1. CSC211 Data Structures Lecture 20 Hashing Instructor: Prof. Xiaoyan Li Department of Computer Science Mount Holyoke College

2. Searching: Most Common Methods p Serial Search p simplest p Binary Search p better average-case performance p Search by Hashing p better average-case performance

3. p p Chapter 12 discusses several ways of storing information in an array, and later searching for the information. p p Hash tables are a common approach to the storing/searching problem. p p This presentation introduces hash tables. Hash Tables Data Structures and Other Objects Using C++

4. What is a Hash Table ? p p The simplest kind of hash table is an array of records. p p This example has 701 records. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] An array of records... [ 700]

5. What is a Hash Table ? p p Each record has a special field, called its key.   In this example, the key is a long integer field called Number. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ]... [ 700] [ 4 ] Number 506643548

6. What is a Hash Table ? p p The number might be a person's identification number, and the rest of the record has information about the person. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ]... [ 700] [ 4 ] Number 506643548

7. What is a Hash Table ? p p When a hash table is in use, some spots contain valid records, and other spots are "empty". [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322...

8. Inserting a New Record p p In order to insert a new record, the key must somehow be converted to an array index. p p The index is called the hash value of the key. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685

9. Inserting a New Record p p Typical way create a hash value: [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 (Number mod 701) What is (580625685 mod 701) ?

10. Inserting a New Record p p Typical way to create a hash value: [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 (Number mod 701) What is (580625685 mod 701) ? 3

11. Inserting a New Record p p The hash value is used for the location of the new record. Number 580625685 [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... [3]

12. Inserting a New Record p p The hash value is used for the location of the new record. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685

13. Collisions p p Here is another new record to insert, with a hash value of __ ? [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868

14. Collisions p p Here is another new record to insert, with a hash value of 2. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868 My hash value is [2].

15. Collisions p p Here is another new record to insert, with a hash value of 2. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868 My hash value is [2]. When a collision occurs, move forward until you find an empty spot. When a collision occurs, move forward until you find an empty spot.

16. Collisions p p This is called a collision, because there is already another valid record at [2]. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868 When a collision occurs, move forward until you find an empty spot. When a collision occurs, move forward until you find an empty spot.

17. Collisions p p This is called a collision, because there is already another valid record at [2]. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868 When a collision occurs, move forward until you find an empty spot. When a collision occurs, move forward until you find an empty spot.

18. Collisions p p This is called a collision, because there is already another valid record at [2]. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868 When a collision occurs, move forward until you find an empty spot. When a collision occurs, move forward until you find an empty spot.

19. Collisions p p This is called a collision, because there is already another valid record at [2]. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868 The new record goes in the empty spot. The new record goes in the empty spot.

20. A Quiz Where would you be placed in this table, if there is no collision? Use your social security number or some other favorite number. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322 Number 580625685 Number 701466868...

21. Another Kind of Collision Where would you be placed in this table, if there is no collision? Use your social security number or some other favorite number. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322 Number 580625685 Number 701466868... Number 155779023 My hash value is [ ? ].

22. Another Kind of Collision Where would you be placed in this table, if there is no collision? Use your social security number or some other favorite number. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322 Number 580625685 Number 701466868... Number 155779023 My hash value is [700].

23. Another Kind of Collision Where would you be placed in this table, if there is no collision? Use your social security number or some other favorite number. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322 Number 580625685 Number 701466868... Number 155779023 My hash value is [700].

24. Searching for a Key p p The data that's attached to a key can be found fairly quickly. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868

25. Searching for a Key p p Calculate the hash value. p p Check that location of the array for the key. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868 My hash value is [2]. Not me.

26. Searching for a Key p p Keep moving forward until you find the key, or you reach an empty spot. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868 My hash value is [2]. Not me.

27. Searching for a Key p p Keep moving forward until you find the key, or you reach an empty spot. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868 My hash value is [2]. Not me.

28. Searching for a Key p p Keep moving forward until you find the key, or you reach an empty spot. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868 My hash value is [2]. Yes!

29. Searching for a Key p p When the item is found, the information can be copied to the necessary location. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868 My hash value is [2]. Yes!

30. Deleting a Record p p Records may also be deleted from a hash table. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 506643548 Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868 Please delete me.

31. Deleting a Record p p Records may also be deleted from a hash table. p p But the location must not be left as an ordinary "empty spot" since that could interfere with searches. [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868

32. Deleting a Record [ 0 ][ 1 ][ 2 ][ 3 ][ 4 ][ 5 ] [ 700] Number 233667136 Number 281942902 Number 155778322... Number 580625685 Number 701466868 p p Records may also be deleted from a hash table. p p But the location must not be left as an ordinary "empty spot" since that could interfere with searches. p p The location must be marked in some special way so that a search can tell that the spot used to have something in it.

33. Time Analysis p Without any collisions p With collisions

34. Time Analysis p Without any collisions p constant p With collisions p O(k) where k is the average collisions for items p k << n, size of the problem

35. Improving Hashing: ? p Size of the hashing table when using division hash function p prime number in the form of 4k+3 p Other hashing functions p mid-square, multiplicative p Double hashing (instead of linear probing) p the 2 nd hash function for stepping through the array p Chained hashing p using a linked list for each component of the hash table Exercise; size = 9? 10? 11?19? What is the best size given n items?

36. p p Hash tables store a collection of records with keys. p p The location of a record depends on the hash value of the record's key. p p When a collision occurs, the next available location is used. p p Searching for a particular key is generally quick. p p When an item is deleted, the location must be marked in a special way, so that the searches know that the spot used to be used. p p Applications? Summary

37. Next Lecture: Sorting

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