An Analysis and Enhancement of the Gap Heuristic for the Pancake Puzzle
Bibliographic record
Abstract
The pancake puzzle is a standard benchmark domain used to test search algorithms, and the gap heuristic is the state-of-the-art heuristic function most often used in such tests. In this work, we analyze the accuracy of this heuristic and identify ways to enhance it. We begin by showing that in the worst-case, the amount that the gap heuristic underestimates the optimal cost of a pancake puzzle state can be linear in the number of pancakes in the stack. However, empirical analysis suggests that it is extremely rare that the gap heuristic underestimates the optimal cost by more than two. We then identify several simple methods that can be used to generate large sets of problems on which the gap heuristic underestimates the optimal cost by a larger amount than it typically does on random permutations. In doing so, we provide new pancake puzzle test sets that can be used to evaluate how search algorithms behave when the heuristic is inaccurate. We also formally characterize states according to the size of the heuristic plateaus around them. This characterization allows us to efficiently compute a two-step look ahead of the gap heuristic on any state, which we can use alongside a state's dual to further improve heuristic accuracy. These enhancements substantially improve the performance of an IDA*-based pancake problem solver on both the existing benchmarks and the new ones proposed in this paper.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.006 | 0.044 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".