FA-73176 / Probabilistic sketches / Open access
Weighted reservoir sampling by exponential keys: negative weights enter the sample · case 01
An item with negative weight gets a key above every legitimate item and is always sampled.
ROOT CAUSE
Only zero weights are excluded; negative weights produce inverted keys.
THE FAILURE
Only zero weights are excluded; negative weights produce inverted keys.
Unsuccessful approach: Substituting a tiny weight keeps ineligible items in the sample whenever fewer than k items are eligible.
Case contract
Input {k, items} where each item is [id, weight, num, den] with u = num/den in (0,1). Items with weight <= 0 are ineligible. Each eligible item gets key u^(1/weight), compared as ln(u)/weight. Return the ids of the k largest keys, largest first, ties broken by smaller id.
Why this case matters
Weighted sampling without replacement (A-Res) chooses representative traces or accounts in proportion to traffic; key or eligibility mistakes skew the sample.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
k = x['k']
keyed = []
for ident, wt, num, den in x['items']:
if wt == 0:
continue
key = math.log(num / den) / wt
keyed.append((key, -ident))
keyed.sort(reverse=True)
return [-negid for key, negid in keyed[:k]]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[['weighted sample 0 k=2',
{'items': [[0, 0.5, 91, 99],
[1, 1, 42, 99],
[2, 5, 91, 99],
[3, 0.5, 48, 99],
[4, 0.5, 13, 99],
[5, 3, 4, 99],
[6, 3, 63, 99],
[7, 2, 87, 99]],
'k': 2},
[2, 7]],
['weighted sample 1 k=3',
{'items': [[0, 1, 65, 99],
[1, 0.5, 34, 99],
[2, 2, 22, 99],
[3, 1, 43, 99],
[4, 1, 77, 99],
[5, 0.5, 50, 99],
[6, 5, 2, 99],
[7, 3, 76, 99]],
'k': 3},
[7, 4, 0]],
['weighted sample 2 k=4',
{'items': [[0, 1, 74, 99],
[1, 3, 25, 99],
[2, 3, 5, 99],
[3, 0.5, 53, 99],
[4, 1, 61, 99],
[5, 1, 6, 99],
[6, 5, 34, 99],
[7, 3, 56, 99]],
'k': 4},
[7, 6, 0, 1]],
['zero and negative weights',
{'items': [[1, 0, 50, 99], [2, -1, 30, 99], [3, 2, 41, 99], [4, 1, 90, 99]], 'k': 3},
[4, 3]],
['equal keys tie',
{'items': [[7, 2, 49, 100], [3, 2, 49, 100], [5, 1, 11, 100]], 'k': 2},
[3, 7]],
['heavier weight wins', {'items': [[1, 1, 60, 100], [2, 4, 21, 100]], 'k': 1}, [2]],
['light weight small uniform',
{'items': [[1, 0.25, 90, 100], [2, 3, 31, 100], [3, 1, 70, 100]], 'k': 2},
[3, 2]]],
[['weighted sample 0 k=2',
{'items': [[0, 1, 8, 99],
[1, 5, 66, 99],
[2, 5, 27, 99],
[3, 5, 16, 99],
[4, 0.5, 70, 99],
[5, 3, 4, 99],
[6, 0.5, 5, 99],
[7, 0.5, 32, 99]],
'k': 2},
[1, 2]],
['weighted sample 1 k=3',
{'items': [[0, 0.5, 34, 99],
[1, 0.5, 41, 99],
[2, 5, 97, 99],
[3, 3, 35, 99],
