Prefix Sharing Is a Sorting Problem

• URL: http://arxiv.org/abs/2609.13692v1
• Date: Sat, 12 Sep 2026 04:02:31 GMT
• Title: Prefix Sharing Is a Sorting Problem
• Abstract summary: serving reuses KV cache by exact prefix match, so when a prompt is assembled from a set of reusable pieces, the order chosen determines how much computation can be shared.<n>We prove this is optimal only when requests contain at most two pieces, and achievable wrong in general.<n>Our main result is a structure theorem: the minimum prefix-trie cost equals min_H sum_x w(x) t_x(H) over binary hierarchies H on the requests.
• Score: 0.304585143845864
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: LLM serving reuses KV cache by exact prefix match, so when a prompt is assembled from a set of reusable pieces -- retrieved passages, tool definitions, few-shot exemplars -- the order chosen for those pieces determines how much computation can be shared. Every deployed system fixes that order by a single global convention. We prove this is optimal only when requests contain at most two pieces, and asymptotically wrong in general. Our main result is a structure theorem: the minimum prefix-trie cost equals min_H sum_x w(x) t_x(H) over binary hierarchies H on the requests, where t_x(H) is the canonical decomposition size of the set of requests needing chunk x. Choosing chunk orders is therefore equivalent to choosing one hierarchy over requests. The identity yields an O(3^m) exact algorithm, identifies the two-chunk case as minimum vertex cover, and shows that on the leave-one-out family the optimum is the minimum external path length of a binary tree -- the merge-sort recursion -- so a global order pays Theta(n^2) against a true cost of Theta(n log n). Agglomerative clustering by common intersection is a tight 1/2-approximation for the achievable saving. On BM25 retrieval traces over three BEIR corpora the resulting layout reduces prefill by 17-36% against production RAG ordering, and the margin widens with retrieval depth as the theory predicts. Serving requests in the hierarchy's DFS order finally lets a cache holding one request's context attain the unbounded-cache optimum exactly, so cache capacity and reorder window act as substitutes.

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