Searching. Constant time access. Hash function. Use an array? Better hash function? Hash function 4/18/2013. Chapter 9

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1 Constant time access Searching Chapter 9 Linear search Θ(n) OK Binary search Θ(log n) Better Can we achieve Θ(1) search time? CPTR Use an array? Use random access on a key such as a string? Hash function Map a key to an array index h(k) = i, where i { 0, 1, 2,..., n 1 } h( mary ) = Hash function Better hash function? Let n be 10,007 (a prime number) Is this a good hash function? Keys has at least three characters English is not random Only 2,851 combinations 5 6 1

2 Better hash function Another Hash Function int hashpjw(char *s) { char *p; unsigned h = 0, g; for ( p = s; *p; p++ ) { h = (h << 4) + (*p); if ( g = h & 0xf ) { h ^= g >> 24; h ^= g; } } return h % PRIME; } Aho, Sethi, and Ullman 7 Perfect Hash Functions Hash function is injective (1-1) Possible only when values to store are fixed and known in advance To find a perfect hash function for n keys: Θ(n) Such a function can be evaluated in Θ(1) time Minimal perfect hash function Range is n What about collisions? What if two keys map to the same location? Strategies: Open hashing (separate chaining) Closed hashing (open addressing, or probing) Linear probing Quadratic probing Rehashing 10 Separate chaining hash(x) = x % 10 Load factor α (some authors use λ) The ratio of the number of s in the hash table to the table size Average cost 1 + α/

3 Disadvantages of separate chaining Probing hash tables Uses linked lists, so time to dynamically allocate and deallocate nodes Requires implementing a second data structure (the linked list) Space required to store the pointers Given a collision, find an empty cell via a collision resolution strategy Try cells h 0 (x), h 1 (x), h 2 (x),... until an empty cell is found h i (x) = (h(x) + f(i)) % n f(0) = 0 Load factor should be α < Linear probing f is a linear function Commonly, f(i) = i Try sequential cells until an open cell is found Linear probing hash(x) = x % 10 Add 89, 18, 49, 58, Problem with linear probing Primary clustering Number of probes Insertion/unsuccessful search: (1 α) 2 Successful search: (1 α) 17 3

4 Quadratic probing f is a quadratic function Commonly, f(i) = i 2 Eliminates the primary clustering problem Other strategies Double hashing Apply second hash function f(i) = i hash 2 (x) Rehashing When the load factor gets high create a new table twice the size with new hash function Rehash the existing values into the new table Can be used with quadratic probing Expensive to rehash, but amortized cost is low 20 CPTR 314 s in Uned list Θ(n) Θ(n 2 ) Θ(n) s in Uned list Θ(n) Θ(n 2 ) Θ(n) s in Uned list Θ(n) Θ(n 2 ) Θ(n) s in Uned list Θ(n) Θ(n 2 ) Θ(n) 4

5 s in Uned list Θ(n) Θ(n 2 ) Θ(n) s in Uned list Θ(n) Θ(n 2 ) Θ(n) s in Uned list Θ(n) Θ(n 2 ) Θ(n) s in Uned list Θ(n) Θ(n 2 ) Θ(n) s in Uned list Θ(n) Θ(n 2 ) Θ(n) s in Uned list Θ(n) Θ(n 2 ) Θ(n) 5

6 s in Uned list Θ(n) Θ(n 2 ) Θ(n) s in Uned list Θ(n) Θ(n 2 ) Θ(n) s in Uned list Θ(n) Θ(n 2 ) Θ(n) s in Uned list Θ(n) Θ(n 2 ) Θ(n) Hash table Θ(1) Θ(n 2 ) Θ(n) 6

Array-based Hashtables

Array-based Hashtables For simplicity, we will assume that we only insert numeric keys into the hashtable hash(x) = x % B; where B is the number of 5 Implementation class Hashtable { int [B]; bool occupied[b];