Hash tables are data structures used to store and retrieve data in an efficient and fast manner. In some programming languages they can also be called objects, they are one of the most useful data structures and can be seen in a variety of different fields like databases, software engineering, cryptography, and data analysis.

A hash table is a collection of items that can be stored and retrieved based on a key. The key is used to compute an index into an array of buckets or slots, where the actual data is stored. The basic idea behind hash tables is to take a key, compute an index, and use this index to store or retrieve data in the corresponding bucket.

Design of Hash Tables

The design of hash tables starts with the selection of a hash function. A hash function is a mathematical function that takes a key and maps it to an index in the array of buckets. The hash function should have the following properties:

1. It should be deterministic, so given a key, the function should always map it to the same index in the array.
2. It should distribute the keys uniformly across the array. This makes it so that there are less collisions and improves performance because there is less risk of data overlaping on the different array buckets.
3. It should be efficient, meaning that it should have a low time complexity and require few operations.

The hash table then uses the hash function to map keys to different index values in the array of buckets, when a key is used to retrieve or store data in the hash table, the hash function is applied to the key to compute the index in the array of buckets, so it is similar to a linked list where every key points to a specific bucket in the array, the difference is the key 'points' to the index through the hash function.

Implementation of Hash Tables

arrays

Arrays are the most straightforward data structure for implementing hash tables. The array of buckets is simply an array of items, where each item is a key-value pair. The hash function maps the key to an index in the array of buckets, and the corresponding item can be retrieved or stored in the corresponding bucket.

Linked Lists

The Linked Lists implementation is similar to the array one, the hash function maps the key to an index in the array of buckets, the corresponding bucket is a linked list of items, where each item is a key-value pair. The linked list allows for dynamic resizing of the hash table, as the number of items in the hash table can change over time.

Trees

Trees are a more complex data structure that can be used for implementing hash tables. In this implementation, the hash function maps the key to an index in the array of buckets. The corresponding bucket is a tree, where each node in the tree is a key-value pair. The tree structure allows for more efficient search and retrieval operations, as the search time complexity is logarithmic in the number of items in the tree.

Applications of Hash Tables

Hash table are pretty much essential for almost every application that exists, they are a widely used data structure to store and retrieve information that is dependent on some sort of connection, for example names and passwords on websites authentication systems. They can be used for data analysis to organize data based on differen categories, on cryptography for creating passwords based on strings or other patterns depending on the function and many other ways.

Python Implementation

class HashTable:
    def __init__(self):
        self.size = 11
        self.slots = [None] * self.size
        self.data = [None] * self.size
    
    def put(self, key, data):
        hashvalue = self.hashfunction(key, len(self.slots))
        
        if self.slots[hashvalue] == None:
            self.slots[hashvalue] = key
            self.data[hashvalue] = data
        else:
            if self.slots[hashvalue] == key:
                self.data[hashvalue] = data  # replace
            else:
                nextslot = self.rehash(hashvalue, len(self.slots))
                while self.slots[nextslot] != None and self.slots[nextslot] != key:
                    nextslot = self.rehash(nextslot, len(self.slots))

                if self.slots[nextslot] == None:
                    self.slots[nextslot] = key
                    self.data[nextslot] = data
                else:
                    self.data[nextslot] = data # replace

    def hashfunction(self, key, size):
        return key%size

    def rehash(self, oldhash, size):
        return (oldhash+1)%size
    
    def get(self, key):
        startslot = self.hashfunction(key,len(self.slots))

        data = None
        stop = False
        found = False
        position = startslot
        while self.slots[position] != None and  not found and not stop:
            if self.slots[position] == key:
                found = True
                data = self.data[position]
            else:
                position=self.rehash(position,len(self.slots))
                if position == startslot:
                    stop = True
        return data

    def __getitem__(self,key):
        return self.get(key)

    def __setitem__(self,key,data):
        self.put(key,data)

Conclusion

Hash tables are a powerful and versatile tool that can be used in a variety of applications, from database indexing and software engineering, to cryptography and data analysis. With their ability to provide fast and efficient lookups, updates, and grouping of data, hash tables are an essential component of many modern computing systems.