Data and File Structures Laboratory
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1 Hashing Assistant Professor Machine Intelligence Unit Indian Statistical Institute, Kolkata September, 2018
2 1 Basics 2 Hash Collision 3 Hashing in other applications
3 What is hashing? Definition (Hashing) Hashing is the process of indexing and retrieving data items in a data structure to provide faster way (preferably O(1)) of finding the element using the hash function.
4 Hash function Definition (Hash function) A hash function h projects a value from a set with many (or even an infinite number of) data items to a value from a set with a fixed number of (fewer) data elements.
5 Hash function Definition (Hash function) A hash function h projects a value from a set with many (or even an infinite number of) data items to a value from a set with a fixed number of (fewer) data elements. Note: The hashed values are kept in a data structure known as hash tables.
6 Hash function Some popular examples of hash function are given below. k = h(m) = k%10, k = h(m) = k/2, k = h(m) = log k, etc.
7 Hash function Some popular examples of hash function are given below. k = h(m) = k%10, k = h(m) = k/2, k = h(m) = log k, etc. Definition (Preimage) For a hash value k = h(k), k is defined as the preimage of k.
8 What makes a good hash function? We want to design a hash function h : [n] [m] (n > m) that satisfies the following requirements: Searching (lookup) is worst-case O(1). Deletions are worst-case O(1). Insertions are amortized, expected O(1). Each data item is equally likely to hash to any of the m positions The function h is computationally collision free.
9 What makes a good hash function? We want to design a hash function h : [n] [m] (n > m) that satisfies the following requirements: Searching (lookup) is worst-case O(1). Deletions are worst-case O(1). Insertions are amortized, expected O(1). Each data item is equally likely to hash to any of the m positions The function h is computationally collision free. Note: Depending on the application, there might be additional requirements.
10 Family of hash functions Definition (k-independent family of hash functions) A family of hash functions is said to be k-independent if selecting a function at random from the family guarantees that the hashed values of any designated k keys are independent random variables.
11 Family of hash functions Definition (k-independent family of hash functions) A family of hash functions is said to be k-independent if selecting a function at random from the family guarantees that the hashed values of any designated k keys are independent random variables. Note: Some efficient approaches are required to satisfy k-independence property.
12 What is hash collision? If multiple data items (keys) hash to the same position then it is termed as a hash collision. Formally, a pair of keys k 1 and k 2 are said to have hash collision with respect to the hash function h() if h(k 1 ) = h(k 2 ). For example, the keys 121 and will have hash collision with respect to the hash function h(k) = k%11.
13 Dealing with hash collision Strategy 1: Resolution Closed addressing: Store all the elements with hash collisions in an auxiliary data structure (e.g., linked list, BST, etc.) outside the hash table. Open addressing: Store all the elements with hash collisions by strategically moving them from preferred to the other positions in the hash table itself.
14 Dealing with hash collision Strategy 1: Resolution Closed addressing: Store all the elements with hash collisions in an auxiliary data structure (e.g., linked list, BST, etc.) outside the hash table. Open addressing: Store all the elements with hash collisions by strategically moving them from preferred to the other positions in the hash table itself. Strategy 2: Avoidance Perfect hashing: Ensure that collisions do not happen and if happen relocate the other elements.
15 Dealing with hash collision Strategy 1: Resolution Closed addressing: Store all the elements with hash collisions in an auxiliary data structure (e.g., linked list, BST, etc.) outside the hash table. Open addressing: Store all the elements with hash collisions by strategically moving them from preferred to the other positions in the hash table itself. Strategy 2: Avoidance Perfect hashing: Ensure that collisions do not happen and if happen relocate the other elements. Note: Closed addressing is also termed as chaining.
16 Closed addressing Closed addressing uses an auxiliary data structure whose domain D is defined as follows D := {k i k j H : k i k j, h(k i ) = h(k j )} Here, H denotes the hash table. Note: Use of auxiliary data structures include additional burdens (of pointer dereferencing) that are not cache-friendly,
17 Closed addressing Insertion Let h(k) = k%26. k
18 Closed addressing Insertion Let h(k) = k%26. k INSERTION
19 Closed addressing Insertion Let h(k) = k%26. k CHAINING
20 Closed addressing Insertion Let h(k) = k%26. k INSERTION
21 Closed addressing Insertion Let h(k) = k%26. k CHAINING
22 Closed addressing Insertion Let h(k) = k%26. k CHAINING
23 Closed addressing Insertion Let h(k) = k%26. k CHAINING
24 Closed addressing Implementation Traditional: typedef struct node{ unsigned hash_val; DATA data; struct node *next; }HNODE;
25 Closed addressing Implementation Traditional: typedef struct node{ unsigned hash_val; DATA data; struct node *next; }HNODE; Alternative: Using unrolled linked lists!!! #define HASH_UNROLL 10 typedef struct node{ unsigned hash_val[hash_unroll]; DATA data[hash_unroll]; // Array of elements at a node struct node *next; }HNODE;
26 Closed addressing Implementation Unrolled linked list: This is a variant of linked list containing nodes of small arrays (of same size), which are large enough to fill the cache line. An iterator pointing into the list comprises both a pointer to a node and an index into that node containing an array. Note: Unrolled linked lists are conceptually related to B-trees.
