Dynamic Searchable Symmetric Encryption (DSSE) enables users to search on the encrypted database stored on a semi-trusted server while keeping the search and update information under acceptable leakage. However, most existing DSSE schemes are not efficient enough in practice due to the complex structures and cryptographic primitives. Moreover, the storage cost on the client side grows linearly with the number of keywords in the database, which induces unaffordable storage cost when the size of keyword set is large. In this article, we focus on secure dynamic searchable symmetric encryption with constant client storage cost. Our framework is boosted by fish-bone chain, a novel two-level structure which consists of Logical Keyword Index Chain (LoKIC) and Document Index Chain (DIC). To instantiate the proposed framework, we propose a forward secure DSSE scheme, called CLOSE-F, and a forward and backward secure DSSE scheme, called CLOSE-FB. Experiments showed that the computation cost of CLOSE-F and CLOSE-FB are as efficient as the state-of-the-art solutions, while the storage costs on the client side are constant in both CLOSE-F and CLOSE-FB, which are much smaller than existing schemes.