[4, 3, 98, 99],
[5, 3, 86, 99],
[6, 0.5, 26, 99],
[7, 1, 66, 99]],
'k': 3},
[4, 2, 5]],
['weighted sample 2 k=4',
{'items': [[0, 2, 63, 99],
[1, 5, 82, 99],
[2, 0.5, 17, 99],
[3, 3, 50, 99],
[4, 3, 90, 99],
[5, 5, 18, 99],
[6, 1, 85, 99],
[7, 1, 8, 99]],
'k': 4},
[4, 1, 6, 0]],
['zero and negative weights',
{'items': [[1, 0, 50, 99], [2, -1, 30, 99], [3, 2, 42, 99], [4, 1, 90, 99]], 'k': 3},
[4, 3]],
['equal keys tie',
{'items': [[7, 2, 49, 100], [3, 2, 49, 100], [5, 1, 12, 100]], 'k': 2},
[3, 7]],
['heavier weight wins', {'items': [[1, 1, 60, 100], [2, 4, 22, 100]], 'k': 1}, [2]],
['light weight small uniform',
{'items': [[1, 0.25, 90, 100], [2, 3, 32, 100], [3, 1, 70, 100]], 'k': 2},
[3, 2]]],
[['weighted sample 0 k=2',
{'items': [[0, 2, 38, 99],
[1, 2, 4, 99],
[2, 5, 4, 99],
[3, 5, 92, 99],
[4, 5, 71, 99],
[5, 1, 36, 99],
[6, 2, 24, 99],
[7, 3, 79, 99]],
'k': 2},
[3, 4]],
['weighted sample 1 k=3',
{'items': [[0, 1, 74, 99],
[1, 5, 71, 99],
[2, 3, 25, 99],
[3, 0.5, 97, 99],
[4, 5, 51, 99],
[5, 1, 86, 99],
[6, 1, 82, 99],
[7, 5, 81, 99]],
'k': 3},
[7, 3, 1]],
['weighted sample 2 k=4',
{'items': [[0, 5, 98, 99],
[1, 2, 23, 99],
[2, 5, 17, 99],
[3, 2, 35, 99],
[4, 0.5, 16, 99],
[5, 3, 77, 99],
[6, 3, 56, 99],
[7, 0.5, 87, 99]],
'k': 4},
[0, 5, 6, 7]],
['zero and negative weights',
{'items': [[1, 0, 50, 99], [2, -1, 30, 99], [3, 2, 43, 99], [4, 1, 90, 99]], 'k': 3},
[4, 3]],
['equal keys tie',
{'items': [[7, 2, 49, 100], [3, 2, 49, 100], [5, 1, 13, 100]], 'k': 2},
[3, 7]],
['heavier weight wins', {'items': [[1, 1, 60, 100], [2, 4, 23, 100]], 'k': 1}, [2]],
['light weight small uniform',
{'items': [[1, 0.25, 90, 100], [2, 3, 33, 100], [3, 1, 70, 100]], 'k': 2},
[3, 2]]],
[['weighted sample 0 k=2',
{'items': [[0, 5, 55, 99],
[1, 2, 83, 99],
[2, 2, 87, 99],
[3, 3, 17, 99],
[4, 1, 74, 99],
[5, 0.5, 82, 99],
[6, 2, 71, 99],
[7, 1, 7, 99]],
'k': 2},
[2, 1]],
['weighted sample 1 k=3',
{'items': [[0, 1, 55, 99],
[1, 1, 50, 99],
[2, 2, 55, 99],
[3, 1, 34, 99],
[4, 1, 58, 99],
[5, 3, 47, 99],
[6, 0.5, 66, 99],
[7, 2, 69, 99]],
'k': 3},
[7, 5, 2]],
['weighted sample 2 k=4',
{'items': [[0, 3, 42, 99],
[1, 0.5, 69, 99],
[2, 3, 41, 99],
[3, 0.5, 42, 99],
[4, 2, 48, 99],
[5, 5, 91, 99],
[6, 5, 74, 99],
[7, 3, 31, 99]],
'k': 4},
[5, 6, 0, 2]],
['zero and negative weights',
{'items': [[1, 0, 50, 99], [2, -1, 30, 99], [3, 2, 44, 99], [4, 1, 90, 99]], 'k': 3},
[4, 3]],
['equal keys tie',
{'items': [[7, 2, 49, 100], [3, 2, 49, 100], [5, 1, 14, 100]], 'k': 2},
[3, 7]],
['heavier weight wins', {'items': [[1, 1, 60, 100], [2, 4, 24, 100]], 'k': 1}, [2]],
['light weight small uniform',
