CS200: Hash Tables. Prichard Ch CS200 - Hash Tables 1

Transcription

1 CS200: Hash Tables Prichard Ch CS200 - Hash Tables 1

2 Table Implemetatios: average cases Search Add Remove Sorted array-based Usorted array-based Balaced Search Trees O(log ) O() O() O() O(1) O() O(log ) O(log ) O(log ) Ca we build a faster data structure? CS200 - Hash Tables 2

3 Fast Table Access Suppose we have a magical address calculator tableisert(i: ewitem:tableitemtype) // magicalc uses ewitem s search key to // compute a idex i = magicalc(ewitem.key) table[i] = ewitem CS200 - Hash Tables 3

4 Hash Fuctios ad Hash Tables Magical address calculators exist: They are called hash fuctios hash table CS200 - Hash Tables 4

5 Hash Table: early-costat-time A hash table is a array i which the idex of the data is determied directly from the key which provides ear costat time access! locatio of data determied from the key q table implemeted usig array(list) q idex computed from key usig a hash fuctio or hash code close to costat time access if we have a early uique mappig from key to idex q cost: extra space for uused slots CS200 - Hash Tables 5

6 Hash Table: examples q key is strig of 3 letters array of (26 3 ) etries hash code: letters are radix 26 digits a/a -> 0, b/b -> 1,.., z/z -> 25, Example: Joe -> 9*26*26+14*26+4 q key is studet ID or social security # how may likely etries? CS200 - Hash Tables 6

7 Hash Table Issues bat coat dwarf hoax law Uderlyig data-structure q q fixed legth array, usually of prime legth each slot cotais data Addressig q q map key to slot idex (hash code) use a fuctio of key e.g., first letter of key What if we add cap? q q collisio with coat collisio occurs because hashcode does ot give uique slots for each key. CS200 - Hash Tables 7

8 Hash Fuctio Maps Key to Idex Desired Characteristics q uiform distributio, fast to compute q retur a iteger correspodig to slot idex withi array size rage q equivalet objects => equivalet hash codes what is equivalet? Depeds o the applicatio, e.g. upper ad lower case letters equivalet Joe == joe Perfect hash fuctio: guaratees that every search key maps to uique address takes potetially eormous amout of space caot always be achieved (e.g., ubouded legth strigs) CS200 - Hash Tables 8

9 Hash Fuctio Computatio Fuctios o positive itegers q Selectig digits (e.g., select a subset of digits) q Foldig: add together digits or groups of digits, or premultiply with weights, the add q Ofte followed by modulo arithmetic: hashcode % table size CS200 - Hash Tables 9

10 What could be the hash fuctio if selectig digits? h( ) = 5 h( ) = 97 h(225671) =? A. 39 B. 31 C. 21 CS200 - Hash Tables 10

11 Hash fuctio: Foldig Suppose the search key is a 9-digit ID. Sum-of-digits: h( ) = satisfies: 0 <= h(key) <= 81 Groupig digits: = <= h(search key) <=3*999=2997 CS200 - Hash Tables 11

12 Hash fuctio data distributio Assume key is a Strig Pick a hash table size; compute key to ay iteger usig some hash fuctio h idex = h(key)%size h(key) e.g.: Sum(i=0 to le-1) getnumericvalue(strig.charat(i))*radix i q similar to Java built-i hashcode() method This does ot work well for very log strigs with large commo subsets (URLs) or Eglish words. CS200 - Hash Tables 12

13 hashcode o words Letter frequecy is NOT UNIFORM i the Eglish laguage (actually i o laguage) Highest frequecy for e : 12% followed by t : 9% followed by a : 8% The polyomial evaluatio i hashcode followed by takig modulo hashsize gives rise to a o uiform hash distributio. CS200 - Hash Tables 13

14 hashsize = 1000 vs 1009 CS200 - Hash Tables 14

15 Collisios Collisio: two keys map to the same idex Hash fuctio: key%101 both 4567 ad 7597 map to 22 CS200 - Hash Tables 15

16 The Birthday Problem What is the miimum umber of people so that the probability that at least two of them have the same birthday is greater tha ½? Assumptios: q Birthdays are idepedet q Each birthday is equally likely

17 The Birthday Problem What is the miimum umber of people so that the probability that at least two of them have the same birthday is greater tha ½? Assumptios: q Birthdays are idepedet q Each birthday is equally likely p the probability that all people have differet birthdays p = at least two have same birthday: = 23 1 p ( 1) 366

18 The Birthday Problem: Probabilities N: # of people P(N): probability that at least two of the N people have the same birthday % % % % % % % % % CS200 - Hash Tables 18

19 Probability of Collisio How may items do you eed to have i a hash table, so that the probability of collisio is greater tha ½? For a table of size 1,000,000 you oly eed 1178 items for this to happe! CS200 - Hash Tables 19

20 Collisios Collisio: two keys map to the same idex Hash fuctio: key%101 both 4567 ad 7597 map to 22 CS200 - Hash Tables 20

21 Methods for Hadlig Collisios Approach 1: Ope addressig q Probe for a empty (ope) slot i the hash table Approach 2: Restructurig the hash table q Chage the structure of the array table: make each hash table slot a collectio (e.g. ArrayList, or liked list), ofte called separate chaiig CS200 - Hash Tables 21

