FA-73056 / Probabilistic sketches / Open access
Count-Min sketch with conservative update: error bound omits the e factor · case 01
Callers trust estimates to be within a bound that Count-Min does not guarantee.
ROOT CAUSE
The additive error bound is total/w instead of (e/w) * total.
THE FAILURE
The additive error bound is total/w instead of (e/w) * total.
Unsuccessful approach: Scaling by depth instead of width confuses the confidence parameter with the accuracy parameter.
Case contract
Input {w, d, ops}. Row r hashes key to ((A_r*key + B_r) mod 2^31-1) mod w with fixed A and B. add with count <= 0 is "rejected" (conservative update cannot represent deletions). Otherwise the current estimate e is the minimum of the key's cells, and each cell is raised to max(cell, e + count). The stream total counts accepted mass and the overestimate bound is ceil(e/w * total). q returns the minimum of the key's cells. Return [results, total, bound, table].
Why this case matters
Conservative update reduces Count-Min overestimation in heavy-hitter monitors, but only if the raise-to-target rule is applied exactly.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
w = x['w']
A = [1103515245, 1812433253, 1664525017, 2013265921, 1597334677]
B = [12345, 1013904223, 7, 99991, 31337]
P = 2147483647
d = x['d']
t = [[0] * w for _ in range(d)]
def bucket(r, key):
return (A[r] * key + B[r]) % P % w
total = 0
res = []
for op in x['ops']:
if op[0] == 'add':
key, cnt = op[1], op[2]
if cnt <= 0:
res.append('rejected')
continue
est = min(t[r][bucket(r, key)] for r in range(d))
for r in range(d):
b = bucket(r, key)
t[r][b] = max(t[r][b], est + cnt)
total += cnt
res.append('ok')
else:
res.append(min(t[r][bucket(r, op[1])] for r in range(d)))
return [res, total, math.ceil(total / w), t]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[['colliding keys',
{'d': 3,
'ops': [['add', 1, 4], ['add', 6, 2], ['add', 1, 1], ['q', 1], ['q', 6], ['q', 11]],
'w': 5},
[['ok', 'ok', 'ok', 5, 2, 0], 7, 4, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]]],
['zero and negative counts',
{'d': 2, 'ops': [['add', 3, 0], ['add', 3, -2], ['add', 3, 5], ['q', 3]], 'w': 7},
[['rejected', 'rejected', 'ok', 5], 5, 2, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]]],
['huge keys wrap prime',
{'d': 3,
'ops': [['add', 900000007, 3],
['add', 2700000021, 2],
['q', 900000007],
['q', 2700000021],
['q', 7]],
'w': 7},
[['ok', 'ok', 3, 2, 0],
5,
2,
[[0, 3, 0, 0, 0, 0, 0], [2, 0, 0, 0, 3, 0, 0], [2, 0, 0, 3, 0, 0, 0]]]],
['uneven rows',
{'d': 4,
'ops': [['add', 2, 3],
['add', 9, 5],
['add', 2, 2],
['add', 5, 2],
['q', 2],
['q', 9],
['q', 5]],
'w': 4},
[['ok', 'ok', 'ok', 'ok', 5, 5, 2],
12,
9,
