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hash-tables.md

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

Hash Tables

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)