Time vs. Information Tradeoffs for Leader Election in Anonymous Trees
Bibliographic record
Abstract
Leader election is one of the fundamental problems in distributed computing. It calls for all nodes of a network to agree on a single node, called the leader . If the nodes of the network have distinct labels, then agreeing on a single node means that all nodes have to output the label of the elected leader. If the nodes of the network are anonymous, the task of leader election is formulated as follows: every node v of the network must output a simple path, which is coded as a sequence of port numbers, such that all these paths end at a common node, the leader. In this article, we study deterministic leader election in anonymous trees. Our aim is to establish tradeoffs between the allocated time τ and the amount of information that has to be given a priori to the nodes to enable leader election in time τ in all trees for which leader election in this time is at all possible. Following the framework of algorithms with advice , this information (a single binary string) is provided to all nodes at the start by an oracle knowing the entire tree. The length of this string is called the size of advice . For a given time τ allocated to leader election, we give upper and lower bounds on the minimum size of advice sufficient to perform leader election in time τ. For most values of τ, our upper and lower bounds are either tight up to multiplicative constants, or they differ only by a logarithmic factor. Let T be an n -node tree of diameter diam ⩽ D . While leader election in time diam can be performed without any advice, for time diam − 1 we give tight upper and lower bounds of Θ(log D ). For time diam − 2 we give tight upper and lower bounds of Θ(log D ) for even values of diam , and tight upper and lower bounds of Θ(log n ) for odd values of diam . Moving to shorter time, in the interval [β · diam , diam − 3] for constant β > 1/2, we prove an upper bound of O ( n log n / D ) and a lower bound of Ω( n / D ), the latter being valid whenever diam is odd or when the time is at most diam − 4. Hence, with the exception of the special case when diam is even and time is exactly diam − 3, our bounds leave only a logarithmic gap in this time interval. Finally, for time α · diam for any constant α < 1/2 (except for the case of very small diameters), we again give tight upper and lower bounds, this time Θ( n ).
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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 | 0.009 | 0.073 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".