Question

Discuss the advantages and disadvantages of the two clash resolution strategies, open hashing and linear probing (see Exercise 15 ).

Short answer

Open hashing handles collisions with a linked list stored at the hashed index, providing simple implementation and better performance though it requires more memory. Linear probing, on the other hand, places the new key in the next available slot, providing better memory use due to contiguous memory allocation but slower searching and inserting issues when the load factor is high. Both have their own advantages and disadvantages and can be used according to the specific requirements of the problem.

Step-by-step solution

Explanation of Open Hashing

Open hashing, also known as separate chaining, deals with collisions by storing all elements that hash to the same index in a linked list. Whenever a collision occurs, the new element is added to the list. To find an element, we traverse through the list.

Advantages and Disadvantages of Open Hashing

In open hashing, collisions do not impact much on the performance unless the hash function is really poor because elements are stored in a list linked to the index they hash to. Generally, it is simpler to implement than its counterpart, linear probing. However, it requires additional memory for the linked list pointers which might be an overhead in memory constraint systems. Also, cache performance is poor due to the link list traversal.

Explanation of Linear probing

Linear probing handles collisions by placing the new key in the next available table slot. If the hash table slot indexed by the hash function is unoccupied, the key object is inserted in that slot. If the slot is already occupied by another key object, linear probing searches linearly in the table until it finds an empty slot.

Advantages and Disadvantages of Linear Probing

In linear probing, memory usage is better than open hashing because only actual elements need space, not lists. It also provides better cache performance due to contiguous memory allocation. However, searching can be slower compared to Open Hashing due to the clustering issue, where a significant number of consecutive slots become occupied. It may also cause an issue in insert operation when the load factor is high.

Key concepts

Open Hashing

Open hashing, also known as separate chaining, is a technique used in hash tables to resolve collisions—a scenario where multiple keys map to the same hash value. In open hashing, each slot of the hash table is associated with a linked list to hold all the elements that hash into the same slot. When a collision occurs, instead of searching for an empty slot, the element is simply added to the end of the list at that index.

One advantage of open hashing is that the linked lists can grow as needed, theoretically allowing an unlimited number of collisions to be handled. Additionally, removal operations tend to be simpler with open hashing as only a list node needs to be disconnected. However, challenges arise with additional memory overhead due to pointers in each list node and possible degradation in cache performance due to non-contiguous storage of the elements. Moreover, as the size of these lists increase, the time complexity for searching within them grows linearly.

Linear Probing

Linear probing is a form of open addressing used in hash tables to cope with collisions. Unlike open hashing, linear probing does not employ any additional data structures like lists. When a collision occurs in linear probing, the hash table checks the next slot sequentially until an empty slot is found to place the new key. This method takes advantage of the cache since it stores elements contiguously, which can lead to more efficient data retrieval.

As an advantage, linear probing is memory-efficient, as it only requires as much space as there are elements to store and does not require extra pointers. However, this method can suffer from 'clustering', where groups of consecutive slots are filled and thus, it lengthens the search time for an empty slot or a particular element. This inefficiency becomes more pronounced as the load factor of the table increases, where the load factor indicates the fullness of the table.

Hash Table Performance

The performance of a hash table is integral to its functionality and dictates how quickly it can process data insertion, deletion, and retrieval operations. Two major factors influencing hash table performance are the choice of the hash function and the collision resolution strategy.

A good hash function should distribute keys uniformly across the table to minimize the likelihood of collisions. Nonetheless, collisions are inevitable in most cases, which is where collision resolution strategies come into play. Open hashing and linear probing affect performance differently. Open hashing can maintain performance stability for a larger number of collisions, while linear probing may deteriorate more quickly as the hash table fills up. In terms of the overall efficiency, the target is to achieve average-case time complexity of O(1) for the basic operations. However, poor design choices can lead to O(n) complexity, where n is the number of items within the hash table, significantly compromising performance.

Algorithm Analysis

Algorithm analysis is the study of the performance and resource requirements of algorithms, particularly the time complexity and space complexity. These complexities are usually expressed using Big O notation, which describes how the running time or the space requirements grow with the input size.

Understanding the complexities of different hashing collision resolution strategies, such as open hashing and linear probing, allows developers to predict performance under different conditions and optimize their implementations accordingly. For instance, if the expected load factor is high, open hashing might be preferred to avoiding the clustering problem of linear probing. Analyzing these algorithms requires considering the best-case, average-case, and worst-case scenarios. For hash tables, while the average-case is often O(1), the worst-case, typically resulting from many collisions, could be O(n) and diminish the performance of related operations drastically.