{'items': [[1, 0.25, 90, 100], [2, 3, 34, 100], [3, 1, 70, 100]], 'k': 2},
[3, 2]]],
[['weighted sample 0 k=2',
{'items': [[0, 1, 87, 99],
[1, 1, 1, 99],
[2, 2, 6, 99],
[3, 5, 67, 99],
[4, 1, 66, 99],
[5, 3, 45, 99],
[6, 0.5, 25, 99],
[7, 1, 41, 99]],
'k': 2},
[3, 0]],
['weighted sample 1 k=3',
{'items': [[0, 1, 25, 99],
[1, 3, 18, 99],
[2, 5, 2, 99],
[3, 2, 65, 99],
[4, 3, 14, 99],
[5, 3, 89, 99],
[6, 3, 10, 99],
[7, 1, 78, 99]],
'k': 3},
[5, 3, 7]],
['weighted sample 2 k=4',
{'items': [[0, 2, 49, 99],
[1, 2, 66, 99],
[2, 0.5, 24, 99],
[3, 0.5, 88, 99],
[4, 1, 6, 99],
[5, 1, 8, 99],
[6, 1, 93, 99],
[7, 0.5, 41, 99]],
'k': 4},
[6, 1, 3, 0]],
['zero and negative weights',
{'items': [[1, 0, 50, 99], [2, -1, 30, 99], [3, 2, 45, 99], [4, 1, 90, 99]], 'k': 3},
[4, 3]],
['equal keys tie',
{'items': [[7, 2, 49, 100], [3, 2, 49, 100], [5, 1, 15, 100]], 'k': 2},
[3, 7]],
['heavier weight wins', {'items': [[1, 1, 60, 100], [2, 4, 25, 100]], 'k': 1}, [2]],
['light weight small uniform',
{'items': [[1, 0.25, 90, 100], [2, 3, 35, 100], [3, 1, 70, 100]], 'k': 2},
[2, 3]]]]
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 |
|---|---|---|---|
| weighted sample 0 k=2 | [2, 7] | [2, 7] | Passed |
| weighted sample 1 k=3 | [7, 4, 0] | [7, 4, 0] | Passed |
| weighted sample 2 k=4 | [7, 6, 0, 1] | [7, 6, 0, 1] | Passed |
| zero and negative weights | [2, 4, 3] | [4, 3] | Failed |
| equal keys tie | [3, 7] | [3, 7] | Passed |
| heavier weight wins | [2] | [2] | Passed |
| light weight small uniform | [3, 2] | [3, 2] | Passed |
SHA-256 / f7182fbc41800821b2fc676d8ab0d19a45382d979c5c47252b2aa0b246903e5f
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
k = x['k']
keyed = []
for ident, wt, num, den in x['items']:
if wt <= 0:
wt = 1e-9
key = math.log(num / den) / wt
keyed.append((key, -ident))
keyed.sort(reverse=True)
return [-negid for key, negid in keyed[:k]]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
cases = [[['weighted sample 0 k=2',
{'items': [[0, 0.5, 91, 99],
[1, 1, 42, 99],
[2, 5, 91, 99],
[3, 0.5, 48, 99],
[4, 0.5, 13, 99],
[5, 3, 4, 99],
[6, 3, 63, 99],
[7, 2, 87, 99]],
'k': 2},
[2, 7]],
['weighted sample 1 k=3',
{'items': [[0, 1, 65, 99],
[1, 0.5, 34, 99],
[2, 2, 22, 99],
[3, 1, 43, 99],
[4, 1, 77, 99],
[5, 0.5, 50, 99],
[6, 5, 2, 99],
[7, 3, 76, 99]],
'k': 3},
[7, 4, 0]],
['weighted sample 2 k=4',
{'items': [[0, 1, 74, 99],
[1, 3, 25, 99],
[2, 3, 5, 99],
[3, 0.5, 53, 99],
[4, 1, 61, 99],
[5, 1, 6, 99],
[6, 5, 34, 99],
[7, 3, 56, 99]],
'k': 4},
[7, 6, 0, 1]],
['zero and negative weights',
{'items': [[1, 0, 50, 99], [2, -1, 30, 99], [3, 2, 41, 99], [4, 1, 90, 99]], 'k': 3},
