CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees

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1 CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 259] B-Trs Th AVL r and rd-black r allowd som variaion in h lnghs of h diffrn roo-o-laf pahs. An alrnaiv ida is o mak sur ha all roo-o-laf pahs hav xacly h sam lngh and allow variaion in h numbr of childrn. Th dfiniion of a B-r uss a paramr m: æ vry laf has h sam dph æ h roo has a mos m childrn æ vry non-roo nod has from m=2 o m childrn Kys ar placd ino nods lik his: æ Each non-laf nod has on fwr kys han i has childrn. Each ky is bwn wo child poinrs. æ Each laf nod has bwn m=2,1 and m,1 kys in i (unlss i is also h roo, in which cas i has bwn 1 and m, 1 kys in i). æ Th kys wihin a nod ar lisd in incrasing ordr.

2 CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 260] B-Trs (con d) And w rquir h xndd sarch r propry: æ For ach nod x, h i-h ky in x is largr han all h kys in x s i-h subr and is smallr han all h kys in x s èi +1è-s subr B-rs ar xnsivly usd in h ral world, for insanc, daabas applicaions. In pracic, m is vry larg (such as 512 or 1024). Thorm: Th dph of a B-r r is Oèlog nè. Insr and dl algorihms ar qui involvd.

3 CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 261] Tris In h prvious sarch rs, ach ky is indpndn of h ohr kys in h r, xcp for hir rlaiv posiions. For som kinds of kys, on ky migh b a prfix of anohr r. For xampl, if h kys ar srings, hn h ky a is a prfix of h ky alas. Th nx kind of r aks advanag of prfix rlaionships bwn kys o sor hm mor fficinly. A ri is a (no ncssarily binary) r in which æ ach nod corrsponds o a prfix of a ky, and æ prfix for ach nod xnds prfix of is parn. Th ri soring a, al, an, bd, b, b : a b l n d

4 CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 262] Insring ino a Tri To insr ino a ri: insr(x,s): // x is nod, s is sring o insr if lngh(s) = 0 hn mark x as holding a compl ky ls c := firs characr in s if no ougoing dg from x is labld wih c hn cra a nw child nod of x labl h dg o h nw child nod wih c pu h dg in h corrc sord ordr among all of x s ougoing dgs ndif x := child of x rachd by dg labld c s := rsul of rmoving firs characr from s insr(x,s) ndif Sar h rcursion wih h roo. To insr an and bp : a b l n d

5 CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 263] Sarching in a Tri To sarch in a ri: sarch(x,s): // x is nod, s is sring o sarch for if lngh(s) = 0 hn if x holds a compl ky hn rurn x ls rurn null // s is no in h ri ls c := firs characr in s if no ougoing dg from x is labld wih c hn rurn null // s is no in h ri ls x := child of x rachd by dg labld c s := rsul of rmoving firs characr from s sarch(x,s) ndif ndif Sar h rcursion wih h roo. To sarch for ar and b : a b l n d

6 CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 264] Hash Tabl Implmnaion of Dicionary ADT Anohr implmnaion of h Dicionary ADT is a hash abl. Hash abls suppor h opraions æ insr an lmn æ dl an arbirary lmn æ sarch for a paricular lmn wih consan avrag im prformanc. This is a significan advanag ovr vn balancd sarch rs, which hav avrag ims of Oèlog nè. Th disadvanag of hash abls is ha h opraions min, max, prd, succ ak Oènè im; and prining all lmns in sord ordr aks Oèn log nè im.

7 CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 265] Main Ida of Hash Tabl Main ida: xploi random accss faur of arrays: h i-h nry of array A can b accssd in consan im, by calculaing h addrss of A[i], which is offs from h saring addrss of A. Simpl xampl: Suppos all kys ar in h rang 0 o 99. Thn sor lmns in an array A wih 100 nris. Iniializ all nris o som mpy indicaor. æ To insr x wih ky k: A[k] := x. æ To sarch for ky k: chck if A[k] is mpy. æ To dl lmn wih ky k: A[k] := mpy. All ims ar Oè1è x0 x2... x99 ky is 0 ky is 2 Bu his ida dos no scal wll. ky is 99

8 CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 266] Hash Funcions Suppos æ lmns ar sudn rcords æ school has 40,000 sudns, æ kys ar social scuriy numbrs ( ). Sinc hr ar 1 billion possibl SSN s, w nd an array of lngh 1 billion. And mos of i will b wasd, sinc only 40,000/1,000,000,000 = 1/25,000 fracion is nonmpy. Insad, w nd a way o condns h kys ino a smallr rang. L M b h siz of h array w ar willing o provid. Us a hash funcion, h, o convr ach ky o an array indx. Thn h maps ky valus o ingrs in h rang 0oM, 1.

9 CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 267] Simpl Hash Funcion Exampl Suppos kys ar ingrs. L h hash funcion b hèkè = k mod M. Noic ha his always givs you somhing in h rang 0 o M, 1 (an array indx). æ To insr x wih ky k: Aëhèkèë := x æ To sarch for lmn wih ky k: chck if Aëhèkèë is mpy æ To dl lmn wih ky k: saëhèkèë o mpy. All ims ar Oè1è, assuming h hash funcion can b compud in consan im x... ky is k and h(k) = 2 Th ky o making his work is o choos hash funcion h and abl siz M proprly (hy inrac).

10 CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 268] Collisions In raliy, any hash funcion will hav collisions: whn wo diffrn kys hash o h sam valu: hèk 1 è=hèk 2 è, alhough k 1 6= k 2. This is inviabl, sinc h hash funcion is squashing down a larg domain ino a small rang. For xampl, if hèkè = k mod M, hn k 1 = 0 and k 2 = M collid sinc hy boh hash o 0 (0 modmis 0, and M mod M is also 0). Wha should you do whn you hav a collision? Two common soluions ar 1. chaining, and 2. opn addrssing

11 CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 269] Chaining Kp all daa ims ha hash o h sam array locaion in a linkd lis: all hav kys ha hash o 1 M-1 æ o insr lmn x wih ky k: of linkd lis a Aëhèkèë pu x a bginning æ o sarch for lmn wih ky k: scan h linkd lis a Aëhèkèë for an lmn wih ky k æ o dl lmn wih ky k: do sarch, if sarch is succssful hn rmov lmn from h linkd lis Wors cas ims, assuming compuing h is consan: æ insr: Oè1è. æ sarch and dl: Oènè. Wors cas is if all n lmns hash o sam locaion.