Data Structure and Algorithms Hashing

Data Structure and Algorithms Hashing

• 1.
  Hashing Introduction Hash Functions HashTable Closed hashing(open addressing) Linear Probing Quadratic Probing Double hashing Open hashing(separate chaining)
• 2.
  Hashing  The searchtime of each algorithm discussed so far depends on number n depends on the number n of items in the collection S of data. A searching Technique, called Hashing or Hash addressing, which is independent of number n. We assume that 1. there is a file F of n records with a set K of keys which unlikely determine the record in F. 2. F is maintained in memory by a Table T of m memory locations and L is the set of memory addresses of locations in T. 3. For notational convenience, the keys in K and Address L are Integers.
• 3.
  Example Suppose A companywith 250 employees assign a 5- digit employee number to each employee which is used as primary key in company’s employee file. We can use employee number as a address of record in memory. The search will require no comparisons at all. Unfortunately, this technique will require space for 1,00,000 memory locations, where as fewer locations would actually used. So, this trade off for time is not worth the expense.
• 4.
  Hashing  The generalidea of using the key to determine the address of record is an excellent idea, but it must be modified so that great deal of space is not wasted. This modification takes the form of a function H from the set K of keys in to set L of memory address.  H: K L , Is called a Hash Function or  Unfortunately, Such a function H may not yield distinct values: it is possible that two different keys k1 and k2 will yield the same hash address. This situation is called Collision, and some method must be used to resolve it.
• 5.
  Hash Functions  thetwo principal criteria used in selecting a hash function H: K L are as follows: 1. The function H should be very easy and quick to compute. 2.The function H should as far as possible, uniformly distribute the hash address through out the set L so that there are minimum number of collision.
• 6.
  Hash Functions 1. Divisionmethod: choose a number m larger than the number n of keys in K. (m is usually either a prime number or a number without small divisor) the hash function H is defined by H(k) = k (mod m) or H(k) = k (mod m) + 1. here k (mod m) denotes the reminder when k is divided by m. the second formula is used when we want a hash address to range from 1 to m rather than 0 to m-1. 2. Midsquare method: the key k is squared. Then the hash function H is defined by H(k) = l. where l is obtained by deleting digits from both end of k^2. 3. Folding Method: the key k is portioned into a number of parts, k1, k2, ……,kr, where each part is added togather, ignoring the last carry. H(k) = k1+k2+ ……………+Kr. Sometimes, for extra “milling”, the even numbered parts, k2, k4, …. Are each reversed befor addition.
• 7.
  Example of HashFunctions  consider a company with 68 employees assigns a 4-digit employee number to each employee. Suppose L consists of 100 two-digit address: 00, 01, 02 , ……….99. we apply above hash functions to each of following employee numbers: 3205, 7148,2345. 1. Division Method: choose a prime number m close to 99, m=97. H(k)=k(mod m): H(3205)=4, H(7148)=67, H(2345)=17. 2. Midsquare Method: k= 3205 7148 2345 k^2= 10272025 51093904 5499025 H(k)= 72 93 99 3. Folding Method: chopping the key k into two parts and adding yield the following hash address: H(3205)=32+05=37, H(7148)=71+48=19, H(2345)=23+45=68 Or, H(3205)=32+50=82, H(7148)=71+84=55, H(2345)=23+54=77
• 8.
  Collision Resolution Suppose wewant to add a new record R with key K to our file F, but suppose the memory location address H(k) is already occupied. This situation is called Collision. There are two general ways to resolve collisions :  Open addressing,(array method)  Separate Chaining (linked list method) The particular procedure that one choose depends on many factors. One important factor is load factor (λ=n/m)i.e. ratio of number n of keys in K (number of records in F) to m of hash address in L. e.g. suppose a student class has 24 students and table has space for 365 records. The efficiency of hash function with a collision resolution procedure is measured by the average number of probes (key comparison) needed to find the location of record with a given k. The efficiency mainly depend on load factor. Specially we are interested in following two quantities:  S(λ) = average number of probes for a successful search  U(λ) = average number of probes for an unsuccessful search
• 9.
  Open Addressing: LinerProbing and Modifications  Suppose that new record R with key k is added to memory table T, but that the memory location with hash address H(k)=h is already filled. One natural way to resolve the collision is to assign R to the first variable locating following T[h] (we assume that the table T with m location is circular i.e. T[1] comes after T[m]). With such a collision procedure, we will search for record R in table T by linearly search the locations T[h], T[h+1], T[h+2], …………. Until finding R or meeting empty location, which indicates an unsuccessful search. The above collision resolution is called Linear probing. The average number of probes for load factor (λ =n/m) are:
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  EX: Linear Probing
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  Open Addressing One maindisadvantage of linear probing is that records tend to cluster, that is, appear next to one another, when the load factor is greater then 50%. The two technique that minimize the clustering are as: 1. Quadratic probing: Suppose the record R with key K has the hash address H(k)=h. Then instead of searching the locations with address h, h+1, h+2, …….. ., we search the location with address h,h+1,h+4,h+9, …………..,h+i^2,…… 2. Double hashing: here the second hash function H’ is used for resolving a collision, as follows. Suppose a record R with key k has a hash address H(k)=h and H’(k)=h’≠m. Then we linearly search the locations with address h, h+h’, h+2h’, h+3h’, …………
• 12.
  Chaining Chaining involves maintainingtwo tables in memory. First of all, as before, there is a table T in memory which contains the records in F, except that T now has an additional field LINK which is used so that all record in T with same hash address h may be linked together to form a linked list. Second, there is a hash address table LIST which contain pointers to linked lists in T. Suppose a new record R with key k is added to the file F. we place R in the first available location in the table T and then add R to the linked list with pointer LIST[H(k)]. The average number of probes for load factor (λ =n/m may be greater than 1) are: S(λ)≈1+ λ/2 and U(λ)≈e^(- λ)+ λ.
• 13.
  Ex: Chaining Using chaining,the record will appear in memory as: