The complexity of hash table operations varies depending on several factors, including the specific operation being performed, the state of the hash table (e.g., the number of elements stored and the capacity of the table), and the method used for handling collisions. Here's a detailed breakdown of the complexities associated with hash table operations:
Time Complexity
Average Case Complexity
Search: The average complexity for searching an element in a hash table is $$O(1)$$. This means that, on average, it takes a constant amount of time to find an element, regardless of the size of the hash table[1][3][4][5].
Insert: Similarly, the average complexity for inserting an element into a hash table is $$O(1)$$. This indicates that the time required to add a new element is constant on average[1][3][4][5].
Delete: The deletion of an element from a hash table also has an average complexity of $$O(1)$$, meaning that removing an element takes a constant amount of time on average[1][3][4][5].
Worst Case Complexity
Search, Insert, and Delete: In the worst-case scenario, all these operations can have a complexity of $$O(n)$$, where $$n$$ is the number of elements in the hash table. This situation occurs when many elements are hashed to the same index, leading to a long chain or cluster that must be searched linearly[1][2][3][4][5].
Space Complexity
Open Addressing: For hash tables that use open addressing for collision resolution, the space complexity remains $$O(n)$$ in all cases, as each element stored in the table must have some memory associated with it[1].
Closed Addressing (Chaining): For hash tables that use chaining for collision resolution, the space complexity is $$O(m+n)$$, where $$m$$ is the size of the hash table and $$n$$ is the number of items inserted. This accounts for the memory used by the table itself and the linked nodes allocated outside the hash m...
