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Tree-size estimation by partial backtracking

Algorithms AIMech:PartialBacktrackingTreeEstimation DRAFT
Aliases: Purdom partial-backtracking estimator
A randomized backtracking-tree estimator that explores a uniformly selected subset of successors at each visited branching point and uses inverse inclusion weights to estimate the total cost of candidate examinations in the corresponding complete fixed search.
Identity Notes & Concept Scope Compared all 442 pinned AIO and 643 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 20, 0, 0 lines. All distinct match contents were inspected. Semantic matching covered 66 AIO and 347 native definitions, with every matched definition read. 13 full annotated AIO node/edge/generated contexts and 5 full native records were inspected. Retain the basic and modified partial-backtracking estimators as one algorithm identity with distinct candidate-testing order. Sampling several successors without replacement extends single-path tree probing without Chen-style stratum merging. Conditional-model extrapolation, search-threshold control and score-based beam pruning have different mechanisms.

Claims & Grounded Evidence

5 Claims
Mechanism #basic-subset-weighting
The basic variant chooses m of a candidate values without replacement, with 1 <= m <= a when a is positive. Its sequential remaining-count selection produces a uniform subset. Each selected candidate receives its parent weight multiplied by a/m, contributes its examination cost and is tested for feasibility before recursion. For a finite fixed tree under the stated conditional sampling rule, the weighted estimate preserves the expected complete examination cost. A branch with no candidates terminates.
Mechanism #modified-pretesting
The modified variant first tests every candidate at a visited parent and charges these tests using the parent weight. It then forms the feasible set, samples m of its a members uniformly without replacement and propagates weight a/m into their descendants. Testing all candidates can avoid choosing a failed branch, but adds work even when only a few feasible children are followed. Basic and modified variants have different sampling populations and cost-accounting order.
Scope #count-target-and-scope
The target cost may include failed feasibility tests, so it need not equal the number of feasible expanded nodes. The displayed algorithm initializes the accumulated cost to zero and omits a root test, while the later theoretical singleton-tree convention counts a root. Count conventions must therefore be specified. Sampling multiple branches retains omitted-branch uncertainty: a finite partial traversal does not certify that no solution or deep subtree exists, and node-test counts are not universal wall-clock costs.
Limitation #efficiency-and-variance-scope
The report studies a finite random binary-tree family and a policy that sometimes explores both successors instead of one. Its efficiency measure combines expected work with expected sampling variance conditional on a tree. This is distinct from variance of true tree sizes across the random family. Full traversal can remove sampling variance while remaining costly; the report’s depth-scheduling advice and historical experiments do not establish a universally optimal sampling policy or finite-sample accuracy guarantee.
Limitation #printed-analysis-and-version
The inspected February 1977 report contains printed analysis inconsistencies. Its displayed tree-size variance recurrence fails the two-level binary-tree case, and the work-distribution recurrence’s literal bracketing loses branching probability mass when both successors are always selected. Displayed policy expressions also conflict with the stated balance equation. These are qualifications on this scanned report, not findings about uninspected implementation code or the later journal version; the record does not certify those formulas or an optimal schedule.

⚠️ Curated Limitations & Negative Results

Artificial Intelligence Ontology (AIO) Grounding

Explicit AIO Snapshot Assessment: No Match

Rationale: No suitable pinned AIO counterpart was established for this subset-weighted backtracking-tree estimator. The inspected classification, bias, learning-task and probabilistic-network identities have different scopes; the no-match assessment is bounded to the inspected snapshot.

Search Scope: Compared all 442 pinned AIO and 643 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 20, 0, 0 lines. All distinct match contents were inspected. Semantic matching covered 66 AIO and 347 native definitions, with every matched definition read. 13 full annotated AIO node/edge/generated contexts and 5 full native records were inspected. Retain the basic and modified partial-backtracking estimators as one algorithm identity with distinct candidate-testing order. Sampling several successors without replacement extends single-path tree probing without Chen-style stratum merging. Conditional-model extrapolation, search-threshold control and score-based beam pruning have different mechanisms.
Examined AIO Candidate Rejection Reason
AIO:DecisionTree The pinned class is a classification task. This method estimates additive properties of a backtracking search tree rather than fitting a classifier.
AIO:MarkovChain Random subset selection does not identify the complete fixed-tree additive-cost estimator with a generic stochastic-network class. No appropriate counterpart was established.
AIO:SelectionAndSamplingBias The record is a sampling algorithm with an expectation claim, not the computational bias phenomenon defined by this class.
AIO:MathematicalFunction The additive tree functional is an input; the maintained identity is the randomized estimation procedure.
AIO:MachineLearningTask The algorithm samples an existing implicit tree and need not learn a model or perform a machine-learning task.
AIO:RegressionAnalysis No response-surface regression is required for the weighted tree estimate.
AIO:ReinforcementLearning Sampling tree nodes to estimate a sum does not learn a reward-maximizing policy or define exploration–exploitation control.
AIO:ProbabilisticGraphicalModel The search tree is not asserted to factor a joint distribution of random variables; no graphical-model learning task is required.
AIO:BayesianNetwork An implicit rooted search tree is not by itself a Bayesian-network factorization or a sampler from its joint law.
AIO:GraphLanguageModel No graph-based language model is required.
AIO:ReasoningLLM The procedure is not a language-model reasoning architecture.
AIO:RecursiveLanguageModel Tree-structured neural language composition is distinct from randomized additive tree-functional estimation.
AIO:ComputationalBias Statistical bias is a property discussed when assessing estimators, not the identity of this sampling procedure.

Evidence Sources & Primary Literature

1 Source
Source ID Publication / Document Type / Access Reference & Checksum
purdom1977tr60
Paul W. Purdom
primary paper
full_text
ID: https://scholarworks.iu.edu/dspace/items/5e00ff20-ab58-41fc-a866-d45b2988521c
sha256:9e6ad24e9ec6…

Curation Provenance

Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-13T19:22:29Z
Updated: 2026-10-13T19:22:29Z
Source File: data/curated/PartialBacktrackingTreeEstimation.yaml