Balanced largest differencing method
Algorithms AIMech:BalancedLargestDifferencing DRAFT
Aliases: BLDM, Balanced LDM
A two-way number-partitioning heuristic that first differences disjoint adjacent pairs in the sorted original list, then repeatedly differences the two largest residuals, preserving subset cardinalities that differ by at most one.
Identity Notes & Concept Scope
Compared all 442 pinned AIO and 593 native IDs, names, aliases and scoped synonyms; no normalized identity match. Ignored- and hidden-inclusive fixed-string searches of the recovery, older discovery and original Documents checkouts returned 54, 0, 0 lines. All distinct match contents were inspected. Semantic matching covered 67 AIO and 100 native definitions, with every matched definition read. 6 full annotated AIO node/edge/generated contexts and 4 full native records were inspected. BLDM fixes one initial round of adjacent sorted pairs, then uses largest-residual differencing. This differs from PDM whole rounds throughout, unconstrained LDM/KK, and complete balanced or unconstrained search. Existing text hits are prior records and pending leads, not an existing identity. The fully inspected AIO contexts have no suitable counterpart.
Claims & Grounded Evidence
6 Claims
Mechanism
#paired-initialization
Sort a finite nonempty multiset of nonnegative weights. Difference the first and second largest original weights, the third and fourth, and so on, carrying the smallest unpaired original weight when the input size is odd. Only this first round uses fixed adjacent pairs. Then use largest differencing on the resulting residual list, reordering after each pair, rather than continuing PDM’s whole-round processing.
Scope
#cardinality-invariant
Each initial paired residual has signed effective cardinality difference 0, because its two original items are committed to opposite sides. An odd unpaired item contributes the only signed unit. Later residual differencing adds or subtracts these effective differences, giving final absolute cardinality difference 0 for even input size and 1 for odd size. This invariant includes zero-valued original items; it does not guarantee equal subset sums.
Mechanism
#largest-phase-and-reconstruction
For each largest-residual step with a>=b, replace the pair by a-b and combine their represented original groups with opposite orientations. Store merge history or memberships to reconstruct the final subsets. A zero residual is still a represented group. On the constructed list 200,100,90,80,70,61,50,42, the first round produces 100,10,9,8. BLDM continues to residue 73, whereas PDM’s additional paired rounds give 89; both outputs satisfy cardinality balance. This demonstrates distinct procedures, not universal BLDM dominance.
Limitation
#not-an-optimality-certificate
BLDM follows one sequence of pair differences and does not explore alternatives to the initial opposite-side commitments. On the constructed list 8,7,6,5,4 it returns balanced loads 16 and 14, while a balanced partition with loads 15 and 15 exists. A feasible heuristic result alone does not certify minimum residue; the separately named complete BLDM must explore discarded choices.
Scope
#resource-cost
Initial sorting, one linear paired round and repeated largest-residual operations can be implemented in O(n log n) unit-cost comparison/arithmetic work, with O(n) storage for residuals and a merge tree. These bounds assume a suitable ordered or heap representation and exclude free arbitrary-precision arithmetic. The supplied indexed examples are not an author-code or resource-use replication.
Scope
#empirical-evidence-scope
The source studies its BLDM implementation on random integer inputs. Its high-precision heuristic experiment uses 2n-bit integers, rescales residues to the unit interval and averages 1000 instances per plotted point; the fitted expression is presented as a conjecture. Such measurements do not establish a universal finite-input error bound. The original Yakir proof cited by the source was not inspected, so no asymptotic expected-residue guarantee is adopted here.
⚠️ Curated Limitations & Negative Results
- The guarantee concerns balanced cardinalities in two interchangeable bins, not arbitrary prescribed size differences or multiway partitioning.
- Tie choices can affect memberships or traces; the constructed calculation uses stable ordering and does not claim an author implementation match.
- The primary Mertens preprint was read in full and supplies method and experiment evidence. Yakir publisher metadata and abstract were inspected, but the public PDF returned HTTP403 and its full proof was not read.
- No average-case experiment, fitted scaling or hardware timing was independently replicated. The record remains DRAFT.
Artificial Intelligence Ontology (AIO) Grounding
Explicit AIO Snapshot Assessment: No Match
Rationale: No inspected AIO class denotes this deterministic two-phase arithmetic heuristic or its cardinality invariant.
Search Scope: Compared all 442 pinned AIO and 593 native IDs, names, aliases and scoped synonyms; no normalized identity match. Ignored- and hidden-inclusive fixed-string searches of the recovery, older discovery and original Documents checkouts returned 54, 0, 0 lines. All distinct match contents were inspected. Semantic matching covered 67 AIO and 100 native definitions, with every matched definition read. 6 full annotated AIO node/edge/generated contexts and 4 full native records were inspected. BLDM fixes one initial round of adjacent sorted pairs, then uses largest-residual differencing. This differs from PDM whole rounds throughout, unconstrained LDM/KK, and complete balanced or unconstrained search. Existing text hits are prior records and pending leads, not an existing identity. The fully inspected AIO contexts have no suitable counterpart.
| Examined AIO Candidate | Rejection Reason |
|---|---|
AIO:GraphLanguageModel
|
Graph-structured language modeling is distinct from arithmetic subpartition manipulation. |
AIO:ReasoningLLM
|
A language model with explicit reasoning is not a number-partitioning algorithm. |
AIO:NeuroSymbolicLLM
|
A neural-symbolic language-model architecture is not this nonlearning search or heuristic. |
AIO:MachineLearningTask
|
Balancing a given multiset does not learn a method from data. |
AIO:MathematicalFunction
|
An algorithm evaluating differences and choosing a partition is distinct from the mathematical functions it uses. |
AIO:ReinforcementLearning
|
There is no reward-based learned policy or exploration/exploitation learning in the curated deterministic procedure. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
mertens1999
|
Stephan Mertens
|
primary paper
full_text
|
ID: https://arxiv.org/abs/cs/9903011v1
sha256:fe0bcf132782…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-13T08:44:39Z
Updated: 2026-10-13T08:44:39Z
Source File:
data/curated/BalancedLargestDifferencing.yaml