[[5, 0, 5, 0], [5, 0, 0, 5], [0, 0, 5, 2], [5, 0, 5, 0]]]],
['single row',
{'d': 1, 'ops': [['add', 1, 2], ['add', 4, 3], ['q', 1], ['add', 0, 0]], 'w': 3},
[['ok', 'ok', 2, 'rejected'], 5, 5, [[2, 3, 0]]]],
['many keys',
{'d': 3,
'ops': [['add', 0, 1],
['add', 1, 2],
['add', 2, 3],
['add', 3, 4],
['add', 4, 1],
['add', 5, 2],
['add', 6, 3],
['add', 7, 4],
['add', 8, 1],
['add', 9, 2],
['add', 10, 3],
['add', 11, 4],
['q', 0],
['q', 3],
['q', 6],
['q', 9],
['add', 2, 0]],
'w': 5},
[['ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
1,
4,
4,
2,
'rejected'],
30,
17,
[[5, 2, 2, 4, 4], [3, 4, 5, 5, 2], [0, 4, 1, 5, 4]]]]],
[['colliding keys',
{'d': 3,
'ops': [['add', 1, 4], ['add', 6, 2], ['add', 1, 1], ['q', 1], ['q', 6], ['q', 11]],
'w': 5},
[['ok', 'ok', 'ok', 5, 2, 0], 7, 4, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]]],
['zero and negative counts',
{'d': 2, 'ops': [['add', 3, 0], ['add', 3, -2], ['add', 3, 5], ['q', 3]], 'w': 7},
[['rejected', 'rejected', 'ok', 5], 5, 2, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]]],
['huge keys wrap prime',
{'d': 3,
'ops': [['add', 900000014, 3],
['add', 2700000042, 2],
['q', 900000014],
['q', 2700000042],
['q', 7]],
'w': 8},
[['ok', 'ok', 3, 2, 0],
5,
2,
[[0, 0, 0, 2, 0, 0, 0, 3], [0, 0, 2, 3, 0, 0, 0, 0], [3, 0, 2, 0, 0, 0, 0, 0]]]],
['uneven rows',
{'d': 4,
'ops': [['add', 2, 3],
['add', 9, 5],
['add', 2, 2],
['add', 5, 3],
['q', 2],
['q', 9],
['q', 5]],
'w': 4},
[['ok', 'ok', 'ok', 'ok', 5, 5, 3],
13,
9,
[[5, 0, 5, 0], [5, 0, 0, 5], [0, 0, 5, 3], [5, 0, 5, 0]]]],
['single row',
{'d': 1, 'ops': [['add', 1, 2], ['add', 4, 3], ['q', 1], ['add', 0, 0]], 'w': 3},
[['ok', 'ok', 2, 'rejected'], 5, 5, [[2, 3, 0]]]],
['many keys',
{'d': 3,
'ops': [['add', 0, 1],
['add', 1, 3],
['add', 2, 1],
['add', 3, 3],
['add', 4, 1],
['add', 5, 3],
['add', 6, 1],
['add', 7, 3],
['add', 8, 1],
['add', 9, 3],
['add', 10, 1],
['add', 11, 3],
['q', 0],
['q', 3],
['q', 6],
['q', 9],
['add', 2, 0]],
'w': 5},
[['ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
1,
3,
3,
3,
'rejected'],
24,
14,
[[4, 3, 3, 3, 3], [3, 3, 4, 4, 3], [0, 3, 1, 4, 3]]]]],
[['colliding keys',
{'d': 3,
'ops': [['add', 1, 4], ['add', 6, 2], ['add', 1, 1], ['q', 1], ['q', 6], ['q', 11]],
'w': 5},
[['ok', 'ok', 'ok', 5, 2, 0], 7, 4, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]]],
['zero and negative counts',
{'d': 2, 'ops': [['add', 3, 0], ['add', 3, -2], ['add', 3, 5], ['q', 3]], 'w': 7},
[['rejected', 'rejected', 'ok', 5], 5, 2, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]]],
['huge keys wrap prime',
{'d': 3,
'ops': [['add', 900000021, 3],
['add', 2700000063, 2],
['q', 900000021],
['q', 2700000063],
['q', 7]],
