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Database indexing is a technique used to improve the speed of data retrieval operations in a database. Think of it as a book's index that lets you quickly find the specific page containing the information you need. In databases, an index stores the data's location based on one or more columns – these columns are known as keys.

For instance, imagine a database table that contains information about books, such as title, author, and ISBN number. By creating an index on the ISBN number, the database can quickly find all the details associated with a particular book without having to go through each row of the table one by one. This is especially crucial for large databases where a full table scan would consume a lot of resources and time.

However, it's worth noting that while indexes speed up search operations, they require additional storage space and can slow down write operations, like updates and inserts, because the index itself must be updated. Effective indexing strategies are essential for maintaining balance between read and write performance within the database.

Search efficiency in databases refers to the ability to retrieve the desired data quickly and with minimal computational resources. The goal is to optimize the time taken to perform search operations and reduce the overhead on the storage system. Search efficiency is influenced by various factors including database structure, indexing strategy, and the nature of query operations.

Using indexing, databases can achieve logarithmic search complexity, which means that the time taken to search through data grows much slower than the size of the data itself. It's similar to how we can speedily locate a word in the dictionary by exploiting the fact that it's sorted. In databases, if data is sorted by an index and you're looking for a specific value, you can quickly disregard large portions of data where the value cannot possibly be.

For complex queries involving multiple criteria, search efficiency is still paramount. Compound indexes, which use multiple keys, can significantly improve search performance for such queries. Optimizing database queries and ensuring efficient search mechanisms are key to ensuring that applications powered by those databases remain responsive and effective.

Index hierarchy is an essential aspect of multilevel indexing in databases, representing a structured approach to data lookup through multiple tiers of indexing. Instead of a flat, single-level index, where all index entries exist in one large index, index hierarchy breaks down the index into several smaller levels.

The top level of the hierarchy might only contain a summary of the index entries, such as the highest or lowest key values found in each lower level index block. When searching for data, the database first references the highest level index to find the relevant range, then moves to the next level index to refine the search, and continues this process down through the hierarchy until the actual data is located.

This method is analogous to filtering through a category, then a subcategory, and so forth, until you find the desired item. In databases, multilevel indexes reduce the number of disk I/O operations and consequently accelerate search operations. For databases that hold vast amounts of data, this approach is especially effective as it avoids exhaustive searches through potentially millions of records, guiding the search operation through a progressively narrowing path.