4. Hashing | Highlights and Annotations by Gistr.

This segment explains the limitations of comparison-based search algorithms, setting the stage for exploring faster alternatives. The speaker introduces the concept of a decision tree to visualize the algorithm's branching based on comparisons, ultimately proving that logarithmic time is a lower bound for find operations in this model. The speaker formally defines the comparison model of computation, where the only operation allowed on items is comparing their keys. This segment introduces the concept of a decision tree to represent the algorithm's execution flow based on comparisons, laying the groundwork for proving lower bounds on search time. This lecture covers hashing, a faster alternative to log n time for set operations (find, insert, delete). It proves a log n lower bound for comparison-based find, then introduces direct addressing and hashing to achieve (expected) constant time. Hashing addresses the space limitations of direct addressing by mapping keys to a smaller array using a hash function, handling collisions via chaining. A universal hash family ensures short chains, yielding expected constant time complexity. This segment builds upon the decision tree representation, demonstrating that a correct comparison-based search algorithm requires at least n+1 leaves in its decision tree to handle all possible inputs and outputs. The speaker then connects the minimum height of this tree to the lower bound on the worst-case search time. The speaker discusses the drawbacks of using linked lists to resolve hash collisions. It highlights the conflict between the desired constant-time access of hash tables and the linear-time access inherent in linked lists for get and set operations, emphasizing the importance of maintaining efficient data retrieval. The speaker addresses the major drawback of direct access arrays: their massive space requirements when dealing with large key ranges. The segment discusses the trade-off between space efficiency and search time, setting the stage for the introduction of hashing as a solution. This segment introduces direct access arrays as a data structure that allows for constant-time search operations. The speaker explains how storing items at indices directly corresponding to their keys enables immediate retrieval, but also points out the limitations of this approach in terms of space usage. This segment explains the core problem of hash table implementation: collisions. It clearly illustrates why storing multiple items at the same index is problematic and introduces the concept of collision handling, outlining two primary approaches: using linked lists or open addressing to manage multiple keys mapping to the same hash value. The speaker highlights the limitations of the comparison model in achieving faster-than-logarithmic search times. The segment then introduces the concept of random access memory as a powerful operation that allows for non-constant branching factors, paving the way for faster search techniques. This segment compares two collision resolution techniques: open addressing and chaining. It explains open addressing, where collisions are handled by finding another open space within the existing array, and notes its complexity. The speaker then introduces chaining as a simpler, more manageable alternative for this course.This segment details the chaining method for resolving hash collisions. It explains how to use an additional data structure (like a linked list or dynamic array) at each index to store multiple keys that hash to the same value, improving efficiency compared to open addressing. The speaker introduces the division method, a simple hashing technique using the modulo operator. It explains how this method works and highlights the importance of uniformly distributed keys for optimal performance. The segment emphasizes that the effectiveness of this method depends on the distribution of input keys. This segment introduces the concept of universal hash functions as a solution to the limitations of deterministic hash functions. It explains that by randomly selecting a hash function from a family of functions, the probability of collisions can be controlled, leading to improved performance. The segment introduces the concept of a "universal hash family" and its properties.