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
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?
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
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)
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
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 …)
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?
Search time • Notation • n items • m slots • load factor α = 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?
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
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
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 > m. Why? • ha,b for any a in Zp* and b in Zp • ha,b(k) = ((ak+b) 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
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)
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
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)+ci+di2) 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
Search time • Preliminaries • n elements, m slots, α = 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,i+1) is empty? • What is expected search time then? • Expected insertion time is essentially the same. Why? • Expected search time on a hit • If entry was ith element added, expected search time is 1/(1 – i/m) = m/(m-i) • Sum this over all m, divide by n, and you get 1/ α (Hm – Hm-n) • This can be bounded by 1/ α ln 1/(1- α)
