Hashing Algorithm 羅正鴻 林彥廷 戴嘉宏

Introduction Hashing, a ubiquitous information retrieval strategy for providing efficient access to information based on a key Information can usually be accessed in constant time Hashing ’ s drawbacks

Concept of hashing The problem at hand is to define and implement a mapping from a domain of keys to a domain of locations From the performance standpoint, the goal is to avoid collisions (A collision occurs when two or more keys map to the same location) From the compactness standpoint, no application ever stores all keys in a domain simultaneously unless the size of the domain is small

Concept of hashing (con ’ t) The information to be retrieved is stored in a hash table which is best thought of as an array of m locations, called buckets The mapping between a key and a bucket is called the hash function The time to store and retrieve data is proportional to the time to compute the hash function

Hashing function The ideal function, termed a perfect hash function, would distribute all elements across the buckets such that no collisions ever occurred h(v) = f(v) mod m Knuth(1973) suggests using as the value for m a prime number

Hashing function (con ’ t) It is usually better to treat v as a sequence of bytes and do one of the following for f(v) ： (1) Sum or multiply all the bytes. Overflow can be ignored (2) Use the last (or middle) byte instead of the first (3) Use the square of a few of the middle bytes

Implementing hashing The following operations are usually provided by an implementation of hashing ： (1) Initialization (2) Insertion (3) Retrieval (4) Deletion

Chained hashing

Chained hashing (con ’ t) In the worst case (where all n keys map to a single location), the average time to locate an element will be proportional to n/2. In the best case (where all chains are of equal length), the time will be proportional to n/m.

Open addressing

Minimal perfect hash functions Minimal perfect hash function (MPHF) is a perfect hash function with the property that is hashed m keys to m buckets with no collisions Cichelli(1980) and of Cercone et al.(1983) proposed two important concepts ： (1)using tables of values as the parameters (2)using a mapping, ordering, and searching (MOS) approach

Minimal perfect hash functions (con ’ t) Mapping ： transform the key set from an original to a new universe Ordering ： place the keys in a sequence that determines the order in which hash values are assigned to keys Searching ： assign hash values to the keys of each level Mapping → Ordering → Searching

Sager ’ s method and improvement Sager(1984,1985) formalizes and extends Cichelli ’ s approach In the mapping step, three auxiliary(hash) functions are defined on the original universe of keys U ： h 0 ： U → { 0, ……, m - 1 } h 1 ： U → { 0, ……, r - 1 } h 2 ： U → { r, ……, 2r – 1 }

Sager ’ s method and improvement The class of functions searched is h(k) = ( h 0 (k) + g(h 1 (k)) + g(h 2 (k)) (mod m) Sager uses a graph that represents the constraints among keys The mapping step goes from keys to triples to a special bipartite graph, the dependency graph, whose vertices are the h 1 (k) and h 2 (k) values and whose edges represent the words

Sager ’ s method and improvement

The algorithm The mapping step

The algorithm (con ’ t) The ordering step

The algorithm (con ’ t) The searching step

Discussion Hashing algorithm is a constant-time algorithm, and there are always advantages to being able to predict the time needed to locate a key The MPHF uses a large amount of space