| """Block stacking (overhang) problem, closed form of the constraints; checked in exact rational arithmetic. | |
| Upstream (block_stacking_problem.ipynb) used a float tolerance of 100*eps on every constraint, which lets a | |
| submission exceed the true optimum H_n/2 by accumulated slack; here all constraints are exact (non-strict). | |
| """ | |
| from fractions import Fraction | |
| from run import Invalid, float_list | |
| def check(inst, ans): | |
| n = inst["n"] | |
| x = [Fraction(v) for v in float_list(ans.get("x"), n, "x")] | |
| xs = [Fraction(0)] + x # x_0 = 0 | |
| for i in range(n): | |
| if xs[i + 1] < xs[i]: | |
| raise Invalid(f"need x_{i} <= x_{i + 1}") | |
| tail = Fraction(0) | |
| for i in range(n - 1, -1, -1): # tail = x_{i+1} + ... + x_n | |
| tail += xs[i + 1] | |
| if tail > (n - i) * (xs[i] + Fraction(1, 2)): | |
| raise Invalid(f"stability constraint violated at i = {i}") | |
| return float(xs[n]), {} | |