Large Language Models (LLMs) are gaining popularity and improving rapidly. Tokenizers are crucial components of natural language processing, especially for LLMs. Tokenizers break down input text into tokens that models can easily process while ensuring the text is accurately represented, capturing its meaning and structure. Effective tokenizers enhance the capabilities of LLMs by improving a model's understanding of context and semantics, ultimately leading to better performance in various downstream tasks, such as translation, classification, sentiment analysis, and text generation. Most pre-trained tokenizers are suitable for high-resource languages like English but perform poorly for low-resource languages. Dzongkha, Bhutan's national language spoken by around seven hundred thousand people, is a low-resource language, and its linguistic complexity poses unique NLP challenges. Despite some progress, significant research in Dzongkha NLP is lacking, particularly in tokenization. This study evaluates the training and performance of three common tokenization algorithms in comparison to other popular methods. Specifically, Byte-Pair Encoding (BPE), WordPiece, and SentencePiece (Unigram) were evaluated for their suitability for Dzongkha. Performance was assessed using metrics like Subword Fertility, Proportion of Continued Words, Normalized Sequence Length, and execution time. The results show that while all three algorithms demonstrate potential, SentencePiece is the most effective for Dzongkha tokenization, paving the way for further NLP advancements. This underscores the need for tailored approaches for low-resource languages and ongoing research. In this study, we presented three tokenization algorithms for Dzongkha, paving the way for building Dzongkha Large Language Models.