22 Ope addressig Whe collidig with a locatio i the hash table that is already occupied q Probe for some other empty, ope, locatio i which to place the item. q Probe sequece The sequece of locatios that you examie Liear probig uses a costat step, ad thus probes loc, (loc+step)%size, (loc+2*step)%size, etc. I the sequel we use step=1 for liear probig examples CS200 - Hash Tables 22

23 Liear Probig, step = 1 Use first char. as hash fuctio q Iit: ale, bay, egg, home Where to search for q q egg ik Where to add gift age hash code 4 hash code 8 6 empty 0 full, 1 full, 2 empty Questio: Durig the process of liear probig, if there is a empty spot, A. Item ot foud? or B. There is still a chace to fid the item? ale bay age egg gift home

24 Ope addressig: Liear Probig Deletio: The empty positios created alog a probe sequece could cause the retrieve method to stop, icorrectly idicatig failure. Resolutio: Each positio ca be i oe of three states occupied, empty, or deleted. Retrieve the cotiues probig whe ecouterig a deleted positio. Isert ito empty or deleted positios. CS200 - Hash Tables 24

25 Liear Probig (cot.) isert q bay q age q acre remove q bay q age retrieve q acre ale egg gift home Questio: Where does almod go ow?

26 Ope Addressig 1: Liear Probig Primary Clusterig Problem keys startig with a, b, c, d all compete for same ope slot (3) ale bay age egg gift home

27 Ope Addressig: Quadratic Probig check h(key) + 1 2, h(key) + 2 2, h(key) + 3 2, Elimiates the primary clusterig pheomeo But secodary clusterig: two items that hash to the same locatio have the same probe sequece is ot solved CS200 - Hash Tables 27

28 Ope Addressig: Double Hashig Use two hash fuctios: h 1 (key) determies,as always, the iitial positio h 2 (key) determies the liear step size for probig q the secodary hash h 2 eeds to satisfy: h 2 (key) 0 h 2 h 1 (bad distributio characteristics) So which locatios are ow probed? h 1, h 1 +h 2, h 1 +2*h 2,, h 1 +i*h 2, Now two differet keys that hash with h 1 to the same locatio most likely (but ot for sure, see ext slide) have differet secodary hash h 2 CS200 - Hash Tables 28

29 Double Hashig, example POSITION: h 1 (key) = key % 11 STEP: h 2 (key) = 7 (key % 7) Isert 58, 14, 91 h1(58) = 3, put it there h1(14) = 3 collisio h2(14) = 7-(14%7) = 7 put it i (3+7)%11 = 10 1 collisio here h1(91) = 3 collisio h2(91) = 7-(91%7) = = 10 collisio put it i (10+7)%11 = 6 2 collisios here CS200 - Hash Tables 29

30 Ope Addressig: Icreasig the table size Icreasig the size of the table: as the table fills the likelihood of a collisio icreases. q Caot simply icrease the size of the table eed to ru the hash fuctio agai CS200 - Hash Tables 30

31 Restructurig the Hash Table: Hybrid Data Structures Separate Chaiig: elemets i hash table become collectios q elemets hashig to same slot grouped together i a collectio (or chai ) q the chai is a separate structure e.g., ArrayList or liked-list, or BST a good hash fuctio keeps a ear uiform distributio, ad hece the collectios small chaiig does ot eed special case for removal as ope addressig does

32 Separate Chaiig Example Hash fuctio q first char Locate q q Add q egg gift bee? Remove q bay? bay elk gate egg

33 The Efficiecy of Hashig Cosider a hash table with items q Load factor α = / tablesize q : curret umber of items i the table q tablesize: maximum size of array q α : a measure of how full the hash table is. measures difficulty of fidig empty slots Efficiecy decreases as ad thus α icreases CS200 - Hash Tables 33

34 Size of Table Determiig the size of Hash table q Estimate the largest possible q Select the size of the table to get the load factor small. q Rule of thumb: load factor should ot exceed 2/3. CS200 - Hash Tables 34

35 Hashig: Legth of Probe Sequece Average umber of comparisos (i.e. collisios) q Liear Probig 1 " successful % # $ 1 α & ' WORST usuccessful usuccessful q Separate chaiig 1 " % $ # (1 α) 2 ' & q Quadratic Probig ad Double Hashig successful log e ( 1 α) α 1 1 α successful: 1 + α/2 usuccessful: α Note that α ca be > 1 for chaiig BETTER BEST From D.E. Kuth, Searchig ad Sortig, Vol. 3 of The Art of Computer Programmig CS200 - Hash Tables 35

36 Average legth of probe sequece Liear Quadratic, double hashig Separate Chaiig Liear Quadratic, Double hashig Separate Chaiig successful search usuccessful search CS200 - Hash Tables 36

37 Hash Tables i Java From the JAVA API: A map is a object that maps keys to values The HashMap class is roughly equivalet to HashTable, except that it is usychroized ad permits ulls. Both provide methods to create ad maitai a hash table data structure with key lookup. Load factor (default 75%) specifies whe the hash table capacity is automatically icreased. public class Hashtable<K,V> exteds Dictioary<K,V> implemets Map<K,V> public Hashtable(it iitialcapacity, float loadfactor) public Hashtable(it iitialcapacity) //default loadfactor: 0.75 public class HashMap<K,V> exteds AbstractMap<K,V> implemets Map<K,V> public HashMap(it iitialcapacity, float loadfactor) public HashMap(it iitialcapacity) //default loadfactor: 0.75 CS200 - Hash Tables 37