'w': 9},
[['ok', 'ok', 3, 2, 0],
5,
2,
[[0, 0, 2, 0, 0, 3, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 2, 0, 0]]]],
['uneven rows',
{'d': 4,
'ops': [['add', 2, 3],
['add', 9, 5],
['add', 2, 2],
['add', 5, 4],
['q', 2],
['q', 9],
['q', 5]],
'w': 4},
[['ok', 'ok', 'ok', 'ok', 5, 5, 4],
14,
10,
[[5, 0, 5, 0], [5, 0, 0, 5], [0, 0, 5, 4], [5, 0, 5, 0]]]],
['single row',
{'d': 1, 'ops': [['add', 1, 2], ['add', 4, 3], ['q', 1], ['add', 0, 0]], 'w': 3},
[['ok', 'ok', 2, 'rejected'], 5, 5, [[2, 3, 0]]]],
['many keys',
{'d': 3,
'ops': [['add', 0, 1],
['add', 1, 4],
['add', 2, 3],
['add', 3, 2],
['add', 4, 1],
['add', 5, 4],
['add', 6, 3],
['add', 7, 2],
['add', 8, 1],
['add', 9, 4],
['add', 10, 3],
['add', 11, 2],
['q', 0],
['q', 3],
['q', 6],
['q', 9],
['add', 2, 0]],
'w': 5},
[['ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
1,
3,
3,
4,
'rejected'],
30,
17,
[[5, 4, 4, 3, 3], [4, 3, 5, 3, 4], [0, 4, 1, 5, 4]]]]],
[['colliding keys',
{'d': 3,
'ops': [['add', 1, 4], ['add', 6, 2], ['add', 1, 1], ['q', 1], ['q', 6], ['q', 11]],
'w': 5},
[['ok', 'ok', 'ok', 5, 2, 0], 7, 4, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]]],
['zero and negative counts',
{'d': 2, 'ops': [['add', 3, 0], ['add', 3, -2], ['add', 3, 5], ['q', 3]], 'w': 7},
[['rejected', 'rejected', 'ok', 5], 5, 2, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]]],
['huge keys wrap prime',
{'d': 3,
'ops': [['add', 900000028, 3],
['add', 2700000084, 2],
['q', 900000028],
['q', 2700000084],
['q', 7]],
'w': 10},
[['ok', 'ok', 3, 2, 0],
5,
2,
[[0, 0, 0, 0, 2, 0, 0, 0, 3, 0],
[0, 0, 0, 0, 0, 3, 2, 0, 0, 0],
[0, 0, 0, 0, 3, 0, 0, 0, 0, 0]]]],
['uneven rows',
{'d': 4,
'ops': [['add', 2, 3],
['add', 9, 5],
['add', 2, 2],
['add', 5, 5],
['q', 2],
['q', 9],
['q', 5]],
'w': 4},
[['ok', 'ok', 'ok', 'ok', 5, 5, 5],
15,
11,
[[5, 0, 5, 0], [5, 0, 0, 5], [0, 0, 5, 5], [5, 0, 5, 0]]]],
['single row',
{'d': 1, 'ops': [['add', 1, 2], ['add', 4, 3], ['q', 1], ['add', 0, 0]], 'w': 3},
[['ok', 'ok', 2, 'rejected'], 5, 5, [[2, 3, 0]]]],
['many keys',
{'d': 3,
'ops': [['add', 0, 1],
['add', 1, 1],
['add', 2, 1],
['add', 3, 1],
['add', 4, 1],
['add', 5, 1],
['add', 6, 1],
['add', 7, 1],
['add', 8, 1],
['add', 9, 1],
['add', 10, 1],
['add', 11, 1],
['q', 0],
['q', 3],
['q', 6],
['q', 9],
['add', 2, 0]],
'w': 5},
[['ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
1,
1,
1,
1,
'rejected'],
12,
7,
[[2, 1, 1, 1, 1], [1, 1, 2, 2, 1], [0, 1, 1, 2, 1]]]]],
[['colliding keys',
{'d': 3,
'ops': [['add', 1, 4], ['add', 6, 2], ['add', 1, 1], ['q', 1], ['q', 6], ['q', 11]],
'w': 5},
[['ok', 'ok', 'ok', 5, 2, 0], 7, 4, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]]],
['zero and negative counts',
{'d': 2, 'ops': [['add', 3, 0], ['add', 3, -2], ['add', 3, 5], ['q', 3]], 'w': 7},
[['rejected', 'rejected', 'ok', 5], 5, 2, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]]],
['huge keys wrap prime',
{'d': 3,
'ops': [['add', 900000035, 3],
['add', 2700000105, 2],
['q', 900000035],
['q', 2700000105],
['q', 7]],
'w': 11},
[['ok', 'ok', 3, 2, 0],
5,
2,
[[0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2],
[0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0]]]],
['uneven rows',
{'d': 4,
'ops': [['add', 2, 3],
['add', 9, 5],
['add', 2, 2],
['add', 5, 6],
['q', 2],
['q', 9],
['q', 5]],
'w': 4},
[['ok', 'ok', 'ok', 'ok', 5, 5, 6],
16,
11,
[[6, 0, 5, 0], [6, 0, 0, 5], [0, 0, 5, 6], [6, 0, 5, 0]]]],
['single row',
{'d': 1, 'ops': [['add', 1, 2], ['add', 4, 3], ['q', 1], ['add', 0, 0]], 'w': 3},
[['ok', 'ok', 2, 'rejected'], 5, 5, [[2, 3, 0]]]],
['many keys',
{'d': 3,
'ops': [['add', 0, 1],
['add', 1, 2],
['add', 2, 3],
['add', 3, 4],
['add', 4, 1],
['add', 5, 2],
['add', 6, 3],
['add', 7, 4],
['add', 8, 1],
['add', 9, 2],
['add', 10, 3],
['add', 11, 4],
['q', 0],
['q', 3],
['q', 6],
['q', 9],
['add', 2, 0]],
'w': 5},
[['ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
1,
4,
4,
2,
'rejected'],
30,
17,
[[5, 2, 2, 4, 4], [3, 4, 5, 5, 2], [0, 4, 1, 5, 4]]]]]]
for label, args, expected in cases[N - 1]:
check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| colliding keys | [['ok', 'ok', 'ok', 5, 2, 0], 7, 2, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]] | [['ok', 'ok', 'ok', 5, 2, 0], 7, 4, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]] | Failed |
| zero and negative counts | [['rejected', 'rejected', 'ok', 5], 5, 1, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]] | [['rejected', 'rejected', 'ok', 5], 5, 2, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]] | Failed |
| huge keys wrap prime | [['ok', 'ok', 3, 2, 0], 5, 1, [[0, 3, 0, 0, 0, 0, 0], [2, 0, 0, 0, 3, 0, 0], [2, 0, 0, 3, 0, 0, 0]]] | [['ok', 'ok', 3, 2, 0], 5, 2, [[0, 3, 0, 0, 0, 0, 0], [2, 0, 0, 0, 3, 0, 0], [2, 0, 0, 3, 0, 0, 0]]] | Failed |
| uneven rows | [['ok', 'ok', 'ok', 'ok', 5, 5, 2], 12, 3, [[5, 0, 5, 0], [5, 0, 0, 5], [0, 0, 5, 2], [5, 0, 5, 0]]] | [['ok', 'ok', 'ok', 'ok', 5, 5, 2], 12, 9, [[5, 0, 5, 0], [5, 0, 0, 5], [0, 0, 5, 2], [5, 0, 5, 0]]] | Failed |
| single row | [['ok', 'ok', 2, 'rejected'], 5, 2, [[2, 3, 0]]] | [['ok', 'ok', 2, 'rejected'], 5, 5, [[2, 3, 0]]] | Failed |
| many keys | [['ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 1, 4, 4, 2, 'rejected'], 30, 6, [[5, 2, 2, 4, 4], [3, 4, 5, 5, 2], [0, 4, 1, 5, 4]]] | [['ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 1, 4, 4, 2, 'rejected'], 30, 17, [[5, 2, 2, 4, 4], [3, 4, 5, 5, 2], [0, 4, 1, 5, 4]]] | Failed |
SHA-256 / d2413f9802b88351773c8ecf8931d68d551dd7c42f96f87a841b93eff792be0d
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
w = x['w']
A = [1103515245, 1812433253, 1664525017, 2013265921, 1597334677]
B = [12345, 1013904223, 7, 99991, 31337]
P = 2147483647
d = x['d']
t = [[0] * w for _ in range(d)]
def bucket(r, key):
return (A[r] * key + B[r]) % P % w
total = 0
res = []
for op in x['ops']:
if op[0] == 'add':
key, cnt = op[1], op[2]
if cnt <= 0:
res.append('rejected')
continue
est = min(t[r][bucket(r, key)] for r in range(d))
for r in range(d):
b = bucket(r, key)
t[r][b] = max(t[r][b], est + cnt)
total += cnt
res.append('ok')
else:
res.append(min(t[r][bucket(r, op[1])] for r in range(d)))
return [res, total, math.ceil(math.e / d * total), t]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[['colliding keys',
{'d': 3,
'ops': [['add', 1, 4], ['add', 6, 2], ['add', 1, 1], ['q', 1], ['q', 6], ['q', 11]],
'w': 5},
[['ok', 'ok', 'ok', 5, 2, 0], 7, 4, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]]],
['zero and negative counts',
{'d': 2, 'ops': [['add', 3, 0], ['add', 3, -2], ['add', 3, 5], ['q', 3]], 'w': 7},
[['rejected', 'rejected', 'ok', 5], 5, 2, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]]],
['huge keys wrap prime',
{'d': 3,
'ops': [['add', 900000007, 3],
['add', 2700000021, 2],
['q', 900000007],
['q', 2700000021],
['q', 7]],
'w': 7},
[['ok', 'ok', 3, 2, 0],
5,
2,
[[0, 3, 0, 0, 0, 0, 0], [2, 0, 0, 0, 3, 0, 0], [2, 0, 0, 3, 0, 0, 0]]]],
['uneven rows',
{'d': 4,
'ops': [['add', 2, 3],
['add', 9, 5],
['add', 2, 2],
['add', 5, 2],
['q', 2],
['q', 9],
['q', 5]],
'w': 4},
[['ok', 'ok', 'ok', 'ok', 5, 5, 2],
12,
9,
[[5, 0, 5, 0], [5, 0, 0, 5], [0, 0, 5, 2], [5, 0, 5, 0]]]],
['single row',
{'d': 1, 'ops': [['add', 1, 2], ['add', 4, 3], ['q', 1], ['add', 0, 0]], 'w': 3},
[['ok', 'ok', 2, 'rejected'], 5, 5, [[2, 3, 0]]]],
['many keys',
{'d': 3,
'ops': [['add', 0, 1],
['add', 1, 2],
['add', 2, 3],
['add', 3, 4],
['add', 4, 1],
['add', 5, 2],
['add', 6, 3],
['add', 7, 4],
['add', 8, 1],
['add', 9, 2],
['add', 10, 3],
['add', 11, 4],
['q', 0],
['q', 3],
['q', 6],
['q', 9],
['add', 2, 0]],
'w': 5},
[['ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
1,
4,
4,
2,
'rejected'],
30,
17,
[[5, 2, 2, 4, 4], [3, 4, 5, 5, 2], [0, 4, 1, 5, 4]]]]],
[['colliding keys',
{'d': 3,
'ops': [['add', 1, 4], ['add', 6, 2], ['add', 1, 1], ['q', 1], ['q', 6], ['q', 11]],
'w': 5},
[['ok', 'ok', 'ok', 5, 2, 0], 7, 4, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]]],
['zero and negative counts',
{'d': 2, 'ops': [['add', 3, 0], ['add', 3, -2], ['add', 3, 5], ['q', 3]], 'w': 7},
