File size: 1,622 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 | """Sum-difference problem III (AlphaEvolve problem 44).
A set U of non-negative integers with min U = 0 gives the lower bound
C >= 1 + log(|U-U| / |U+U|) / log(2 max U + 1)
(AlphaEvolve's compute_lower_bound). Upstream checked |U-U| <= 2 max U + 1, which always holds when
U is inside [0, max U]; we check the hypotheses (integers, distinct, min 0) instead.
"""
import math
import numpy as np
from run import Invalid, int_list
def check(inst, ans):
u = ans.get("U")
if not isinstance(u, list) or not (2 <= len(u) <= inst["max_size"]):
raise Invalid(f"U must be a list of 2..{inst['max_size']} integers")
int_list(u, None, 0, inst["max_value"], "U")
if min(u) != 0:
raise Invalid("U must contain 0 and be non-negative")
if len(set(u)) != len(u):
raise Invalid("U must consist of distinct integers")
x = np.array(sorted(u), dtype=np.int64)
m = int(x[-1])
# bitmap sweeps with a buffer of m+1 bytes: U-U is symmetric, so count non-negative differences;
# U+U lies in [0, 2m], handled as two windows [0, m] and [m+1, 2m] (x sorted -> contiguous slices)
buf = np.zeros(m + 1, dtype=bool)
for i, v in enumerate(x):
buf[v - x[:i + 1]] = True
d = 2 * int(buf.sum()) - 1
s = 0
for lo in (0, m + 1):
buf[:] = False
for v in x:
t = v + x
a, b = np.searchsorted(t, lo), np.searchsorted(t, lo + m, side="right")
buf[t[a:b] - lo] = True
s += int(buf.sum())
del buf
return 1.0 + math.log(d / s) / math.log(2 * m + 1), {"size": len(u), "max": m, "diffset": d, "sumset": s}
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