The database holds the records, fields, cells ..with data.
The DBMS is the "tools" used to manipulate the data in the database. An example of a DBMS is SQL Server or Oracle. The DBMS is the database engine itself. This is the core, the actual program that contains the logic to execute your SQL statements, performs that actual data access based upon these commands , whereas DS is linked with improving the speed of traversing and searching the data in main memory. As we all know that the processor can only process the data that is available in main memory. Therefore the different models used to organize data in main memory are collectively referred as DS.
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