Hash Tables

1
Hash Tables

• How well do hash tables support dynamic set
  operations?
• Implementations
• Direct address
• Hash functions
• Collision resolution methods
• Universal hash functions
• Detailed analysis of hashing methods

2
Hash Tables and Dynamic Sets

• Dynamic set operations and hash tables
• Search(S,k)
• Insert(S,x)
• Delete(S,x)
• Minimum or Maximum(S)
• Successor or Predecessor (S,x)
• Merge(S1,S2)
• When is a hash table good for dynamic sets?

3
Direct-address hash table

• Assumptions
• Universe of keys is small (size m)
• Set of keys can be mapped to 0, 1, , m-1
• No elements have the same key
• Use an array of size m
• Array contents can be pointer to element
• Array can directly store element

4
Hash Functions

• Problem with direct-addressed tables
• Universe of possible keys U is too large
• Set of keys used K may be much smaller
• Hash function
• Use an array of size Q(m)
• Use function h(k) x to determine slot x
• h U ? 0, 1, , m-1
• Collision
• When h(k1) h(k2)

5
Good Hash Functions

• Each key is equally likely to hash to any of the
  m slots independently of where any other key has
  hashed to
• Difficult to achieve as this requires knowledge
  of distribution of keys
• Good characteristics
• Must be able to evaluate quickly
• May want keys that are close to map to slots
  that are far apart

6
Collision Resolution

• Collisions are unavoidable even if we have a good
  hash function
• Resolution mechanisms
• Chaining
• Linked list of items that share same hash value
• Universal hash functions
• Open addressing
• Hash again (and again, and again )

7
Chaining

• Create a linked list to store all elements that
  map to same table slot
• Running time
• Insert(T,x) how long? what assumptions?
• Search(T,k) how long? (next slide)
• Delete(T,x) pointer to element x, how long, what
  assumptions?

8
Search time

• Notation
• n items
• m slots
• load factor a n/m
• Worst-case search time?
• What is worst case?
• Expected search time
• Simple uniform hashing each element is equally
  likely to hash to any of the m slots, independent
  of where any other element has hashed to.
• Expected search time?

9
Universal hashing

• In the worst-case, for any hash function, the
  keys may be exactly the worst-case for your
  function
• Avoid this by choosing the hash function randomly
  independent of the keys to be hashed
• Key distinction from probabilistic analysis
• Universal hash function will work well with high
  probability on EVERY input instance but may
  perform poorly with low probability on EVERY
  input instance
• Probabilistic analysis of static hash function h
  says h will work well on most input instances
  every time but may perform poorly on some input
  instances every time

10
Definition and analysis

• Let H be a finite collection of hash functions
  that map U into 0, , m-1
• This collection is universal if for each pair of
  distinct keys k and q in U, the number of hash
  functions h in H for which h(k) h(q) is at most
  H/m.
• If we choose our hash function randomly from H,
  this implies that there is at most a 1/m chance
  that h(k) h(q).
• This leads to the expected length of a chain
  being n/m
• If you are looking for item with key k, what is
  the expected number of items that will share the
  same hash value h(k)?
• Note we assume chaining and not open addressing
  in analysis

11
An example of universal hash functions

• Choose prime p larger than all possible keys
• Let Zp 0, , p-1 and Zp 1, , p-1
• Clearly p gt m. Why?
• ha,b for any a in Zp and b in Zp
• ha,b(k) ((akb) mod p) mod m
• Hp,m ha,b a in Zp and b in Zp
• This family has a total of p(p-1) hash functions
• This family of hash functions is universal
• Proof involves number-theoretic properties

12
Open addressing

• Store all elements in the table
• Probe the hash table in event of a collision
• Key idea probe sequence is NOT the same for each
  element, depends on initial key
• h U x 0, 1, , m-1 ? 0, 1, , m-1
• Permutation requirement
• h(k,0), h(k,1), , h(k,m-1) is a permutation of
  (0, , m-1)

13
Operations

• Insert, search straightforward
• Why can we not simply mark a slot as deleted?
• If keys need to be deleted, open addressing may
  not be the right choice

14
Probing schemes

• uniform hashing each of m! permutations equally
  likely
• not typically achieved
• linear probing h(k,i) (h(k) i) mod m
• Clustering effect
• Only m possible probe sequences are considered
• quadratic probing h(k,i) (h(k)cidi2) mod m
• constraints on c, d, m
• better than linear probing as clustering effect
  is not as bad
• Only m possible probe sequences are considered,
  and keys that map to same position do have
  identical probe sequences
• double hashing h(k,i) (h(k) iq(k)) mod m
• q(k) must be relatively prime wrt m
• m2 probe sequences considered
• Much closer to uniform hashing

15
Search time

• Preliminaries
• n elements, m slots, a n/m with n m
• Assumption of uniform hashing
• Expected search time on a miss
• Given that h(k,i) is non-empty, what is the
  probability that h(k,i1) is empty?
• What is expected search time then?
• Expected insertion time is essentially the same.
  Why?
• Expected search time on a hit
• Expected search time for ith element added is
• If entry was i1st element added, expected search
  time is 1/(1 i/m) m/(m-i)
• Sum this for all i, divide by n, and you get 1/ a
  (Hm Hm-n)
• This can be bounded by 1/ a ln 1/(1- a)