HASH TABLES. Hash Workshop Applet (pages ) The Hash Workshop applet demonstrates the Java implementation of a hash table using linear probing.

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1 HASH TABLES Open Addressing Hash Tables (clsed hashing) pp Table rganizatin (pages ) An pen addressing hash table is an array structure with B elements, called buckets, which are numbered 0 thrugh B 1. Hashing Functins (pages ) Assume that database recrds are being stred in the hash table and that each recrd has a primary key. A hashing functin inputs a value fr the key and returns an index (0 thrugh B 1) within the hash table. The index returned by is called the hash address r hash cde. When the key values are integers, such as Scial Security numbers r credit card accunt numbers, the simplest hashing functin scales the key t the range 0 thrugh B 1 by dividing it mdul B: public int ( int key ) return key % B; Algrithm fr peratins find and insert (page ) Suppse yu wish t find r insert a recrd with key X int a hash table. Cmpute (X) = b and inspect bucket b. If bucket b is empty then place the recrd at that psitin. If bucket b is nt empty then a cllisin has ccurred and a cllisin reslutin (r prbing) strategy must be emplyed. Linear Prbing (page 528) The simplest prbing strategy is linear prbing, in which yu successively search buckets b+1, b+2, b+3, (wrap arund t bucket 0 if necessary) until: the recrd with key X is fund, meaning yu ve fund the recrd, OR an empty slt is encuntered, in which case the recrd is nt in the table and s yu insert it int the table at that psitin, OR yu return t bucket b in which case the table is full and must be increased in size Since the same algrithm is used t find and insert, a recrd is always fund even if it is nt placed int bucket given by its hash address. Hash Wrkshp Applet (pages ) The Hash Wrkshp applet demnstrates the Java implementatin f a hash table using linear prbing. Deletin (pages ) T delete an item in a hash table, verwrite it with a special value. The find algrithm treats this bucket as nt empty; the insert treats it as empty. Hash Tables Page 1

2 Clustering Linear prbing suffers frm primary clustering where large grups f cnsecutive e buckets becme ccupied. Duble Hashing (pages ) Uses this prbing sequence: If bucket b is full, successively prbe buckets b+h, b+2h, b+3h, where h is an increment value given by a secndary hashing functin. Thus, even if tw keys hash t the same bucket address b using the primary hashing functin, they wn t hash t the same increment h using the secndary ne and, therefre, their prbe sequences will differ. We need a guarantee that we d nt keep searching the same buckets ver and ver again (see page 551). Duble Hashing Therem: If table size B is relatively prime t the increment value h, then the duble hashing prbing sequence b, b+h, b+2h, b+3h, etc. will eventually examine every bucket f the hash table. T satisfy the requirement if the Duble Hashing Therem, make the table size B a prime number and use this secndary hashing functin (where c is sme small cnstant): public int ( int key ) return c - (key % c); Yur authr (n page 545) says that the cnstant c must be a prime number. Weiss 1 als suggests chsing a prime number fr c but desn t imply that it is necessary. Sedgwick 2 suggests chsing c = 8 s that yu can efficiently implement the secndary hashing functin by simply extracting the rightmst three bits f the key. Rehashing (page 541) If the hash array becmes t full, yu cannt simply create a larger ne and cpy the cntents f the ld array int it because each recrd s psitin is derived frm the array size. T enlarge a hash table, yu must allcate a new array using the larger size and, by lping thrugh the ld table, take each recrd, rehash its key with the new size and insert it int the new table. 1 Mark Allen Weiss, Data Structures and Algrithm Analysis in Java, Addisn-Wesley, 1999, p Rbert Sedgewick, Algrithms, Addisn-Wesley Publishing Cmpany, 1988, p Hash Tables Page 2

3 Clsed Addressing Hash Tables (pen hashing) pp Each bucket f the hash table is a linked list. T insert a recrd with key X int the hash table cmpute (X) = b and search the linked list at bucket b fr the recrd, inserting it int the list if it isn t already present. Alphanumeric Keys (pages ) Fr recrds with alphanumeric keys (such as VIN numbers) the key must be cnverted t an integer as part f the hashing functin. The cnversin algrithm uses the fact that each character f a string is stred as an integer such as its ASCII cde (American Standard Cde fr Infrmatin Interchange) where each character is stred as an eight-bit binary number. Fr example: Letter ASCII Cde Decimal Hex Binary A B C D E F A gd cnversin algrithm must use as many characters f the key as pssible and, preferably, all f them. Fr example, if nly the first 3 characters are used then FZZ000 thrugh FZZ999 wuld all hash t the same address. Furthermre, gd cnversin algrithm must accunt fr the psitin f each character. If it didn t, fr example, TAR, ART and RAT wuld all hash t the same address. The best strategy guarantees that n tw different strings are cnverted t the same integer. This is dne by using the binary string that represents the key. C A B E Hash Tables Page 3

4 Here s the algrithm fr cnstructing this number. String t Integer Cnversin Algrithm Given: key, a string f alphanumeric characters key[0], key[1], etc. Steps: Initialize integer result t key[0] fr k=1 t last character in the string d shift bits in result left 8 bits place key[k] int rightmst 8 bits end fr Shifting the integer 8 bits t the left can be accmplished by multiplying it by 256 (2 8 ). The final result must be divided mdul the hash table size B but it is better t d this within the lp as the result is calculated s as t prevent verflwing the cmputer s integer range befre all the characters have been prcessed. Here s the algrithm cded int Java. 3 public int hashstring( String key, int B ) // Hash the given alphanumeric key int an address // fr a hash table with B buckets. int hashval = key.charat( 1 ); fr ( int k = 1; k < key.length( ); k++ ) hashval *= 256; // shift left 8 bits hashval += key.charat( k ); // place char int the 8 bits hashval %= B; // md by table size return hashval; 3 Yur textbk (n page 565) presents a similar algrithm. His algrithm uses a radix f 27 and subtracts 96 frm the ASCII cde because he is assuming keys are made up nly f blanks and the twenty-six lwer-case letters. The ASCII cde fr the lwer-case letter a is 97. Thus, by subtracting 96 frm the ASCII cde in the first statement, he scales each lwer-case letter t an integer in the range 1 thrugh 26. Hash Tables Page 4

5 Expert Recmmendatins n Hashing Techniques Bth yur authr (n page 552) and Sedgewick 4 claim that duble hashing is the pen addressing methd f chice. Weiss 5, n the ther hand, refers t duble hashing as theretically interesting and recmmends quadratic prbing. Sedgewick 6 recmmends: 1. If the number f recrds t be placed int the table is unknwn then use separate chaining. 2. If yu can rughly estimate in advance the number f recrds t be placed int the table then use duble hashing. Make the table abut twice as large as the number f recrds (remember that it must be a prime number). Never let it becme mre than 60% full. If it reaches 60% full, rehash it t at least twice its current size. 4 Rbert Sedgewick, p. cit., p Mark Allen Weiss, p. cit., p Rbert Sedgewick, lc. cit. Hash Tables Page 5