[['rejected', 'rejected', 'ok', 5], 5, 2, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]]],
['huge keys wrap prime',
{'d': 3,
'ops': [['add', 900000014, 3],
['add', 2700000042, 2],
['q', 900000014],
['q', 2700000042],
['q', 7]],
'w': 8},
[['ok', 'ok', 3, 2, 0],
5,
2,
[[0, 0, 0, 2, 0, 0, 0, 3], [0, 0, 2, 3, 0, 0, 0, 0], [3, 0, 2, 0, 0, 0, 0, 0]]]],
['uneven rows',
{'d': 4,
'ops': [['add', 2, 3],
['add', 9, 5],
['add', 2, 2],
['add', 5, 3],
['q', 2],
['q', 9],
['q', 5]],
'w': 4},
[['ok', 'ok', 'ok', 'ok', 5, 5, 3],
13,
9,
[[5, 0, 5, 0], [5, 0, 0, 5], [0, 0, 5, 3], [5, 0, 5, 0]]]],
['single row',
{'d': 1, 'ops': [['add', 1, 2], ['add', 4, 3], ['q', 1], ['add', 0, 0]], 'w': 3},
[['ok', 'ok', 2, 'rejected'], 5, 5, [[2, 3, 0]]]],
['many keys',
{'d': 3,
'ops': [['add', 0, 1],
['add', 1, 3],
['add', 2, 1],
['add', 3, 3],
['add', 4, 1],
['add', 5, 3],
['add', 6, 1],
['add', 7, 3],
['add', 8, 1],
['add', 9, 3],
['add', 10, 1],
['add', 11, 3],
['q', 0],
['q', 3],
['q', 6],
['q', 9],
['add', 2, 0]],
'w': 5},
[['ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
1,
3,
3,
3,
'rejected'],
24,
14,
[[4, 3, 3, 3, 3], [3, 3, 4, 4, 3], [0, 3, 1, 4, 3]]]]],
[['colliding keys',
{'d': 3,
'ops': [['add', 1, 4], ['add', 6, 2], ['add', 1, 1], ['q', 1], ['q', 6], ['q', 11]],
'w': 5},
[['ok', 'ok', 'ok', 5, 2, 0], 7, 4, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]]],
['zero and negative counts',
{'d': 2, 'ops': [['add', 3, 0], ['add', 3, -2], ['add', 3, 5], ['q', 3]], 'w': 7},
[['rejected', 'rejected', 'ok', 5], 5, 2, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]]],
['huge keys wrap prime',
{'d': 3,
'ops': [['add', 900000021, 3],
['add', 2700000063, 2],
['q', 900000021],
['q', 2700000063],
['q', 7]],
'w': 9},
[['ok', 'ok', 3, 2, 0],
5,
2,
[[0, 0, 2, 0, 0, 3, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 2, 0, 0]]]],
['uneven rows',
{'d': 4,
'ops': [['add', 2, 3],
['add', 9, 5],
['add', 2, 2],
['add', 5, 4],
['q', 2],
['q', 9],
['q', 5]],
'w': 4},
[['ok', 'ok', 'ok', 'ok', 5, 5, 4],
14,
10,
[[5, 0, 5, 0], [5, 0, 0, 5], [0, 0, 5, 4], [5, 0, 5, 0]]]],
['single row',
{'d': 1, 'ops': [['add', 1, 2], ['add', 4, 3], ['q', 1], ['add', 0, 0]], 'w': 3},
[['ok', 'ok', 2, 'rejected'], 5, 5, [[2, 3, 0]]]],
['many keys',
{'d': 3,
'ops': [['add', 0, 1],
['add', 1, 4],
['add', 2, 3],
['add', 3, 2],
['add', 4, 1],
['add', 5, 4],
['add', 6, 3],
['add', 7, 2],
['add', 8, 1],
['add', 9, 4],
['add', 10, 3],
['add', 11, 2],
['q', 0],
['q', 3],
['q', 6],
['q', 9],
['add', 2, 0]],
'w': 5},
[['ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
1,
3,
3,
4,
'rejected'],
30,
17,
[[5, 4, 4, 3, 3], [4, 3, 5, 3, 4], [0, 4, 1, 5, 4]]]]],
[['colliding keys',
{'d': 3,
