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1ca0407 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | """Ring loading problem with 15 pairs (EinsteinArena ring-loading-15, MIT).
A(u, v) = min over z_i in {v_i, -u_i} of max_k |sum_{i<=k} z_i - sum_{i>k} z_i| (maximize).
Audit: upstream verifier parses exact rationals and enumerates all 2^15 adversary choices; sound.
Re-implemented over a common denominator with Python integers (same exact value, faster)."""
import math
import re
from fractions import Fraction
from run import Invalid
PAT = re.compile(r"^(?:0|[1-9]\d*)(?:(?:\.\d+)|(?:/[1-9]\d*))?$")
def _rat(s):
if not isinstance(s, str) or not 1 <= len(s) <= 80 or PAT.fullmatch(s) is None:
raise Invalid("values must be nonnegative decimal or fraction strings such as '0.25' or '3/7'")
r = Fraction(s)
if r.numerator.bit_length() > 64 or r.denominator.bit_length() > 64:
raise Invalid("reduced numerator and denominator must fit in 64 bits")
return r
def check(inst, ans):
k = inst["pairs"]
p = ans.get("pairs")
if not isinstance(p, list) or len(p) != k:
raise Invalid(f"pairs must be a list of exactly {k} pairs")
uv = []
for q in p:
if not isinstance(q, list) or len(q) != 2:
raise Invalid("each entry must be a pair [u, v]")
u, v = _rat(q[0]), _rat(q[1])
if u + v > 1:
raise Invalid("every pair must satisfy u + v <= 1")
uv.append((u, v))
den = 1
for u, v in uv:
for r in (u, v):
den = den * r.denominator // math.gcd(den, r.denominator)
U = [int(u * den) for u, _ in uv]
V = [int(v * den) for _, v in uv]
best = None
for mask in range(1 << k):
z = [-U[i] if mask >> i & 1 else V[i] for i in range(k)]
tot = sum(z)
pre, worst = 0, 0
for x in z:
pre += x
worst = max(worst, abs(2 * pre - tot))
if best is None or worst < best:
best = worst
return float(Fraction(best, den)), {"exact": str(Fraction(best, den))}
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