Basic Idea of Hashing Hashing

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1 Principles of Imperative Computation V. Adamchik CS 5-22 Carnegie Mellon University Hashing Data Structure Searching Worst-case runtime complexity Unsorted Array Sorted Array Unsorted Linked List Sorted Linked List Hash Table linear logarithmic linear linear Constant (on average) Library of Congress Classification How to implement search() that will run at O()? Each book in the library has a unique call number. A call number is like an address: it tells us where the book is located in the library. Basic Idea of Hashing Hashing array[ H(data) ] = data We need to create a function H that will map data into integers. Such function H is called a hash function. We will use these integers as table indexes. array[ H(data) % m ] = data

2 Hash Function Hash function has the following properties: Hash Function for Strings in Java s[0]*3 n- + s[] *3 n s[n-] it always returns a number for a key. two equal keys will always have the same number two unequal keys not always have different numbers "ABC" is mapped to 64578=65* *3+67 'A' has an ASCII value 65 Examples of hash function int java_hash_function(string str) { int h = 0; for (int i = 0; i < string_length(str); i++) h = 3*h + char_ord(string_charat(str, i)); Hash Collisions h(key) = h(key2) key not equal to key2 Some keys map to the same location. } return h; s[0]*3 n- + s[] *3 n s[n-] Example, "Aa" = 'A' * 3 + 'a' = 22 "BB" = 'B' * 3 + 'B' = 22 Pseudo-Random Number Generator X n+ = (a*x n + b) mod 2 32 a = b = X 0 = random seed Examples of randomized hash function int hash_function(string str) { int a = ; int b = ; int r = 0x337beef; int h = 0; for (int i = 0; i < string_length(str);; i++) { h = r * h + char_ord(string_charat(str, i)); r = r*a + b; } return h; } 2

3 Examples of cryptographic hash function SHA- is a cryptographic hash function (60 bits) SHA("The quick brown fox jumps over the lazy dog") = 2fd4ec6 7a2d28fc ed849ee bb76e739 b93eb2 Pigeonhole principle: Collisions If n items are put into m < n pigeonholes, then at least one pigeonhole must contain more than one item. SHA("The quick brown fox jumps over the lazy cog") = de9f2c7f d25eb3a fad3e85a 0bd7d9b 00db4b3 The Birthday Paradox The Birthday Paradox What probability that two people in this room have the same birthday? The first person has a 00% chance of a unique number The second has a ( - /) chance (all but ) The third has a ( - 2/) chance (all but 2) The 40th has a ( - 39/) chance The Birthday Paradox The probability that NO two have the same birthday is Then, with probability 90% two people out of 40 have the same birthday. Applying this to hashing Given an m-element hashtable. The probability of NO collisions after k insertions is m m m k... m m m This tends to 0, for large k. 3

4 Probability of NO collisions Collisions No mater how good our hash function is, we better be prepared for collisions. h (A) A h (B) B For N=00 and M=30 the probability is < % Separate chaining Collision resolution by chaining We put the objects that collide in a linked list. A hashtable is an array of linked lists. Load Factor Exercise A hashtable is an array of linked lists. The ratio between the number of elements M in the table to the table size N is called a load factor. M N 2 Consider a hashtable of size 5. Insert the following sequence {0, 6, 2, 5,, 2, 6} into the table. What is the load factor? 4

5 Separate chaining Consider a hashtable of size 5. Insert the following sequence {0, 6, 2, 5,, 2, 6} into the table load factor or average length Worst-case complexity Insertion O(n) Searching O(n) Deletion O(n) Average case analysis Average case analysis An unsuccessful search A successful search the computation of h(k) takes O() each list has average length l Thus, search takes O( + l) on average Even faster than unsuccessful search Disadvantages of Separate Chaining We can end-up with pretty long linked list at some indexes this will slow down searching. Linked lists are very expensive in external memory. Rehashing When the load factor is getting greater than a chosen value, we rehash. Rehashing is not so expensive as it seems. We can prove that rehashing runs at amortized constant time. M N 3 5

6 typedef struct list_node* list; typedef struct hset_header* hset; struct list_node { elem data; list next; }; struct hset_header{ int size; int cap; list[] array; }; 6