| """Spencer discrepancy constant witness (EinsteinArena spencer-discrepancy, MIT). |
| Score disc(A)/sqrt(n) with disc(A) = min_{x in {-1,1}^n} ||Ax||_inf (maximize). |
| |
| Audit: upstream verifier enumerates all 2^n sign vectors exactly; sound. Re-implemented with numpy |
| (vectorised popcounts; x and -x give the same value so half the vectors suffice).""" |
| import math |
|
|
| import numpy as np |
|
|
| from run import Invalid, int_list |
|
|
|
|
| def _popcount(a): |
| a = a - ((a >> 1) & 0x55555555) |
| a = (a & 0x33333333) + ((a >> 2) & 0x33333333) |
| a = (a + (a >> 4)) & 0x0F0F0F0F |
| return (a * 0x01010101 & 0xFFFFFFFF) >> 24 |
|
|
|
|
| def check(inst, ans): |
| m = ans.get("matrix") |
| cap = inst["max_order"] |
| if not isinstance(m, list) or not 1 <= len(m) <= cap: |
| raise Invalid(f"matrix order must be between 1 and {cap}") |
| n = len(m) |
| masks = [] |
| for row in m: |
| int_list(row, n, -1, 1, "matrix row") |
| if any(v == 0 for v in row): |
| raise Invalid("entries must be exactly -1 or 1") |
| masks.append(sum(1 << j for j, v in enumerate(row) if v == -1)) |
| disc = n |
| total = 1 << max(n - 1, 0) |
| step = 1 << 18 |
| for s0 in range(0, total, step): |
| x = np.arange(s0, min(total, s0 + step), dtype=np.uint64) |
| worst = np.zeros(len(x), dtype=np.int64) |
| for r in masks: |
| pc = _popcount((x ^ np.uint64(r)) & np.uint64(0xFFFFFFFF)).astype(np.int64) |
| worst = np.maximum(worst, np.abs(n - 2 * pc)) |
| disc = min(disc, int(worst.min())) |
| return disc / math.sqrt(n), {"disc": disc, "order": n} |
|
|
