# Counting and Min-Cost Encoding for Tokenization in Large Language Models

arXiv: https://arxiv.org/abs/2610.01127 · HTML: https://mlsanity.com/paper/2610.01127/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 41 references that are in the map.
- First posted on arXiv 1 Oct 2026.

## Builds on (most cited first, 30 of 33)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [TriviaQA: A Large Scale Distantly Supervised Challenge Dataset for Reading Comprehension](https://mlsanity.com/paper/1705.03551/) (2017)
- [Neural Machine Translation of Rare Words with Subword Units](https://mlsanity.com/paper/1508.07909/) (2015)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [Can a Suit of Armor Conduct Electricity? A New Dataset for Open Book Question Answering](https://mlsanity.com/paper/1809.02789/) (2018)
- [CommonsenseQA: A Question Answering Challenge Targeting Commonsense Knowledge](https://mlsanity.com/paper/1811.00937/) (2018)
- [No Language Left Behind: Scaling Human-Centered Machine Translation](https://mlsanity.com/paper/2207.04672/) (2022)
- [DeepSeek-V4: Towards Highly Efficient Million-Token Context Intelligence](https://mlsanity.com/paper/2606.19348/) (2026)
- [The LAMBADA dataset: Word prediction requiring a broad discourse context](https://mlsanity.com/paper/1606.06031/) (2016)
- [Subword Regularization: Improving Neural Network Translation Models with Multiple Subword Candidates](https://mlsanity.com/paper/1804.10959/) (2018)
- [Byte Pair Encoding is Suboptimal for Language Model Pretraining](https://mlsanity.com/paper/2004.03720/) (2020)
- [FineWeb2: One Pipeline to Scale Them All -- Adapting Pre-Training Data Processing to Every Language](https://mlsanity.com/paper/2506.20920/) (2025)
- [Skywork: A More Open Bilingual Foundation Model](https://mlsanity.com/paper/2310.19341/) (2023)
- [Tokenization Is More Than Compression](https://mlsanity.com/paper/2402.18376/) (2024)
- [SuperBPE: Space Travel for Language Models](https://mlsanity.com/paper/2503.13423/) (2025)
- [The MiniMax-M2 Series: Mini Activations Unleashing Max Real-World Intelligence](https://mlsanity.com/paper/2605.26494/) (2026)
- [Fishing for Magikarp: Automatically Detecting Under-trained Tokens in Large Language Models](https://mlsanity.com/paper/2405.05417/) (2024)
- [Nemotron-CC-Math: A 133 Billion-Token-Scale High Quality Math Pretraining Dataset](https://mlsanity.com/paper/2508.15096/) (2025)
- [Enhancing Multilingual LLM Pretraining with Model-Based Data Selection](https://mlsanity.com/paper/2502.10361/) (2025)
- [Greed is All You Need: An Evaluation of Tokenizer Inference Methods](https://mlsanity.com/paper/2403.01289/) (2024)
- [Boundless Byte Pair Encoding: Breaking the Pre-tokenization Barrier](https://mlsanity.com/paper/2504.00178/) (2025)
- [BPE Gets Picky: Efficient Vocabulary Refinement During Tokenizer Training](https://mlsanity.com/paper/2409.04599/) (2024)
- [Multi-word Tokenization for Sequence Compression](https://mlsanity.com/paper/2402.09949/) (2024)
- [Scaffold-BPE: Enhancing Byte Pair Encoding for Large Language Models with Simple and Effective Scaffold Token Removal](https://mlsanity.com/paper/2404.17808/) (2024)
- [MinGram: A Minimalist Unigram Tokenizer with High Compression and Competitive Morphological Alignment](https://mlsanity.com/paper/2606.27019/) (2026)

## Abstract (arXiv)

Mainstream large language models rely on a tokenizer to encode text into a token sequence. Different tokenizers may yield token sequences of substantially different lengths for the same text. With a fixed model architecture, shorter token sequences correspond to lower inference time. We propose a tokenizer training approach named Counting and Filtering (CNF) and a text encoding algorithm called Min-Cost Encoding (MCE). MCE defines a cost function over a text segment, and determines the best segmentation by globally minimizing the overall segmentation cost. CNF builds a raw vocabulary by directly counting valid substrings, and then constructs the final vocabulary through a filtering step based on actual token usage when segmenting the training corpus with MCE. The CNF-MCE conbination offers several advantages over BPE, including higher token efficiency, greater scalability, and lower dependency. Across six text categories and two vocabulary-size groups, CNF-MCE consistently achieves better compression than the evaluated BPE tokenizers. With a 250K vocabulary, CNF-MCE increases compression rate by 26% and 30% on English web text over the o200k_base and qwen250k tokenizers. Experiments scaling the vocabulary to 1M entries on English web text demonstrate sustained improvements over BPE, with a token efficiency improvement of over 60% and vocabulary utilization rising from 52.9% to 96.9%. The MCE algorithm does not depend on a merge list (as in BPE) or token probability (as in UnigramLM), making it applicable to a wide range of vocabularies, including those built from BPE, UnigramLM, CNF, and others. Language models trained from scratch at the 1.8B and 8B scales achieve comparable average performance to models using the BPE tokenizers across 11 benchmarks. These results demonstrate that CNF-MCE can improve token efficiency significantly while maintaining competitive downstream performance.