Hashing. Prof. Dr. Debora Weber-Wulff Informatik 2

Transcription

1 Hashing Prof. Dr. Debora Weber-Wulff Informatik 2

2 Literature This lecture is based on Michael Waite, Data Structures and Algorithms in Java 2

3 Not this kind of hash.. Flickr, cc-by-nc-nd, Emma and Kunley,

4 nor this kind.. Flickr, cc-by-sa, avixyz,

5 Introduction to Hashing Sometimes we can use key values directly as the indices for an array. Other times we have a large range of key values that need to be transformed into a range of array index values. For example: matriculation numbers 5

6 Example: Matriculation Nr How are matriculation numbers given out? What kind of a data structure shall we use to store all the IMI students? How many IMI students are there? What are typical matriculation numbers? 6

7 Example: Matriculation Number If the numbers were sequential from the start we could just write: StudentRecord stud = allimis[72]; if we knew that Frieda Meyer was student number 72. 7

8 Example: Immatriculation Just remember the last one stored allimis[lastimi++]= new Student ( Jackie Chan ); We have to make sure the array has a bit more space for expansions. 8

9 Well, what s the Problem? This only works because the keys are orderly, they start at 1, increase by 1, and don t have holes. The array has no deletions, no gaps. 9

10 Not-so-well-behaved-keys Dictionary: Keys run from a to zyzzyva. Symbol Tables from scanning - access to the symbol table needs to be fast!... Zyzzyva is a genus of tropical American weevil often found in association with palms. It is also the last word in many English language dictionaries. 10

11 Converting Words to Numbers An easy solution: turn a word into a number! We can use the ASCII code for the characters. Or we can say: a=1, b=2,... z=26 But how do you add characters? 11

12 Example: Add the digits Cats: c = 3, a = 1, t = 20, s = = 43, so we can store it in index 43 Let s restrict words to 10 letters. Then a = 1 and zzzzzzzzzz = 260 But there are 50,000 words! How do we fit them into 260 cells? 12

13 Collisions! There are probably about 200 collisions for each index. Was, tin, give, tend, moan, tick, bails, dredge, etc. all have the value of 43. Too few elements in the array, we need to spread out the range of possible indices 13

14 Example: Multiply by Power Well, we could take a base 27 number (26 letters plus blank) and set: 3*27^3 + 1*27^2 + 20*27^1 + 19*27^0 which would be 60,337 This generates a unique number for every potential word! But: 27^9 is > 7,000,000,000,000 14

15 Goldilocks Goldilocks sat in the first chair to rest her feet. "This chair is too big!" she exclaimed. So she sat in the second chair. "This chair is too small!" she whined. So she tried the last chair. "Ahhh, this chair is just right," she sighed. 15

16 Modulo Operator We can map a huge range of numbers to a reasonably sized array index by using the modulo operator, which returns the remainder: smallnumber = largenumber%smallrange; 16

17 Hash Function This is an example of a hash function. It hashes (converts) a number in a large range into a number in a smaller range. Hash is a dish made of chopped up meat and vegetables. The smaller range corresponds to the index numbers in an array. 17

18 Collision Resolution There are many methods of collision resolution, we ll look at these: Linear probing, quadratic probing, double hashing, pseudo-random probing Separate chaining - a linked list for collecting up the collisions. Open hashing or bucket hashing 18

19 Simple Hash Function with Separate Chaining h(k) = k % m m = 7, K = {0,..., 500} Number sequence: 12, 53, 5, 15, 2, 19, / / / / /

20 Your turn! In groups of 2 or 3, what is the result with separate chaining? h(k) = k % m m = 11, K = {0,..., 500} Number sequence: 13, 53, 5, 24, 102, 222, 74, 2, 18, 73 20

21 Linear Probing If the hash position is already taken, the index is decreased by one (wrapping around) until a free position is found Order: h(k), h(k)-1, h(k)-2,... 0, m-1,... Number sequence: 12, 53, 5, 15, 2, 19,

22 Your turn! In groups of 2 or 3, what is the result with linear probing? h(k) = k % m Order: h(k), h(k)-1, h(k)-2,... 0, m-1,... m = 11, K = {0,..., 500} Number sequence: 13, 53, 5, 24, 102, 222, 74, 2, 18, 73 22

23 Problems with Linear Probing Extending the array is a pain - you have to rehash the entire sequence of data. This takes a lot of time! The more collisions there are, the longer the probing chain becomes 23

24 Quadratic Probing The longer the probing chain, the larger the step - using the square of the step Order: h(k), h(k)+1, h(k)+4, h(k)+9, h(k)+16, h(k)+25,... Always use a prime number as the size of the array! 24

25 Your turn! In groups of 2 or 3, what is the result with quadratic probing? h(k) = k % m Order: h(k), h(k)+1, h(k)+4, h(k)+9, h(k)+16, h(k)+25,... m = 11, K = {0,..., 500} Number sequence: 13, 53, 5, 24, 102, 222, 74, 2, 18, 73 25

26 Problem with Quadratic Probing Similar to the primary clustering problem with linear probing there is a secondary clustering problem with quadratic probing. 26

27 Double Hashing A solution to the secondary clustering problem is given with double hashing (or rehashing). If there is a collision on hashing the key with one function, a different function is applied and this value is taken as the step. Again, the table size must be prime. 27

28 Double Hashing S (j,k) = -j*h (k) Sequence: h(k), h(k) h (k), h(k)-2*h (k)... Try: h(k) = k mod m, h (k) = 1+k mod (m-2) for m=7: 12, 53, 5, 15, 2, 19 28

29 Buckets Each array element is a reference to an array of the kind of data being stored. We can think of there being a bucket for keeping a certain amount of data at that position. 29

30 Load Factor α = # items / size of table is referred to as the load factor. In separate chaining the load factor is often larger than 1. Complexity: finding first one O(1), finding right one O(M), M number of items on list. For other methods, α should be kept below 66% to reduce probing. 30

31 Duplicates Some hash functions do not do very well for duplicates (which will, of course, have the same values). Allowing duplicates will decrease performance. 31

32 Perfect Hash A perfect hash maps every key into a different table location. It is often possible to construct such a function for a given set of data (such as a keyword table). 32

33 Complexity? Insertion: O(1) Searching: O(1) With collisions: + length of chain 33