'ops': [['add', 1, 4], ['add', 6, 2], ['add', 1, 1], ['q', 1], ['q', 6], ['q', 11]],
'w': 5},
[['ok', 'ok', 'ok', 5, 2, 0], 7, 4, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]]],
['zero and negative counts',
{'d': 2, 'ops': [['add', 3, 0], ['add', 3, -2], ['add', 3, 5], ['q', 3]], 'w': 7},
[['rejected', 'rejected', 'ok', 5], 5, 2, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]]],
['huge keys wrap prime',
{'d': 3,
'ops': [['add', 900000028, 3],
['add', 2700000084, 2],
['q', 900000028],
['q', 2700000084],
['q', 7]],
'w': 10},
[['ok', 'ok', 3, 2, 0],
5,
2,
[[0, 0, 0, 0, 2, 0, 0, 0, 3, 0],
[0, 0, 0, 0, 0, 3, 2, 0, 0, 0],
[0, 0, 0, 0, 3, 0, 0, 0, 0, 0]]]],
['uneven rows',
{'d': 4,
'ops': [['add', 2, 3],
['add', 9, 5],
['add', 2, 2],
['add', 5, 5],
['q', 2],
['q', 9],
['q', 5]],
'w': 4},
[['ok', 'ok', 'ok', 'ok', 5, 5, 5],
15,
11,
[[5, 0, 5, 0], [5, 0, 0, 5], [0, 0, 5, 5], [5, 0, 5, 0]]]],
['single row',
{'d': 1, 'ops': [['add', 1, 2], ['add', 4, 3], ['q', 1], ['add', 0, 0]], 'w': 3},
[['ok', 'ok', 2, 'rejected'], 5, 5, [[2, 3, 0]]]],
['many keys',
{'d': 3,
'ops': [['add', 0, 1],
['add', 1, 1],
['add', 2, 1],
['add', 3, 1],
['add', 4, 1],
['add', 5, 1],
['add', 6, 1],
['add', 7, 1],
['add', 8, 1],
['add', 9, 1],
['add', 10, 1],
['add', 11, 1],
['q', 0],
['q', 3],
['q', 6],
['q', 9],
['add', 2, 0]],
'w': 5},
[['ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
1,
1,
1,
1,
'rejected'],
12,
7,
[[2, 1, 1, 1, 1], [1, 1, 2, 2, 1], [0, 1, 1, 2, 1]]]]],
[['colliding keys',
{'d': 3,
'ops': [['add', 1, 4], ['add', 6, 2], ['add', 1, 1], ['q', 1], ['q', 6], ['q', 11]],
'w': 5},
[['ok', 'ok', 'ok', 5, 2, 0], 7, 4, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]]],
['zero and negative counts',
{'d': 2, 'ops': [['add', 3, 0], ['add', 3, -2], ['add', 3, 5], ['q', 3]], 'w': 7},
[['rejected', 'rejected', 'ok', 5], 5, 2, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]]],
['huge keys wrap prime',
{'d': 3,
'ops': [['add', 900000035, 3],
['add', 2700000105, 2],
['q', 900000035],
['q', 2700000105],
['q', 7]],
'w': 11},
[['ok', 'ok', 3, 2, 0],
5,
2,
[[0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2],
[0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0]]]],
['uneven rows',
{'d': 4,
'ops': [['add', 2, 3],
['add', 9, 5],
['add', 2, 2],
['add', 5, 6],
['q', 2],
['q', 9],
['q', 5]],
'w': 4},
[['ok', 'ok', 'ok', 'ok', 5, 5, 6],
16,
11,
[[6, 0, 5, 0], [6, 0, 0, 5], [0, 0, 5, 6], [6, 0, 5, 0]]]],
['single row',
{'d': 1, 'ops': [['add', 1, 2], ['add', 4, 3], ['q', 1], ['add', 0, 0]], 'w': 3},
[['ok', 'ok', 2, 'rejected'], 5, 5, [[2, 3, 0]]]],
['many keys',
{'d': 3,
'ops': [['add', 0, 1],
['add', 1, 2],
['add', 2, 3],
['add', 3, 4],
