| """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))} |
|
|
