Hashing

1. Hashing

2. Motivating Applications • Large collection of datasets • Datasets are dynamic (insert, delete) • Goal: efficient searching/insertion/deletion • Hashing is ONLY applicable for exact-match searching

3. Direct Address Tables • If the keys domain is U  Create an array T of size U • For each key K  add the object to T[K] • Supports insertion/deletion/searching in O(1)

4. Direct Address Tables Alg.: DIRECT-ADDRESS-SEARCH(T, k) return T[k] Alg.: DIRECT-ADDRESS-INSERT(T, x) T[key[x]] ← x Alg.: DIRECT-ADDRESS-DELETE(T, x) T[key[x]] ← NIL • Running time for these operations: O(1) Solution is to use hashing tables Drawbacks >> If U is large, e.g., the domain of integers, then T is large (sometimes infeasible) >> Limited to integer values and does not support duplication

5. Direct Access Tables: Example Example 1: Example 2: U is the domain K is the actual number of keys

6. Hashing • A data structure that maps values from a certain domain or range to another domain or range Hash function 3 15 Domain: String values 20 55 Domain: Integer values

7. Hashing • A data structure that maps values from a certain domain or range to another domain or range Hash function Range 0 ….. 10000 Student IDs 950000 ….. 960000 Domain: numbers [0 … 10,000] Domain: numbers [950,000 … 960,000]

8. Hash Tables • When K is much smaller than U, a hash tablerequires much less space than a direct-address table • Can reduce storage requirements to |K| • Can still get O(1) search time, but on the average case, not the worst case

9. Hash Tables: Main Idea • Use a hash function h to compute the slot for each key k • Store the element in slot h(k) • Maintain a hash table of size m  T [0…m-1] • A hash function h transforms a key into an index in a hash table T[0…m-1]: h : U → {0, 1, . . . , m - 1} • We say that k hashes to slot h(k)

10. Hash Tables: Main Idea Hash Table (of size m) 0 U (universe of keys) h(k1) h(k4) k1 K (actual keys) h(k2) = h(k5) k4 k2 k3 k5 h(k3) m - 1 >> m is much smaller that U (m <<U) >> m can be even smaller than |K|

11. Example • Back to the example of 100 students, each with 9-digit SSN • All what we need is a hash table of size 100

12. What About Collisions 0 U (universe of keys) h(k1) h(k4) k1 K (actual keys) h(k2) = h(k5) k4 k2 Collisions! k3 k5 h(k3) m - 1 • Collision means two or more keys will go to the same slot

13. Handling Collisions • Many ways to handle it • Chaining • Open addressing • Linear probing • Quadratic probing • Double hashing

14. Chaining: Main Idea • Put all elements that hash to the same slot into a linked list (Chain) • Slot j contains a pointer to the head of the list of all elements that hash to j

15. Chaining - Discussion • Choosing the size of the hash table • Small enough not to waste space • Large enough such that lists remain short • Typically 10% -20% of the total number of elements • How should we keep the lists: ordered or not? • Usually each list is unsorted linked list

16. Insertion in Hash Tables Alg.:CHAINED-HASH-INSERT(T, x) insert x at the head of list T[h(key[x])] • Worst-case running time is O(1) • May or may not allow duplication based on the application

17. Deletion in Hash Tables Alg.:CHAINED-HASH-DELETE(T, x) delete x from the list T[h(key[x])] • Need to find the element to be deleted. • Worst-case running time: • Deletion depends on searching the corresponding list

18. Searching in Hash Tables Alg.:CHAINED-HASH-SEARCH(T, k) search for an element with key k in list T[h(k)] • Running time is proportional to the length of the list of elements in slot h(k) What is the worst case and average case??

19. T 0 chain m - 1 Analysis of Hashing with Chaining:Worst Case • All keys will go to only one chain • Chain size is O(n) • Searching is O(n) + time to apply h(k)

20. T 0 chain chain chain chain m - 1 Analysis of Hashing with Chaining:Average Case • With good hash function and uniform distribution of keys • Any given element is equally likely to hash into any of the m slots • All chain will have similar sizes • Assume n (total # of keys), m is the hash table size • Average chain size  O (n/m) Average Search Time O(n/m): The common case

21. Analysis of Hashing with Chaining:Average Case • If m (# of slots) is proportional to n (# of keys): • m = O(n) • n/m = O(1)  Searching takes constant time on average

22. Hash Functions

23. Hash Functions • A hash function transforms a key (k) into a table address (0…m-1) • What makes a good hash function? (1) Easy to compute (2) Approximates a random function: for every input, every output is equally likely (simple uniform hashing) (3) Reduces the number of collisions

24. Hash Functions • Goal: Map a key k into one of the m slots in the hash table • Make table size (m) a prime number • Avoids even and power-of-2 numbers • Common function h(k) = F(k) mod m Some function or operation on K (usually generates an integer) The output of the “mod” is number [0…m-1]

25. Examples of Hash Functions Collection of images F(k): Sum of the pixels colors h(k) = F(k) mod m Collection of strings F(k): Sum of the ascii values h(k) = F(k) mod m Collection of numbers F(k): just return k h(k) = F(k) mod m