File size: 1,587 Bytes
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 | """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) # fix the top sign (x ~ -x)
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}
|