Time Complexity of Hash Set Operations

In a hash set, the time complexity of inserting, searching, and deleting elements is generally $O(1)$ on average. This means that these operations are performed in constant time, regardless of the size of the hash set.

Insertion

When inserting an element into a hash set, the hash function is applied to the element to determine its bucket index. If there are no collisions (i.e., no other elements in the same bucket), the insertion is done in constant time. However, if collisions occur and the bucket already contains elements, the insertion time may increase slightly due to the need to resolve collisions.

Example of inserting an element into a hash set:

1#include <unordered_set>
2
3int main() {
4  std::unordered_set<int> set;
5  set.insert(10);// O(1) on average
6  set.insert(20);// O(1) on average
7  set.insert(30);// O(1) on average
8}

Search

Searching for an element in a hash set is also an $O(1)$ operation on average. The hash function is applied to the element to determine its bucket index, and then the bucket is searched for the element. If there are no collisions, the search is done in constant time. If collisions exist, the search time may increase slightly, but it still remains constant on average.

Example of searching for an element in a hash set:

1#include <unordered_set>
2
3int main() {
4  std::unordered_set<int> set = {10, 20, 30};
5
6  // O(1) on average
7  bool found = set.find(20) != set.end();
8}

Deletion

Deleting an element from a hash set follows a similar process as searching. The hash function is applied to the element to locate its bucket, and then the element is removed from the bucket. The deletion operation is also $O(1)$ on average.

Example of deleting an element from a hash set:

1#include <unordered_set>
2
3int main() {
4  std::unordered_set<int> set = {10, 20, 30};
5  set.erase(20);// O(1) on average
6}

It's important to note that the constant time complexity of hash set operations relies on a good hash function that minimizes collisions and ensures an even distribution of elements across the buckets. If the hash function is poorly designed or there are many collisions, the time complexity may degrade to O(n) in the worst case, where n is the number of elements in the hash set.