Download as PDF, PPTX
![22
•Table size [0..99]
•A..Z ---> 1,2, ...26
•0..9 ----> 27,...36
•Key: CS1 --->3+19+28 (concatenate) = 31,928
•(31,928)2 = 1,019,397,184 → 10 digits
•Extract middle 2 digits (5th and 6th) as table size
is 0..99.
•Get 39, so: H(CS1) = 39.
Hashing a string key](https://image.slidesharecdn.com/lect1011-dsalgohashing-220407193337/75/LECT-10-11-DSALGO-Hashing-pdf-22-2048.jpg)
![22
•Table size [0..99]
•A..Z ---> 1,2, ...26
•0..9 ----> 27,...36
•Key: CS1 --->3+19+28 (concatenate) = 31,928
•(31,928)2 = 1,019,397,184 → 10 digits
•Extract middle 2 digits (5th and 6th) as table size
is 0..99.
•Get 39, so: H(CS1) = 39.
Hashing a string key](https://crownmelresort.com/image.slidesharecdn.com/lect1011-dsalgohashing-220407193337/75/LECT-10-11-DSALGO-Hashing-pdf-22-2048.jpg)
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LECT 10, 11-DSALGO(Hashing).pdf
- 1. Lecture 10 &11 HASHING 1 Course Supervisor: Syeda Nazia Ashraf Data Structures & Algorithm CSC-102
- 2. MOTIVATION Linear Search • SimplestAlgorithm to search for a specific target key in a data collection. • Examines each element • Takes 10 times longer to search for an element in an array of 100 elements as compared to the 10 element array O(n). 2
- 3. MOTIVATION Binary Search • Requireselement to be in an order(sorted). • Search time depends on the logarithm of the collection size O(log n). • Takes twice as long on average to search for an element in an array of 100 elements as compared to the 10 element array. 3
- 4. MOTIVATION Conclusion • The timetaken for a search using each of these methods depends on the size of the collection. • Hash data structures – Allow the storage and retrieval of data in an average time which does not depend at all on the collection size. 4
- 5. HASHING • Hashing isthe transformation of a string of characters into a usually shorter fixed- length value or key that represents the original string. • Hashing is used to index and retrieve items in a database because it is faster to find the item using the shorter hashed key than to find it using the original value. 5
- 6. HASHING Hash Tables(Hash Map) •Simplest data structure. • Hash Function – Basis of Hash Tables. Hash Functions • A hash function is any function that can be used to map data of arbitrary size to data of fixed size. 6
- 7. HASHING Hashes • The valuesreturned by a hash function are called hash values, hash codes, hash sums, or simply hashes. • Hash values are used to determine the location in the table for the given element. 7
- 8. HASHING IS THERE ANYPARAMETER FOR A GOOD HASH FUNCTION? • A good hash function is the one that distributes the numbers fairly evenly in the hash tables. 8
- 9. POPULAR HASH FUNCTIONS 1.Division Method • A key (given element) is mapped into one of m slots using the function. h(k) = k mod m Where m is the size of the table and is usually chosen to be a prime number and k is the key. 9 Different types of hash functions are used for the mapping of keys into tables. (a) Division Method (b) Mid-square Method (c) Folding Method
- 10. 10 1. Division Method •Choose a number m larger than the number n of keys in k • The number m is usually chosen to be a prime no. or a number without small divisors • The hash function H is defined as, H(k) = k(mod m) or H(k) = k(mod m) + 1 • Denotes the remainder, when k is divided by m • 2nd formula is used when range is from 1 to m.