[4, 3]],
['equal keys tie',
{'items': [[7, 2, 49, 100], [3, 2, 49, 100], [5, 1, 11, 100]], 'k': 2},
[3, 7]],
['heavier weight wins', {'items': [[1, 1, 60, 100], [2, 4, 21, 100]], 'k': 1}, [2]],
['light weight small uniform',
{'items': [[1, 0.25, 90, 100], [2, 3, 31, 100], [3, 1, 70, 100]], 'k': 2},
[3, 2]]],
[['weighted sample 0 k=2',
{'items': [[0, 1, 8, 99],
[1, 5, 66, 99],
[2, 5, 27, 99],
[3, 5, 16, 99],
[4, 0.5, 70, 99],
[5, 3, 4, 99],
[6, 0.5, 5, 99],
[7, 0.5, 32, 99]],
'k': 2},
[1, 2]],
['weighted sample 1 k=3',
{'items': [[0, 0.5, 34, 99],
[1, 0.5, 41, 99],
[2, 5, 97, 99],
[3, 3, 35, 99],
[4, 3, 98, 99],
[5, 3, 86, 99],
[6, 0.5, 26, 99],
[7, 1, 66, 99]],
'k': 3},
[4, 2, 5]],
['weighted sample 2 k=4',
{'items': [[0, 2, 63, 99],
[1, 5, 82, 99],
[2, 0.5, 17, 99],
[3, 3, 50, 99],
[4, 3, 90, 99],
[5, 5, 18, 99],
[6, 1, 85, 99],
[7, 1, 8, 99]],
'k': 4},
[4, 1, 6, 0]],
['zero and negative weights',
{'items': [[1, 0, 50, 99], [2, -1, 30, 99], [3, 2, 42, 99], [4, 1, 90, 99]], 'k': 3},
[4, 3]],
['equal keys tie',
{'items': [[7, 2, 49, 100], [3, 2, 49, 100], [5, 1, 12, 100]], 'k': 2},
[3, 7]],
['heavier weight wins', {'items': [[1, 1, 60, 100], [2, 4, 22, 100]], 'k': 1}, [2]],
['light weight small uniform',
{'items': [[1, 0.25, 90, 100], [2, 3, 32, 100], [3, 1, 70, 100]], 'k': 2},
[3, 2]]],
[['weighted sample 0 k=2',
{'items': [[0, 2, 38, 99],
[1, 2, 4, 99],
[2, 5, 4, 99],
[3, 5, 92, 99],
[4, 5, 71, 99],
[5, 1, 36, 99],
[6, 2, 24, 99],
[7, 3, 79, 99]],
'k': 2},
[3, 4]],
['weighted sample 1 k=3',
{'items': [[0, 1, 74, 99],
[1, 5, 71, 99],
[2, 3, 25, 99],
[3, 0.5, 97, 99],
[4, 5, 51, 99],
[5, 1, 86, 99],
[6, 1, 82, 99],
[7, 5, 81, 99]],
'k': 3},
[7, 3, 1]],
['weighted sample 2 k=4',
{'items': [[0, 5, 98, 99],
[1, 2, 23, 99],
[2, 5, 17, 99],
[3, 2, 35, 99],
[4, 0.5, 16, 99],
[5, 3, 77, 99],
[6, 3, 56, 99],
[7, 0.5, 87, 99]],
'k': 4},
[0, 5, 6, 7]],
['zero and negative weights',
{'items': [[1, 0, 50, 99], [2, -1, 30, 99], [3, 2, 43, 99], [4, 1, 90, 99]], 'k': 3},
[4, 3]],
['equal keys tie',
{'items': [[7, 2, 49, 100], [3, 2, 49, 100], [5, 1, 13, 100]], 'k': 2},
[3, 7]],
['heavier weight wins', {'items': [[1, 1, 60, 100], [2, 4, 23, 100]], 'k': 1}, [2]],
['light weight small uniform',
{'items': [[1, 0.25, 90, 100], [2, 3, 33, 100], [3, 1, 70, 100]], 'k': 2},
[3, 2]]],
[['weighted sample 0 k=2',
{'items': [[0, 5, 55, 99],
[1, 2, 83, 99],
[2, 2, 87, 99],
[3, 3, 17, 99],
[4, 1, 74, 99],
[5, 0.5, 82, 99],
[6, 2, 71, 99],
[7, 1, 7, 99]],
'k': 2},
[2, 1]],
['weighted sample 1 k=3',
{'items': [[0, 1, 55, 99],
[1, 1, 50, 99],
[2, 2, 55, 99],
[3, 1, 34, 99],
[4, 1, 58, 99],
