File size: 3,517 Bytes
848d4b7 | 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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | #!/usr/bin/env python3
"""
Validator for problem 041: Improve Upper Bound on Difference Basis Constant
For any natural number n, Δ(n) is the size of the smallest set B of integers such
that every k in {1, ..., n} is expressible as |a-b| for some a,b ∈ B.
We validate a proposed (n, B) and compute ratio = |B|^2 / n, which is a certified
upper bound on C^6.7 = inf_{n>=1} Δ(n)^2/n.
Expected input format:
{"n": <int>, "basis": [b0, b1, b2, ...]}
"""
import argparse
from typing import Any
from . import ValidationResult, load_solution, parse_integer, output_result, success, failure
MAX_BASIS_SIZE = 20000
MAX_N = 50_000_000 # memory guard for bytearray(n+1)
def validate(solution: Any) -> ValidationResult:
try:
if not isinstance(solution, dict):
return failure("Invalid format: expected dict with 'n' and 'basis' keys")
n = parse_integer(solution["n"])
raw_basis = solution["basis"]
if not isinstance(raw_basis, (list, tuple)):
return failure("Invalid format: 'basis' must be a list of integers")
basis = [parse_integer(b) for b in raw_basis]
except (KeyError, ValueError, TypeError) as e:
return failure(f"Failed to parse solution: {e}")
if n < 1:
return failure(f"n must be positive, got {n}")
B = sorted(set(basis))
m = len(B)
if m == 0:
return failure("Basis must contain at least one integer")
if m > MAX_BASIS_SIZE:
return failure(f"Basis too large: |B|={m} exceeds MAX_BASIS_SIZE={MAX_BASIS_SIZE}")
if n > MAX_N:
return failure(f"n too large: n={n} exceeds MAX_N={MAX_N}")
# Necessary condition: at most m*(m-1)/2 distinct positive differences exist.
if m * (m - 1) // 2 < n:
return failure(
f"Impossible coverage: |B|={m} allows at most {m*(m-1)//2} distinct positive differences "
f"but need to cover n={n} values (1..n).",
n=n,
basis_size=m,
)
# Normalize by translation invariance (differences unchanged by shifting).
shift = B[0]
B = [x - shift for x in B]
covered = bytearray(n + 1) # covered[d] = 1 iff difference d is achieved
for i in range(m):
a = B[i]
j = i + 1
while j < m:
d = B[j] - a
if d > n:
break
covered[d] = 1
j += 1
missing_count = 0
examples = []
for d in range(1, n + 1):
if covered[d] == 0:
missing_count += 1
if len(examples) < 5:
examples.append(d)
if missing_count:
return failure(
f"Not a difference basis for {{1..{n}}}: missing {missing_count} values. Examples: {examples}",
missing_count=missing_count,
missing_examples=examples,
n=n,
basis_size=m,
)
ratio = (m * m) / n
return success(
f"Verified difference basis for n={n}: |B|={m}, |B|^2/n = {ratio:.6f}",
n=n,
basis_size=m,
ratio=ratio,
)
def main():
parser = argparse.ArgumentParser(description="Validate difference basis construction")
parser.add_argument("solution", help="Solution as JSON string or path to JSON file")
parser.add_argument("--verbose", "-v', action='store_true", help="Verbose output")
args = parser.parse_args()
solution = load_solution(args.solution)
result = validate(solution)
output_result(result)
if __name__ == "__main__":
main() |