run import Invalid, int_list


def check(inst, ans):
    s = ans.get("set")
    if not isinstance(s, list) or not 1 <= len(s) <= inst["max_size"]:
        raise Invalid(f"set must be a list of 1..{inst['max_size']} integers")
    int_list(s, None, 0, inst["max_value"], "set")
    if len(set(s)) != len(s):
        raise Invalid("elements must be distinct")
    B = sorted(set(s) | {0})
    if len(B) > inst["max_size"]:
        raise Invalid(f"at most {inst['max_size']} elements including 0")
    a = np.array(B, dtype=np.int64)
    d = (a[None, :] - a[:, None])[np.triu_indices(len(a), 1)]
    present = np.zeros(int(a[-1]) + 2, dtype=bool)
    present[d] = True
    present[0] = True
    v = int(np.argmin(present)) - 1  # first missing positive difference minus one
    if v < 1:
        raise Invalid("1 is not a difference of the set")
    return len(B) ** 2 / v, {"size": len(B), "v": v}