IKI 10100: Data Structures & Algorithms Ruli Manurung (acknowledgments to Denny & Ade Azurat) 1 Fasilkom UI Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Hash Tables

2 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Linked List insert, find, delete operations take O(n) Stack & Queue insert, find, delete operations take O(1) but the access is restricted Binary Search Tree insert, find, delete operations take O(log n) in average case, but take O(n) in worst case AVL Tree, Red-Black Tree insert, find, delete operations take O(log n) Review

3 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Review Array all operations take O(1) time data accessed using index (integer) size should be determined first not growable

4 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Hashing Definition Hash function Collision resolution Open hashing Separate chaining Closed hashing (Open addressing) Linear probing Quadratic probing Double hashing Primary Clustering, Secondary Clustering Access: insert, find, delete Outline

5 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Hash Tables Hashing is used for storing relatively large amounts of data in a table called a hash table ADT. Hash table is usually fixed as H-size, which is larger than the amount of data that we want to store. We define the load factor ( ) to be the ratio of data to the size of the hash table. Hash function maps an item into an index in range H-1 key hash function item hash table

6 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Hash Tables (2) Hashing is a technique used to perform insertions, deletions, and finds in constant average time. To insert or find a certain data, we assign a key to the elements and use a function to determine the location of the element within the table called hash function. Hash tables are arrays of cells with fixed size containing data or keys corresponding to data. For each key, we use the hashing function to map key into some number in the range 0 to H-size-1 using hashing function.

7 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Hash Function Hashing function should have the following features: Easy to compute. Two distinct key map to two different cells in array (Not true in general) - why?. This can be achieved by using direct-address table where universal set of keys is reasonably small. Distributes the keys evenly among cells. One simple hashing function is to use mod function with a prime number. Any manipulation of digits, with least complexity and good distribution can be used.

8 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Hash Function: Truncation Part of the key is simply ignored, with the remainder truncated or concatenated to form the index. Phone no:index

9 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Hash Function: Folding The data can be split up into smaller chunks which are then folded together in some form. Phone no: 3-groupindex

10 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Hash Function: Modular arithmetic Convert the data into an integer, divide by the size of the hash table, and take the remainder as the index. 3-groupindex % 100 = % 100 = % 100 = 25

11 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Choosing a hash function A good has function should satisfy two criteria: 1. It should be quick to compute 2. It should minimize the number of collisions

12 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Example of hash function Hash function for string X = 128 A 3 X 3 + A 2 X 2 + A 1 X 1 + A 0 X 0 (((A 3 X) + A 2 ) X + A 1 ) X + A 0 The result of hash function is much larger than the size of table, so we should modulo the result with the size of hash table.

13 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Example of hash function int hash(String key, int tableSize) { int hashVal = 0; for (int i=0; i < key.length(); i++) hashVal = (hashVal * key.charAt(i)) % tableSize; return hashVal % tableSize; } Modulo (A + B) % C = (A % C + B % C) % C (A * B) % C = (A % C * B % C) % C

14 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Example of hash function int hash(String key, int tableSize) { int hashVal = 0; for (int i=0; i < key.length(); i++) hashVal = (hashVal * 37 + key.charAt(i)); hashVal %= tableSize; if (hashVal < 0) hashVal += tableSize; return hashVal; }

15 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Example of hash function int hash(String key, int tableSize) { int hashVal = 0; for (int i=0; i < key.length(); i++) hashVal += key.charAt(i) return hashVal % tableSize; }

16 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Collision resolution When two keys map into the same cell, we get a collision. We may have collision in insertion, and need to set a procedure (collision resolution) to resolve it.

17 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Closed Hashing If collision, try to find alternative cells within table. Closed hashing also known as open addressing. For insertion, we try cells in sequence by using incremented function like: h i (x) = (hash(x) + f(i)) mod H-sizef(0) = 0 Function f is used as collision resolution strategy. The table is bigger than the number of data. Different method to choose function f : Linear probing Quadratic probing Double hashing

18 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Linear probing Use a linear function f(i) = i Find the first position in the table for the key, which is close to the actual position. Least complex function. May result in primary clustering. Elements that hash to the different location probe the same alternative cells The complexity of this probing is dependent on the value of (load factor). We do not use this probing if > 0.5.

