11. Hash Tables. m is not too large. Many applications require a dynamic set that supports only the directory operations INSERT, SEARCH and DELETE.

Transcription

1 11. Hash Tables May applicatios require a dyamic set that supports oly the directory operatios INSERT, SEARCH ad DELETE. A hash table is a geeralizatio of the simpler otio of a ordiary array. Directly addressig ito a ordiary array makes effective use of our ability to examie a arbitrary positio i the array i O(1) time, idepedet of the size of array. Direct addressig is a simple techique that works well whe the uiverse U of the keys (all possible values of k) is reasoably small, where U = {0,1,...,m 1}, m is ot too large o two elemets share the same key. COMP3600/6466: Lecture

2 11.1. Direct-address tables Direct-Addressig (Corme et al., p254) COMP3600/6466: Lecture

3 11.1 Operatios o direct-address tables To represet a dyamic set that has isertio, deletio, ad searchig operatios, we use a direct-address table, deoted by T [0..m 1], i which each positio, or slot, correspods to a key i the uiverse U. DIRECT ADDRESS SEARCH(T,k) retur T [k] DIRECT ADDRESS INSERT(T,x) T [key[x]] x DIRECT ADDRESS DELETE(T,x) T [key[x]] NIL COMP3600/6466: Lecture

4 11.2 Hash tables With hashig, the elemet is stored i slot h(k), i.e., we use a hash fuctio h to compute the slot for the elemet usig key k, where h maps the uiverse U of keys ito the slots of a hash table T [0..m 1]. h:u {0,1,...,m 1}. We say that a elemet with key k hashes to slot h(k). We also say that h(k) is the hash value of key k. Notice that with direct addressig, a elemet with key k is stored i slot k, which is a very special hash table. COMP3600/6466: Lecture

5 11.2 Hash tables Usig a hash fuctio (Corme et al., p256) COMP3600/6466: Lecture

6 11.2 Collisio i Hash tables The drawback of ay hash tables is the collisio whe two differet keys are mapped to the same slot. Resolvig collisios is a key issue i the desig of hash fuctios. Oe effective way to resolve collisios is called chaiig, which works as follows: Put all elemets that hash to the same slot i a liked list. COMP3600/6466: Lecture

7 11.2 Collisio i Hash tables Avoidig collisios usig liked lists (Corme et al., p257) COMP3600/6466: Lecture

8 11.2 Operatios o hash tables The directory operatios o a hash table T are easy to implemet whe collisios are resolved by chaiig. CHAINED HASH SEARCH(T,k) search for a elemet with key k i the liked list T [h(k)] CHAINED HASH INSERT(T,x) isert x at the head of the liked list T [h(key[x])] (Why?) CHAINED HASH DELETE(T,x) delete x from the liked list T [h(key[x])] (How to delete x from the list?) Hashig with chaiig takes O() time i the worst case whe searchig or deletig a elemet from the hash table. COMP3600/6466: Lecture

9 11.2 Aalysis of simple uiform hashig with chaiig Give a hash table with m slots that stores elemets, the load factor of the hash table is α = /m. A simple uiform hashig assumes that ay give elemet is equally likely to hash ito ay of the m slots, idepedetly of where ay other elemet has hashed to. The average behavior of hashig uder the simple uiform hashig assumptio is much better, which takes Θ(1 + α) time. Let the hash table cotai m slots. For j = 0,...,m 1, deote by j the legth of the liked list T [ j], so that = m 1, ad the average value of j is E[ j ] = α = /m. What is the relatioship betwee the load factor α ad the time of searchig/deletio of a elemet? COMP3600/6466: Lecture

10 11.2 Aalysis of simple uiform hashig with chaiig (cot.) Theorem Collisios i a hash table are resolved by chaiig, a usuccessful search (or a successful search) takes time Θ(1 + α) i expectatio uder the assumptio of simple uiform hashig. Case 1: Usuccessful search for key k: The liked list T ( j) for hash value h(k) (= j) has to traversed. The expected legth of T ( j) is E[ j ] = α = /m. Case 2: Successful search for key k: Let k i = key[x i ]. For keys k i ad k j, deote by X i j = I{h(k i ) = h(k j )} a radom variable. Pr{h(k i ) = h(k j )} = 1/m. Thus, E[X i j ] = 1/m. Assume that key k i is hashed to a slot h(k i ), whe we retrieve the liked list that cotais key k i is from its head, we ca fid all keys k j i frot of key k i with j > i (Why?). I other words, the amout of time spet o this liked list (to idetify k i ) is proportioal to the umber of keys before it i the liked list, while the probability of a key k j i frot of key k i i the liked list with j > i is 1/m. The average time complexity of successful search for key k i thus is COMP3600/6466: Lecture

11 11.2 Aalysis of simple uiform hashig with chaiig (cot.) E[ 1 i=1 (1 + j=i+1x i j )] = 1 ( + E[ i=1 = 1 ( + i=1 = 1 ( + i=1 = i=1 X i j ]) j=i+1 E[X i j ]) j=i+1 j=i+1 i m 1 m ), as E[X i j] = 1/m = m = 1 + ( 2m m 1 2 ) = 1 + α/2 α/2, sice α = /m = Θ(1 + α). (1) COMP3600/6466: Lecture