- 11. 11 • Example: Elements are:3205, 7148, 2345 Table size: 0 – 99 (prime) m = 97 (prime no. close to 99) H(k)=k(mod m) i.e 3205 mod 97=4 H(3205)= 4, H(7148)=67, H(2345)=17 • For 2nd formula add 1 into the remainders. • H(k)=k(mod m)+1 to obtain: • H(3205)= 4+1=5, H(7148)=67+1=68, H(2345)=17+1=18 DIVISION METHOD 3205 2345 7148 17 67 . 0 . 4 . 99 . 3205 2345 7148 18 68 0 . 5 . 99
- 12. POPULAR HASH FUNCTIONSContd… 2. Folding Method • The key is partitioned into a number of parts k1 + k2 + k3 + … kn • where each part except possibly the last part has the same number of digits as the required hash address. • Then the parts are added together, ignoring the last carry. That is, h(k)= k1 + k2 + k3 + … kn • Sometimes the even numbered parts (k2, k4 …) are reversed before adding. 12
- 13. Folding Method • Herewe are dealing with a hash table with index from 00 to 99, i.e., two-digit hash table • So we divide the K numbers of two digits H(7148) = 71 + 48 = 119, here we will eliminate the leading carry (i.e., 1). So H(7148) = 71 + 48 = 19
- 14. Folding Method • Sometimes,for extra "milling;" the even- numbered parts, k2, k4, . . . , are each reversed before the addition • H(7148) = 71 + 84 = 155, here we will eliminate the leading carry (i.e., 1). So H(7148) = 71 + 84 = 55
- 15. FOLDING METHOD Example • Createa hash table for the Keys 3205, 7148, 2345 by using Folding Method Solution • Partition K into a number of parts. • Each part has the same number of digits as the required address. • Add parts together ignoring the last carry. • h(3205) , h(k)= 32 + 05 , hashed key= 37 • h(7148) , h(k)= 71 + 48 , hashed key= 119 (Discard leading digit 1) = 19 • h(2345) , h(k)= 23 + 45 , hashed key= 68 15
- 16. FOLDING METHOD Contd… •Alternatively , one may want to reverse the second part before adding. • h(3205) = 32 + 50 = 82 • h(7148) = 71 + 84 = 155 (Discard 1) = 55 • h(2345) = 23 + 54 = 77 • Creation of the hash table on board. 16
- 17. POPULAR HASH FUNCTIONSContd… 3. Midsquare Method • The key is squared . The hash function is defined by • h(k) = l where l is obtained by deleting digits from both the ends of k2. 17
- 18. Mid-Square Method • Thekey is squared and the address selected from the middle of the squared number • The hash function H is defined by: h(k) = k2 = l • Where l is obtained by digits from both the end of k2 starting from left • The most obvious limitation of this method is the size of the key • Given a key of 6 digits, the product will be 12 digits, which may be beyond the maximum integer size of many computers • Same number of digits must be used for all of the keys
- 19. Mid-Square Method -Example • Consider following keys in the table and its hash index :
- 20. Mid-Square Method -Example Hash Table with Mid-Square Division
- 21. MID-SQAURE METHOD Example • Createa hash table for the Keys 3205, 7148, 2345 by using Mid-square Method Solution: • Square K. • Strip predetermined digits from front and rear. • e.g., use thousands and ten thousands places • K: 3205 7148 2345 • k2: 10272025 51093904 5499025 • hashed key=h(k):72 93 99 • 4th and 5th digits counting from the right side, are chosen for hash address. 21
- 22. 22 •Table size [0..99] •A..Z---> 1,2, ...26 •0..9 ----> 27,...36 •Key: CS1 --->3+19+28 (concatenate) = 31,928 •(31,928)2 = 1,019,397,184 → 10 digits •Extract middle 2 digits (5th and 6th) as table size is 0..99. •Get 39, so: H(CS1) = 39. Hashing a string key
- 23. Hash Function Examples Leth(k) = k % 15. Then, if k = 25 129 35 2501 47 36 h(k) = 10 9 5 11 2 6 Storing the keys in the array is straightforward: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 _ _ 47 _ _ 35 36 _ _ 129 25 2501 _ _ _ Thus, delete and find can be done in O(1), and also insert, except…
- 24. Hash Function What happenswhen you try to insert: k = 65 ? k = 65 h(k) = 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 _ _ 47 _ _ 35 36 _ _ 129 25 2501 _ _ _ 65(?) This is called a collision.
- 25. 25 • If twokeys map on the same hash table index then we have a collision. • As the number of elements in the table increases, the likelihood of a collision increases - so make the table as large as practical • Collisions may still happen, so we need a collision resolution strategy COLLISION
- 26. COLLISION • When ahash function maps two different keys to the same table address, a collision is said to occur. • Two elements can not be stored at the same location in the hash table. • Two approaches are used to resolve collisions. • Open Hashing : Means that collisions are resolved by storing the colliding object in a separate area. • Separate chaining • Closed Hashing (Open Addressing) : In closed hashing, all keys are stored in the hash table itself. • Linear Probing • Quadratic Probing • Double Hashing 26 What is Probing? If the table position given by the hashed key is already occupied, increase the position by some amount, until an empty position is found
- 27.
- 28. CLOSED HASHING METHODS(COLLISION RESOLUTION TECHNIQUES) Linear Probing • Here we place the elements by using the hash function hi(x) = (h(x) + i) mod TableSize. • One of the methods for dealing with collisions. • If a data element hashes to a location in the table which is already occupied , the table is searched consecutively from that location until an empty location is found. • The key would then be stored in the empty location. • rap around from the last to the first bucket array location if necessary. 28
- 29. LINEAR PROBING Exercise Question •h(K) = K mod 7 • Insert keys: 76 93 40 47 10 55 29
- 30.
- 31.
- 32.
- 33. 33 Linear probing hash tableafter each insertion
- 34.