27 Open addressing with linear probing Linear probing uses a hash function of the form h(k, i) = (h (k) + i)%m. Here, h is an auxiliary hash function and i = 0, 1,..., m 1. Note: The number of collisions tends to grow as a function of the number of existing collisions. This problem is known as primary clustering. It increases the average search time in a hash table.
28 Open addressing with linear probing Insertion Let h (k) = k%26. k
29 Open addressing with linear probing Insertion Let h (k) = k%26. k INSERTION
30 Open addressing with linear probing Insertion Let h (k) = k%26. k INSERTION
31 Open addressing with linear probing Insertion Let h (k) = k%26. k INSERTION
32 Open addressing with linear probing Insertion Let h (k) = k%26. k COLLISION
33 Open addressing with linear probing Insertion Let h (k) = k%26. k PROBING
34 Open addressing with linear probing Insertion Let h (k) = k%26. k COLLISION
35 Open addressing with linear probing Insertion Let h (k) = k%26. k PROBING, COLLISION
36 Open addressing with linear probing Insertion Let h (k) = k%26. k PROBING
37 Open addressing with linear probing Insertion Let h (k) = k%26. k INSERTION
38 Open addressing with linear probing Searching Let h (k) = k%26. s ? LOOKUP, MOVE
39 Open addressing with linear probing Searching Let h (k) = k%26. s ? LOOKUP, MOVE
40 Open addressing with linear probing Searching Let h (k) = k%26. s ? LOOKUP, NOWHERE
41 Open addressing with linear probing Searching Let h (k) = k%26. s ? 9 23 LOOKUP, MOVE
42 Open addressing with linear probing Searching Let h (k) = k%26. s LOOKUP, FOUND 35?
43 Open addressing with linear probing Searching Let h (k) = k%26. s LOOKUP, FOUND ?
44 Open addressing with linear probing Deletion Let h (k) = k%26. s ? 9 23 LOOKUP, DELETE Note: The symbol represents the tombstones.
45 Open addressing with linear probing Deletion Let h (k) = k%26. s ? LOOKUP, DELETE
46 Open addressing with linear probing Deletion Let h (k) = k%26. s ? LOOKUP, DELETE
47 Open addressing with linear probing Deletion Let h (k) = k%26. s ? 9 23 LOOKUP, NOWHERE To trace linearly probed items, we have to keep track of the positions from where items have been deleted!!!
48 Open addressing with linear probing Deletion Let h (k) = k%26. s $ $ $? 9 23 LOOKUP, MOVE Note: The tombstones (denoted with the symbol $) are used to keep track of the positions of deleted items.
49 Open addressing with quadratic probing Quadratic probing uses a hash function of the form h(k, i) = (h (k) + (c 1 i 2 + c 2 i))%m. Here, h is an auxiliary hash function, c 1 and c 2 are auxiliary constants, and i = 0, 1,..., m1. Note: It suffers from a problem known as secondary clustering.
50 Open addressing with double hashing Double hashing uses a hash function of the form h(k, i) = (h 1 (k) + i h 2 (k))%m. The permutations produced have many of the characteristics of randomly chosen permutations. Note: It has the only disadvantage that as soon as the hash table fills up the performance degrades.
51 Robin Hood hashing Robinhood hashing is a variation of open addressing where keys can be moved after they are placed. When an existing key is found during an insertion that is closer to its preferred location than the new key, it is displaced (relocation) to make room for it. This dramatically decreases the variance in the expected number of searchers (lookups). It also makes it possible to terminate searches (lookups) early. Note: Assuming truly random hash functions, the variance of the expected number of probes required in Robin Hood hashing is O(log log n).
52 Robin Hood hashing Insertion Let h (k) = k%26. k
53 Robin Hood hashing Insertion Let h (k) = k%26. k RELOCATION
54 Robin Hood hashing Insertion Let h (k) = k%26. k RELOCATION
55 Robin Hood hashing Insertion Let h (k) = k%26. k INSERTION
56 Cuckoo hashing We choose a pair of hash functions h 1 and h 2 such that h 1 : [n] [m] and h 2 : [n] [m]. We use two tables, each of which can accommodate m items. Every item k R will either be at position h 1 (k) in the first table or at h 2 (k) in the second.
