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CS201: Data Structures and Discrete Mathematics I

CS201: Data Structures and Discrete Mathematics I. Hash Table. Hashing. Dictionary: in many tasks, one needs store a collection of data and find item in it An important and widely useful technique for implementing dictionaries Insert, delete, and search

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CS201: Data Structures and Discrete Mathematics I

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  1. CS201: Data Structures and Discrete Mathematics I Hash Table

  2. Hashing • Dictionary: in many tasks, one needs store a collection of data and find item in it • An important and widely useful technique for implementing dictionaries • Insert, delete, and search • Constant time per operation (on average) • Worst case time proportional to the size of the set for each operation (just like array and chain/list implementation)

  3. Data collection • Data collection is a set of records (static or dynamic) • Each record consists of two parts • A key: a unique identifier of the record. • Data item: it can be arbitrarily complex. • The key is usually a number, but can be a string or any other data type. • Non-numbers are converted to numbers when applying hashing.

  4. Basic ideas of hashing • Use hash functionto map keys into positions in a hash table Ideally • If data item or element e has key k and h is hash function, then e is stored in position h(k) of table • To search for e, compute h(k) to locate position. If no element/item, dictionary does not contain e.

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

  6. Following the example • We want to store a dictionary of Object Records, no more than 701 objects • Keys are Object ID numbers, e.g., 506643548 • Hash function: h(k) maps k(=ID) into distinct table positions 0-700 • Operations: insert, delete, and search

  7. Complexity (ideal case) • Why hashing and hash table? • It is very efficient. • O(D) time to initialize hash table (D number of positions or cells in hash table) • O(1) time to perform insert, remove, search

  8. [ 4 ] Use the Hash Table [ 4 ] • Each record has a special field, i.e., its key. • In this example, the key is a long integer field called Number. 506643548 [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700] . . .

  9. [ 4 ] Use the Hash Table [ 4 ] • The number is a object's identification number, and the rest of the record has information about the object. 506643548 [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700] . . .

  10. . . . Use the Hash Table • When a hash table is in use, some spots contain valid records, and other spots are "empty". [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700] 155778322 233667136 506643548

  11. . . . Inserting a New Record • In order to insert a new record, the key must somehow be mapped toan array index using ahash function. • The index is called the hash valueof the key. 580625685 [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700] 155778322 233667136 506643548

  12. Hash functions • Popular hash functions: hashing by division h(k) = k mod D, where D is number of cells in hash table • Example: hash table with 701 cells h(k) = k mod 701 h(80) = 80 mod 701 = 80 h(1000) = 1000 mod 701 = 299

  13. . . . Inserting a New Record • Let us find the hash value for 580625685 What is (580625685 mod 701) ? 580625685 [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700] 155778322 233667136 506643548

  14. . . . Inserting a New Record • Let us find the hash value for 580625685 580625685 mod 701 = 3 580625685 3 [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700] 155778322 233667136 506643548

  15. . . . Inserting a New Record The hash value is used to find the location of the new record. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700] 580625685 155778322 233667136 506643548

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

  17. . . . Collisions • This is called a collision, because there is already another valid record at [2]. 701466868 [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700] 580625685 155778322 233667136 506643548

  18. Collision Resolution Policies • Two strategies: • (1) Open hashing, a.k.a. separate chaining • (2) Closed hashing, a.k.a. open addressing • Difference has to do with whether collisions are stored outside the table (open hashing) or whether collisions result in storing one of the records at another slot in the table (closed hashing)

  19. Closed hashing • Associated with closed hashing is a rehash strategy: “If we try to place x in cell h(x) and find it occupied, find alternative location h1(x), h2(x), etc. Try each in order. If none empty, table is full,” • h(x) is called home cell • Simplest rehash strategy is called linear hashing hi(x) = (h(x) + i) mod D

  20. . . . Collisions: closed hashing • Collision, there is already another valid record at [2]. 701466868 When a collision occurs, move forward until we find an empty spot. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700] 580625685 155778322 233667136 506643548

  21. . . . Collisions: closed hashing • Collision, there is already another valid record at [2]. 701466868 When a collision occurs, move forward until we find an empty spot. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700] 580625685 155778322 233667136 506643548

  22. . . . Collisions: closed hashing • Collision, there is already another valid record at [2]. 701466868 When a collision occurs, move forward until we find an empty spot. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700] 580625685 155778322 233667136 506643548

  23. . . . Collisions: closed hashing • Collision, there is already another valid record at [2]. The new record goes in the empty spot. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 700] 580625685 155778322 233667136 701466868 506643548

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

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

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

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

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

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

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

  31. Deleting a Record • Records may also be deleted from a hash table. • 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] 580625685 155778322 233667136 701466868

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

  33. Open hashing • Each cell in the hash table is the head of a linked list • All records or elements that hash to a particular cell are placed on that cell’s linked list • Records within a cell can be ordered in several ways • by order of insertion, by key value order, or by frequency of access order

  34. Open hashing: same example [ 0 ] [ 1 ] 233667136 701466868 [ 2 ] 580625685 [ 3 ] [ 4 ] 506643548 [ 5 ] 155778322 [ 700]

  35. Summary • Hash tables store a collection of records with keys. • The location of a record depends on the hash value of the record's key. • When a collision occurs, use a collision resolution strategy to deal with it. • Hash table is a very efficient way to store and to search for data

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