Hash Tables

Hash tables are a corner stone of data structures for most problems, they are useful when there are unique keys with given values

Runtime analysis

1. Insert: $O(1)$

2. Search: $O(1)$

3. Delete: $O(1)$

4. Update: $O(1)$

Take note…

1. Do we need a custom hashing function?

Techniques

Anagram hashing

This technique is particularly when trying to detect all anagrams and store them in buckets of same anagrams

• Multiplicative hash with prime numbers (3 onwards)

• Assign each character a prime number and calculate the hash

• Hash will be unique per anagram

• Does not scale well if more than 52 characters possible or $n > 5000$

primes = [3, 5, 7, ...]
def hash(string):
	value = 1
	for ch in string:
		value *= primes[ord(ch) - ord('a')]
	return value

Fingerprint hash tables/arrays over hash tables

I would recommend using these over hash tables where possible as they can represent the same information while reducing the complexity of the code

freq = [0] * 26
# is the same as 
freq = {}
for ch in s:
    if ch not in freq:
        freq[ch] = 0

Duplicate checking using bitmasks

This is a slight optimization over using hash tables/sets if the only thing that needs to be done is duplicate detection

• Not feasible if the numbers can be very large as size grows by powers of 2

acc = 0
for ch in string:
	mask = 1 << (ord(ch) - ord('a'))
	if acc ^ mask < acc: # duplicate found
		return False
	acc |= mask

Multiplicative hashing

$$h(k) = \lfloor m(\phi \times k - \lfloor \phi \times k \rfloor) \rfloor$$

• Where $k$ is the key to hash, $\phi$ is the golden ratio = $\sqrt{5} - \frac{1}{2}$

Pairing with another data structure

Often used to improve performance at the cost of using more space to store the hash table

• Use Linked Lists to optimize to quickly delete/insert nodes in a certain order

  • Hash table has pointers to the nodes of the linked list

Types of hashing

1. Chaining (linked list per bucket)

2. Open addressing (probing for a blank spot if current bucket is filled)