57 Cuckoo hashing We choose a pair of hash functions h 1 and h 2 such that h 1 : [n] [m] and h 2 : [n] [m]. We use two tables, each of which can accommodate m items. Every item k R will either be at position h 1 (k) in the first table or at h 2 (k) in the second. Note: New hash functions might be required to be introduced in case of critical conditions.
58 Cuckoo hashing Insertion 1 To insert an item k, start by inserting it into Table 1. 2 If h 1 (k) is empty, place k there. 3 Otherwise, place k there, taking out the old item k and relocating it into Table 2 at h 2 (k ). 4 Repeat this process, bouncing between the tables, until all the items stabilize. 5 If the same position is revisited with the same item to insert (known as a cycle), perform rehashing by choosing a new pair of hash functions and insert all items back into the tables.
59 Cuckoo hashing Insertion 1 To insert an item k, start by inserting it into Table 1. 2 If h 1 (k) is empty, place k there. 3 Otherwise, place k there, taking out the old item k and relocating it into Table 2 at h 2 (k ). 4 Repeat this process, bouncing between the tables, until all the items stabilize. 5 If the same position is revisited with the same item to insert (known as a cycle), perform rehashing by choosing a new pair of hash functions and insert all items back into the tables. Note: Multiple rehashes might be necessary before it succeeds.
60 Cuckoo hashing Insertion Let h 1 (k) = k%8 and h 2 (k) = [log 10 k]. k
61 Cuckoo hashing Insertion Let h 1 (k) = k%8 and h 2 (k) = [log 10 k]. k INSERTION
62 Cuckoo hashing Insertion Let h 1 (k) = k%8 and h 2 (k) = [log 10 k]. k INSERTION
63 Cuckoo hashing Insertion Let h 1 (k) = k%8 and h 2 (k) = [log 10 k]. k INSERTION
64 Cuckoo hashing Insertion Let h 1 (k) = k%8 and h 2 (k) = [log 10 k]. k ND CHOICE
65 Cuckoo hashing Insertion Let h 1 (k) = k%8 and h 2 (k) = [log 10 k]. k RELOCATION
66 Cuckoo hashing Insertion Let h 1 (k) = k%8 and h 2 (k) = [log 10 k]. k ND CHOICE
67 Cuckoo hashing Insertion Let h 1 (k) = k%8 and h 2 (k) = [log 10 k]. k RELOCATION
68 Cuckoo hashing The cuckoo graph Note: An insertion in cuckoo hash tables traces a path through the cuckoo graph. An insertion succeeds iff the connected component containing the inserted item contains at most one cycle.
69 Cuckoo hashing Deletion Let h 1 (k) = k%8 and h 2 (k) = [log 10 k]. k LOOKUP, MOVE 4 4?
70 Cuckoo hashing Deletion Let h 1 (k) = k%8 and h 2 (k) = [log 10 k]. k ? LOOKUP, DELETE
71 Cryptography Definition (One-way function) A function y = f (x) that satisfies the following two properties is known as a one-way function. 1 Given x, it is easy to compute y = f (x). 2 Given y, it is computationally infeasible to compute x = f 1 (y). E.g., y = f (x 1, x 2 ) = x 1 x 2, where x 1 and x 2 are two large prime numbers, is a one-way function. Note: A one-way function is not necessarily a single-variable function.
72 Cryptography Requirement Description Variable input size h can be applied to a block of data of any size Fixed output size h produces a fixed-length output h(k) is relatively easy to compute for any Efficiency given k, making both hardware and software implementations practical Preimage resistant For any given hash value k, it is computationally (one-way property) infeasible to find k such that k = h(k) Second preimage For any given key k 1, it is computationally resistant (weak infeasible to find k 2 k 1 with h(k 1) = h(k 2) collision resistant) Collision resistant It is computationally infeasible to find any pair (strong collision (k 1, k 2) such that h(k 1) = h(k 2) resistant) Pseudorandomness Output of k meets standard tests for pseudorandomness
73 Cryptography The MD5 hashing mechanism
74 Dimensionality reduction The locality sensitive hashing (LSH)
75 Problems Day 19 1 Given a string with English alphabets as input, write a program to return its largest substring that contains equal number of vowels and consonants. The only constraint is that the time consumed by your program must be linear to the length of the string. 2 Write a program to implement Cuckoo hashing for storing some given numbers within [1, n] with the following hash functions. h 1 (k) = k/2 h 2 (k) = k%(n/2) Consider that n is a user input.
76 Problems Day 19 3 Given an integer matrix and a column number as input, write a program to efficiently find out all the columns that are permutation of the given column. Input Format: # No. of rows, No. of columns, input column Output: 2, 3 4 Write a program that will take a set of n distinct integers as user input and find out the largest x 4 in O(n 2 ) time such that x 1 + x 2 + x 3 = x 4, where x 1, x 2, x 3 and x 4 are all distinct elements from the input set.
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