Introduction to Algorithms, Second Edition

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The University of Texas at Austin Hashing Department of Computer Sciences Professor Vijaya Ramachandran Lecture 20 CS357: ALGORITHMS, Spring 2006 1 Hashing Hashing is a widely-used class of data structures that support the operations of insert, delete and search on a dynamic set of keys S . The keys are drawn from a universe U ( | U | >> | S | ), which for our purposes will be a set of positive integers. These keys are mapped into a hash table using a hash function ; the size of the hash table is usually within a constant factor of the number of elements in the current set S . There are two main methods used to implement hashing: hashing with chain- ing and hashing with open addressing . We will study hashing with chaining here. 1.1 Hashing with Chaining In hashing with chaining , the elements in S are stored in a hash table T [0 .. m 1] of size m , where m is somewhat larger than n , the size of S . The hash table is said to have m slots . Associated with the hashing scheme is a hash function h which is a mapping from U to { 0 , · · · , m 1 } . Each key k S is stored in location T [ h ( k )], and we say that key k is hashed into slot h ( k ). If more than one key in S hashes into the same slot, then we have a collision . In such a case, all keys that hash into the same slot are placed in a linked list associated with that slot; this linked list is called the chain at that slot. The load factor of a hash table is defined to be α = n/m ; it represents the average number of keys per slot. We typically operate in the range m = Θ( n ) so α is usually a constant (usually α < 1). If a large number of insertions or deletions destroys this property, a more suitable value of m is chosen and the keys are re-hashed into a new table with a new hash function.
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In simple uniform hashing we assume that we have a ‘good’ hash function h such that any element in U is equally likely to hash into any of the m slots in T , independent of the other keys in the table. We also assume that h ( k ) can be computed in constant time. In hashing with chaining, inserts and deletes can be performed in constant time with a suitable linked list representation for the chains. In the worst case, a search operation can take as long as n steps if all keys hash into the same slot. However, if we assume simple uniform hashing then the expected
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