['add', 4, 1],
['add', 5, 2],
['add', 6, 3],
['add', 7, 4],
['add', 8, 1],
['add', 9, 2],
['add', 10, 3],
['add', 11, 4],
['q', 0],
['q', 3],
['q', 6],
['q', 9],
['add', 2, 0]],
'w': 5},
[['ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
'ok',
1,
4,
4,
2,
'rejected'],
30,
17,
[[5, 2, 2, 4, 4], [3, 4, 5, 5, 2], [0, 4, 1, 5, 4]]]]]]
for label, args, expected in cases[N - 1]:
check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| colliding keys | [['ok', 'ok', 'ok', 5, 2, 0], 7, 7, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]] | [['ok', 'ok', 'ok', 5, 2, 0], 7, 4, [[5, 0, 0, 0, 2], [0, 2, 0, 0, 5], [0, 2, 0, 0, 5]]] | Failed |
| zero and negative counts | [['rejected', 'rejected', 'ok', 5], 5, 7, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]] | [['rejected', 'rejected', 'ok', 5], 5, 2, [[0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0]]] | Failed |
| huge keys wrap prime | [['ok', 'ok', 3, 2, 0], 5, 5, [[0, 3, 0, 0, 0, 0, 0], [2, 0, 0, 0, 3, 0, 0], [2, 0, 0, 3, 0, 0, 0]]] | [['ok', 'ok', 3, 2, 0], 5, 2, [[0, 3, 0, 0, 0, 0, 0], [2, 0, 0, 0, 3, 0, 0], [2, 0, 0, 3, 0, 0, 0]]] | Failed |
| uneven rows | [['ok', 'ok', 'ok', 'ok', 5, 5, 2], 12, 9, [[5, 0, 5, 0], [5, 0, 0, 5], [0, 0, 5, 2], [5, 0, 5, 0]]] | [['ok', 'ok', 'ok', 'ok', 5, 5, 2], 12, 9, [[5, 0, 5, 0], [5, 0, 0, 5], [0, 0, 5, 2], [5, 0, 5, 0]]] | Passed |
| single row | [['ok', 'ok', 2, 'rejected'], 5, 14, [[2, 3, 0]]] | [['ok', 'ok', 2, 'rejected'], 5, 5, [[2, 3, 0]]] | Failed |
| many keys | [['ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 1, 4, 4, 2, 'rejected'], 30, 28, [[5, 2, 2, 4, 4], [3, 4, 5, 5, 2], [0, 4, 1, 5, 4]]] | [['ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 'ok', 1, 4, 4, 2, 'rejected'], 30, 17, [[5, 2, 2, 4, 4], [3, 4, 5, 5, 2], [0, 4, 1, 5, 4]]] | Failed |
SHA-256 / e3c8ca60fef5ed940b6f899357f3f14fb3eb5dfa1fb4b24624af4f7ac53614f9
HELD IN THE MEMBER ARCHIVE
The verified repair and its recorded checks are member-only.
This mechanism has 6 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.
Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.
Member access is invitation-based. Sign in with your invited account to inspect the repair.
Sign in to the archive ↗Verification & scope
A deterministic, bounded teaching model with stipulated constants and pre-hashed or explicitly hashed inputs; it is not a production implementation and makes no claim of conformance to any library or paper beyond the stated contract. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.
Observations recorded using Python 3.12.14 at 2026-09-29T14:48:44.007104+00:00.
Case digest / e86916fbc4b902b7376272bd05d83cd0b787233fcefe73a3641c1720dca0ab56