- 35. LINEAR PROBING Contd… Disadvantage •Clustering- Elements appearing next to one another thus increasing search time. Searching/ lookup • To search for a given key x, the cells of T are examined, beginning with the cell at index h(x) (where h is the hash function) and continuing to the adjacent cells h(x) + 1, h(x) + 2, ..., until finding either an empty cell or a cell whose stored key is x. 35
- 36.
- 37.
- 38. LINEAR PROBING Contd… Deletion •It is also possible to remove a key–value pair from the dictionary. However, it is not sufficient to do so by simply emptying its cell. This would affect searches for other keys that have a hash value earlier than the emptied cell, but that are stored in a position later than the emptied cell. The emptied cell would cause those searches to incorrectly report that the key is not present. • Use Tombstones or markers. 38
- 39.
- 40.
- 41. CLOSED HASHING METHODS(COLLISION RESOLUTION TECHNIQUES) 2. Quadratic Probing • Here we place the elements by using the hash function • hi(x) = (h(x) + i2) mod TableSize. • Fast searching as compared to linear probing. • secondary clustering since keys that have the same hash value also have the same probe sequence 41
- 42. 42 2. Quadratic Probing •Quadratic probing is a solution to the clustering problem – Linear probing adds 1, 2, 3, etc. to the original hashed key – Quadratic probing adds 12, 22, 32 etc. to the original hashed key • However, whereas linear probing guarantees that all empty positions will be examined if necessary, quadratic probing does not
- 43. 43 • If thetable size is prime, this will try approximately half the table slots. • More generally, with quadratic probing, insertion may be impossible if the table is more than half-full! H(k) = h, h+1, h+4, h+9, h+25,……, h+i2
- 44. 44 Quadratic Probing • QuadraticProbing eliminates primary clustering problem of linear probing. • Collision function is quadratic. – The popular choice is f(i) = i2. • If the hash function evaluates to h and a search in cell h is inconclusive, we try cells h + 12, h+22, … h + i2. – i.e. It examines cells 1,4,9 and so on away from the original probe. • Remember that subsequent probe points are a quadratic number of positions from the original probe point.
- 45. QUADRATIC PROBING Cont… Example •h(K) = K mod 7 • Insert keys: 76 93 40 47 10 55 45
- 46.
- 47.
- 48.
- 49. 49 A quadratic probing hashtable after each insertion (note that the table size was poorly chosen because it is not a prime number).
- 50. CLOSED HASHING METHODS(COLLISION RESOLUTION TECHNIQUES) 3. Double Hashing • uses a secondary hash function h’(k) and places the colliding item in the first available cell of the series. • The value calculated by the second hash functions acts as an offset. 50
- 51. 51 3. Double Hashing •2nd hash function H’ is used to resolve the collision. • Suppose a record R with key k has hash address H(k)=h and H’(k) = h’ ≠ m • Therefore we can search the locations with addresses, H’(k) = h, h+h’, h+2h’, h+3h’,……. • If m is prime, then this sequence access all the locations.
- 52. 52 A Good DoubleHash Table
- 53.
- 54.
- 55.
- 56. 56 Let the keysare 76, 93, 40, 47, 10, 55 and table size is 7 then apply Double hashing technique for each insertion.
- 57. Open addressing: storethe key/entry in a different position. Separate Chaining • Chain together several keys/entries in each position. • Instead of storing the data item directly in the hash table, each hash table entry contains a reference to a data structure, e.g. a linked list. • In the worst case scenario, all items hash to the same value . Thus we store them in the data structure ( linked list ). 57 Open addressing
- 58. 58 • The ideais to keep a list of all elements that hash to the same value. – The array elements are pointers to the first nodes of the lists. – A new item is inserted to the front of the list. • Advantages: – Better space utilization for large items. – Simple collision handling: searching linked list. – Overflow: we can store more items than the hash table size. – Deletion is quick and easy: deletion from the linked list. Separate Chaining
- 59. 59 Disadvantages of SeparateChaining • Parts of the array might never be used. • As chains get longer, search time increases to O(n) in the worst case. • Constructing new chain nodes is relatively expensive. • Is there a way to use the “unused” space in the array instead of using chains to make more space?
- 60. 60 0 1 2 3 4 5 6 7 8 9 0 81 1 64 4 25 3616 49 9 Keys: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 hash(key) = key % 10.
- 61. SEPARATE CHAINING • Inour example, we use a linked list: • keys: 5, 17, 37, 20, 42, 3, 11 61
- 62. 62 Applications of Hashing •Compilers use hash tables to keep track of declared variables • A hash table can be used for on-line spelling checkers — if misspelling detection (rather than correction) is important, an entire dictionary can be hashed and words checked in constant time • Game playing programs use hash tables to store seen positions, thereby saving computation time if the position is encountered again • Hash functions can be used to quickly check for inequality — if two elements hash to different values they must be different