Hashing. Algorithm : Design & Analysis [09]

Transcription

1 Hashig Algorithm : Desig & Aalysis [09]

2 I the last class Implemetig Dictioary ADT Defiitio of red-black tree Black height Isertio ito a red-black tree Deletio from a red-black tree

3 Hashig Hashig Collisio Hadlig for Hashig Closed Address Hashig Ope Address Hashig Hash Fuctios Array Doublig ad Amortized Aalysis

4 Hashig: the Idea E[0] E[1] E[k] E[m-1] I feasible size Idex distributio Collisio hadlig Hash Fuctio H(x)=k A calculated array idex for the key Very large, but oly a small part is used i a applicatio x Key Space Value of a specific key

5 Collisio Hadlig: Closed Address Each address is a liked list k 1 k 4 k 1 k 5 k 2 k 7 k 4 k5 k 2 k 3 k 7 k 6 k 6 k 3

6 Closed Address: Aalysis Assumptio: simple uiform hashig: for j=0,1,2,...,m-1, the average legth of the list at E[j] is /m. The average cost of a usuccessful search: Ay key that is ot i the table is equally likely to hash to ay of the m address. The average cost to determie that the key is ot i the list E[h(k)] is the cost to search to the ed of the list, which is /m. So, the total cost is Θ(1+ /m).

7 Closed Address: Aalysis(cot.) For successful search: (assumig that x i is the ith elemet iserted ito the table, i=1,2,...,) For each i, the probability of that x i is searched is 1/. For a specific x i, the umber of elemets examied i a successful search is t+1, where t is the umber of elemets iserted ito the same list as x i, after x i has bee iserted. Ad for ay j, the probability of that x j is iserted ito the same list of x i is 1/m. So, the cost is: Cost for computig hashig i= j= i+ m Expected umber of elemets i frot of the searched oe i the same liked list.

8 Closed Address: Aalysis(cot.) The average cost of a successful search: Defie α=/m as load factor, The average cost of a successful 1 = m i= 1 j = 1+ = i m m i= 1 α α = 1+ = Θ(1 + α ) 2 2 Cost for computig hashig ( i) search is : = 1+ 1 m 1 i= 1 Number of elemets i frot of the searched oe i the same liked list. i

9 Collisio Hadlig: Ope Address All elemets are stored i the hash table, o liked list is used. So, α, the load factor, ca ot be larger tha 1. Collisio is settled by rehashig : a fuctio is used to get a ew hashig address for each collided address, i.e. the hash table slots are probed successively, util a valid locatio is foud. The probe sequece ca be see as a permutatio of (0,1,2,..., m-1)

10 Commoly Used Probig Liear probig: Give a ordiary hash fuctio h, which is called a auxiliary hash fuctio, the hash fuctio is: (clusterig may occur) Quadratic Probig: h(k,i) = (h (k)+i) mod m (i=0,1,...,m-1) Give auxiliary fuctio h ad ozero auxiliary costat c 1 ad c 2, the hash fuctio is: (secodary clusterig may occur) Double hashig: h(k,i) = (h (k)+c 1 i+ c 2 i 2 ) mod m (i=0,1,...,m-1) Give auxiliary fuctios h 1 ad h 2, the hash fuctio is: h(k,i) = (h 1 (k)+ ih 2 (k)) mod m (i=0,1,...,m-1)

11 Liear Probig: a Example Idex 0 H 1776 Hash fuctio: h(x)=5x mod 8 Hash fuctio: h(x)=5x mod 8 1 rehashig hashig hashig Rehash fuctio: rh(j)=(j+1) mod 8 chai of rehashigs

12 Equally Likely Permutatios Assumptio: each key is equally likely to have ay of the m! permutatios of (1,2...,m-1) as its probe sequece. Note: both liear ad quadratic probig have oly m distict probe sequece, as determied by the first probe.

13 Aalysis for Ope Address Hash Assumig uiform hashig, the average umber of probes i a usuccessful search is at most 1/(1-α) (α=/m<1) Note is so, m m the m The, : the, ad 1 1 the probabilit that probabilit m 2 2 of average the y of L y of jth( the umber the j > umber i + 2 m i + 2 of first 1) positio of probe probed m is probe i 1 : i = 1 positio occupied o i = α i α 1 less 1 = beig i = 0 is tha i α -j m-j i = occupied will, 1 1 α be :

14 Aalysis for Ope Address Hash Assumig uiform hashig, the average cost of probes i a 1 1 successful search is at most l (α=/m<1) α 1 α To search for the ( i + 1)th iserted elemet i the table, the cost is the same as the cost for isertig it whe there i are just i elemets i the table. At that time, α =, so, m 1 m the cost is = i For For your your referece: 1- m i m Half Half full: full: 1.387; 90% 90% full: full: So, the cost is : 1 1 m 1 m m m dx 1 m 1 1 = = = = α α l l = m i = m i = + i m x α m α 1 α i 0 i 0 i m 1

15 Hashig Fuctio A good hash fuctio satisfies the assumptio of simple uiform hashig. Heuristic hashig fuctios The divisio method: h(k)=k mod m The multiplicatio method: h(k)= m(ka mod 1) (0<A<1) No sigle fuctio ca avoid the worst case Θ(), so, Uiversal hashig is proposed. Rich resource about hashig fuctio: Goet ad Baeza-Yates: Hadbook of Algorithms ad Data Structures, Addiso-Wesley, 1991

16 Array Doublig Cost for search i a hash table is Θ(1+α), the if we ca keep α costat, the cost will be Θ(1) Space allocatio techiques such as array doublig may be eeded. The problem of uusually expesive idividual operatio.

17 Lookig at the Memory Allocatio hashigisert(hashtable H, ITEM x) iteger size=0, um=0; if size=0 the allocate a block of size 1; size=1; if um=size the allocate a block of size 2size; move all item ito ew table; size=2size; isert x ito the table; um=um+1; retur Isertio with expasio: cost size Elemetary isertio: cost 1

18 Worst-case Aalysis of the Isertio For executio of isertio operatios A bad aalysis: the worst case for oe isertio is the case whe expasio is required, up to So, the worst case cost is i O( 2 ). Note the expasio is required durig the ith operatio oly if i=2 k, ad the cost of the ith operatio c i So, i = 1 the if i total 1is exactly power of 2 otherwise cost is : i= 1 c i + lg j= 0 2 j < + 2 = 3

19 Amortized Time Aalysis Amortized equatio: amortized cost = actual cost + accoutig cost Desig goals for accoutig cost I ay legal sequece of operatios, the sum of the accoutig costs is oegative. The amortized cost of each operatio is fairly regular, i spite of the wide fluctuate possible for the actual cost of idividual operatios.

20 Amortized Aalysis: MultiPop Stack Pop: Cost=1 MultiPop: Cost=mi(s,t) Push: Cost=1 s t Amortized cost: push:2; pop, multipop: 0

21 Amortized Aalysis: Biary Couter Cost measure: bit flip amortized cost: set 1: 2 set 0: 0

22 Accoutig Scheme for Stack Push Push operatio with array doublig No resize triggered: 1 Resize( 2) triggered: t+1 (t is a costat) Accoutig scheme (specifyig accoutig cost) No resize triggered: 2t Resize( 2) triggered: -t+2t So, the amortized cost of each idividual push operatio is 1+2t Θ(1)

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