EXPLORING PLANAR GRAPHS USING UNORIENTED MAPS
Bibliographic record
Abstract
A mobile agent (robot) has to explore an unknown terrain modeled as a planar embedding of an undirected planar connected graph. Exploration consists in visiting all nodes and traversing all edges of the graph, and should be completed using as few edge traversals as possible. The agent has an unlabeled map of the terrain which is another planar embedding of the same graph, preserving the clockwise order of neighbors at each node. The starting node of the agent is marked in the map but the map is unoriented: the agent does not know which direction in the map corresponds to which direction in the terrain. The quality of an exploration algorithm [Formula: see text] is measured by comparing its cost (number of edge traversals) to that of the optimal algorithm having full knowledge of the graph. The ratio between these costs, for a given input consisting of a graph and a starting node, is called the overhead of algorithm [Formula: see text] for this input. We seek exploration algorithms with small overhead. We show an exploration algorithm with overhead of at most 7/5 for all trees, which is the best possible overhead for some trees. We also show an exploration algorithm with the best possible overhead, for any tree with starting node of degree 2. For a large class of planar graphs, called stars of graphs, we show an exploration algorithm with overhead of at most 3/2. Finally, we show a lower bound 5/3 on the overhead of exploration algorithms for some planar graphs.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".