| """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]) |
| |
| |
| 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} |
|
|