[5, 3, 47, 99],
[6, 0.5, 66, 99],
[7, 2, 69, 99]],
'k': 3},
[7, 5, 2]],
['weighted sample 2 k=4',
{'items': [[0, 3, 42, 99],
[1, 0.5, 69, 99],
[2, 3, 41, 99],
[3, 0.5, 42, 99],
[4, 2, 48, 99],
[5, 5, 91, 99],
[6, 5, 74, 99],
[7, 3, 31, 99]],
'k': 4},
[5, 6, 0, 2]],
['zero and negative weights',
{'items': [[1, 0, 50, 99], [2, -1, 30, 99], [3, 2, 44, 99], [4, 1, 90, 99]], 'k': 3},
[4, 3]],
['equal keys tie',
{'items': [[7, 2, 49, 100], [3, 2, 49, 100], [5, 1, 14, 100]], 'k': 2},
[3, 7]],
['heavier weight wins', {'items': [[1, 1, 60, 100], [2, 4, 24, 100]], 'k': 1}, [2]],
['light weight small uniform',
{'items': [[1, 0.25, 90, 100], [2, 3, 34, 100], [3, 1, 70, 100]], 'k': 2},
[3, 2]]],
[['weighted sample 0 k=2',
{'items': [[0, 1, 87, 99],
[1, 1, 1, 99],
[2, 2, 6, 99],
[3, 5, 67, 99],
[4, 1, 66, 99],
[5, 3, 45, 99],
[6, 0.5, 25, 99],
[7, 1, 41, 99]],
'k': 2},
[3, 0]],
['weighted sample 1 k=3',
{'items': [[0, 1, 25, 99],
[1, 3, 18, 99],
[2, 5, 2, 99],
[3, 2, 65, 99],
[4, 3, 14, 99],
[5, 3, 89, 99],
[6, 3, 10, 99],
[7, 1, 78, 99]],
'k': 3},
[5, 3, 7]],
['weighted sample 2 k=4',
{'items': [[0, 2, 49, 99],
[1, 2, 66, 99],
[2, 0.5, 24, 99],
[3, 0.5, 88, 99],
[4, 1, 6, 99],
[5, 1, 8, 99],
[6, 1, 93, 99],
[7, 0.5, 41, 99]],
'k': 4},
[6, 1, 3, 0]],
['zero and negative weights',
{'items': [[1, 0, 50, 99], [2, -1, 30, 99], [3, 2, 45, 99], [4, 1, 90, 99]], 'k': 3},
[4, 3]],
['equal keys tie',
{'items': [[7, 2, 49, 100], [3, 2, 49, 100], [5, 1, 15, 100]], 'k': 2},
[3, 7]],
['heavier weight wins', {'items': [[1, 1, 60, 100], [2, 4, 25, 100]], 'k': 1}, [2]],
['light weight small uniform',
{'items': [[1, 0.25, 90, 100], [2, 3, 35, 100], [3, 1, 70, 100]], 'k': 2},
[2, 3]]]]
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 |
|---|---|---|---|
| weighted sample 0 k=2 | [2, 7] | [2, 7] | Passed |
| weighted sample 1 k=3 | [7, 4, 0] | [7, 4, 0] | Passed |
| weighted sample 2 k=4 | [7, 6, 0, 1] | [7, 6, 0, 1] | Passed |
| zero and negative weights | [4, 3, 1] | [4, 3] | Failed |
| equal keys tie | [3, 7] | [3, 7] | Passed |
| heavier weight wins | [2] | [2] | Passed |
| light weight small uniform | [3, 2] | [3, 2] | Passed |
SHA-256 / 7b6489884df358f4f35104f10e010b74d61853387dfdb8a3bbc9791e58fa8749
HELD IN THE MEMBER ARCHIVE
The verified repair and its recorded checks are member-only.
This mechanism has 7 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.
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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 2026-09-29T14:48:45.185660+00:00.
Case digest / 72b0c236bc7a9d6aef52f8cac79b38823f6266b324919dc0b61871fe98065e41