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Key-Value Store Paradigm

A core design principle in many NoSQL databases, utilizing key-value pairs for efficient data retrieval and storage.
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The statement of the theorem

Define the Key-Value Store S\mathcal{S} as a function S:K→V\mathcal{S}: K \to V, where KK is the key space (domain) and VV is the value space (codomain). The retrieval operation GET(k)\text{GET}(k) is defined as S(k)\mathcal{S}(k), and the update operation PUT(k,v)\text{PUT}(k, v) is defined as S←S∪{(k,v)}\mathcal{S} \leftarrow \mathcal{S} \cup \{(k, v)\}. The efficiency relies on the ability to compute GET(k)\text{GET}(k) in O(1)O(1) time complexity, independent of ∣K∣|K|.