19 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Hashing - insert dawn emerald crystal marigold alpha flamingo hallmark moon

20 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May cobalt? marigold? private? alpha crystal dawn emerald flamingo hallmark moon marigold private Hashing - lookup

21 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Hashing - delete delete emerald delete moon alpha crystal dawn flamingo hallmark marigold private lazy deletion - why?

22 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Hashing - operation after delete custom (insert) marigold? alpha crystal dawn flamingo hallmark marigold private

23 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May canary alpha crystal dawn custom flamingo hallmark marigold private cobalt canary alpha crystal dawn custom flamingo hallmark marigold private dark Primary Clustering Elements that hash to the different location probe the same alternative cells

24 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Quadratic probing Eliminate the primary clustering by selecting f(i) = i 2 There is more problem with a hash table that is more than half full. You have to select appropriate table size that is not square of a number. We can prove that quadratic probing with table size prime number and at least half empty will always find a location for an element. Can use increment to collision by noting that quadratic function f(i) = i 2 = f(i-1) + 2 i - 1. Elements that hash to the same location will probe the same alternative cells (secondary clustering).

25 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Double hashing Collision resolution function is another hash function like f(i) = i * hash2 (x) Each time a factor of hash2 (x) is added to probe. Have to be careful for the choice of second hash function to ensure that it does not come to zero and it probes all the cells. It is essential to have a prime size hash table.

26 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May canary alpha crystal dawn custom flamingo hallmark marigold private cobalt done alpha crystal dawn custom flamingo hallmark marigold private dark Double Hashing

27 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Open Hashing Collision problems is solved by inserting all elements that hash to the same bucket into a single collection of values. Open Hashing: To keep a linked list of all the elements that are hashed to the same cell (separate chaining). Each cell in the hash table contains a pointer to a linked list containing the data. Functions and Analysis of Open Hashing: Inserting a new element in to the table: We add the element at the beginning or the end of the appropriate linked list. Depending if you want to check for duplicates or not. Also depends on how frequent you expect to access the most recently added elements.

28 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May Open Hashing

29 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Open Hashing For search, we use the hash function to determine which linked list holds the element, and then traverse the linked list to find the element. Deletion is done to the element in the appropriate linked list after we find the element to be deleted. We could use other kinds of lists like a tree or another hash table for each cell in the hash table to resolve collision. The main advantage of this method is the fact that it can handle any amount of data (dynamic expansion). The main disadvantage of this method is the memory usage for each cell.

30 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Analysis of Open Hash In general the average length of a list is the load factor. Complexity of insertion depends on hashing function and where insertion is done but in general has the same complexity of insertion to the linked list + time to evaluate the hashing function used. For search, time complexity is the constant time to evaluate the hashing function + traversing the list. Worst case O(n) for search. Average case depends. General rule for open hashing is to make  1. Used for dynamic size data.

31 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Issues Other issues common to all closed hashing resolutions: Confusing after deletion. Simpler than open hashing function Good if we do not expect too many collisions. If search is unsuccessful, we may have to search the whole table. Use of large table compare to number of data expected.

32 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Summary Hash tables: array Hash function: function that maps key into number [0  size of hash table) Collision resolution Open hashing Separate chaining Closed hashing (Open addressing) Linear probing Quadratic probing Double hashing Primary Clustering, Secondary Clustering

33 Ruli Manurung (Fasilkom UI)IKI10100: Lecture8 th May 2007 Summary Advantage Running time O(1) + O(Collision resolution) Disadvantage Difficult (not efficient) to print all elements in hash table Inefficient to find minimum element or maximum element Not growable (for closed hash/open addressing) Waste some space (load factor)