Expected Worst-case Performance of Hash Files

Transcription

1 Expected Worst-cse Performnce of Hsh Files Per-Ake Lrson Deprtment of Informtion Processing, Abo Akdemi, Fnriksgtn, SF-00 ABO 0, Finlnd The following problem is studied: consider hshfilend the longest probe sequence tht occurs when retrieving record. How long is this probe sequence expected to be? The pproch tken differs from trditionl worst-cse considertions, which consider only the longest probe sequence of the worst possiblefileinstnce. Three overflow hndling schemes re nlysed: uniform hshing (rndom probing), liner probing nd seprte chining. The numericl results show tht the worst-cse performnce is expected to be quite resonble. Provided tht the hshing functions used re well-behved, extremely long probe sequences re very unlikely to occur. INTRODUCTION It is well-known tht the worst-cse performnce of hsh file is 0(n), where n is the number of records stored. This is true for ny of the bsic opertions: retrievl, insertion or deletion of record. This result is seen immeditely by considering the longest probe sequence of the worst possible file instnce. However, the probbility tht the worst possible file instnce will ctully occur is incredibly smll. It is n extremely pessimistic view which does not give ny informtion bout the expected worst-cse performnce of 'norml, wellbehved' hsh file. In this pper the worst-cse performnce of hsh file is pproched from different ngle. The problem studied cn be stted s follows: consider certin hsh file nd the longest of the probe sequences required to retrieve record stored in the file. Wht is the expected length of this probe sequence? In other words, not only the longest probe sequence of the worst possible file instnce is of interest, but the whole distribution of the longest probe sequence over ll file instnces. The term 'expected worst-cse' will be used to refer to the ltter type of nlysis. The former, trditionl pproch might perhps be clled the worst-of-the-worst pproch. The nlysis reported here is restricted to the retrievl performnce of externlly stored hshfiles. The length of probe sequence, or simply the serch length, is mesured in the number of ccesses to secondry storge. Three different overflow hndling schemes re studied: uniform hshing (rndom probing) (section two), liner probing (section three) nd seprte chining (section four). The bucket size is one or lrger thn one. Only the cse of initil loding is considered, i.e. no deletions re llowed. So fr only two nlyses of the expected worst-cse performnce of hshing hve been reported: one by Gonnet nd one by Lrson. Gonnet studied internlly stored hsh tbles (bucket size one) ssuming tht overflow records re hndled by rndom probing or by seprte chining. He showed tht the expected length of the longest probe sequence grows logrithmiclly in the cse of rndom probing, nd sublogrithmiclly in the cse of seprte chining. Lrson nlysed new hshing scheme clled Liner Hshing with Prtil Expnsions. Gonnet ws ble to obtin simple pproximte formule for the schemes considered. The modelling of the schemes considered in this pper is considerbly more complex mthemticlly, nd therefore we will hve to be content with numericl results. From elementry probbility theory we recll the following useful theorem. In rndom smple x x, x,..., x n the probbility tht the lrgest observed vlue is less thn or equl to fixed vlue x 0 equls the product of the probbilities tht ech observed vlue is less thn or equl to x 0, i.e. P(mx(Xj) < x o ) = P(JCI < *o) P(* ^*o)- Hx n <x 0 ). This is true if ll the rndom vribles re mutully independent. The nlysis of ech scheme follows the sme bsic pttern. First, the probbility distribution of the length of probe sequence (successful serch) is derived. Therefter, the probbility distribution of the length of the longest probe sequence, herefter bbrevited to lips, in smple of n sequences is redily found by pplying the bove theorem. Finlly the expected vlue of the lips is numericlly computed nd tbulted. The results re pproximte becuse the rndom smple is considered to be drwn from n infinite file. This simplifiction ws necessry becuse, for the schemes considered, either the distribution of the length of probe sequence in finite file is not known or, when known, the numericl computtions involved re fr too time-consuming. The results given here overestimte the expected length of the longest probe sequence. For modertely lrge files the error is no more thn few per cent. To fcilitte comprisons the expected lengths of both successful nd unsuccessful serches hve lso been computed nd compiled into n ppendix. Note tht these expected vlues re for n rbitrry successful or unsuccessful serch, not the longest one. UNIFORM HASHING ~ Consider hshfileconsisting of m buckets, ech bucket hving cpcity of b records, b >. There re n records stored in thefile.the storge utiliztion, denoted by, is then = n/(mb). We shll ssume in this section tht overflow records re hndled by uniform hshing (rndom probing, rehshing); i.e. ech overflowing record is rehshed rndomly until non-full bucket is found (see Ref. ). Let P(L = k),k = \,,..., denote the probbility tht retrievl of record requires k ccesses. For finite m this CCC-OOKM6/8/00-07 $0.00 Heyden & Son Ltd, 98 THE COMPUTER JOURNAL, VOL., NO.,98 7 Downloded from

2 P.-A. LARSON distribution is known only for the cse b =. For b >, the symptotic distribution, obtined by letting m oo keeping the storge utiliztion constnt, ws derived by Lrson. It cn be computed s ) P(L = k) = (I/) P b (yf~ '( - Jo where b-l (ybv/jl nd where the function x() is (implicitly) defined by Eq. (). fr- = exp( xb){\ x) ]T (xbf/kl exp( xb)(xby > ~ /(b )! () Let Q k denote the probbility tht retrievl of record requires t most k ccesses, i.e. Q k P(L < k). This probbility is k Q k = I 0/) P^y-'U - P b (y)) dy j=i Jo which cn be reduced to -l-/oof* 0 ] Jo The integrl must be computed numericlly. The vlue x() must lso be computed numericlly from Eqn (). Consider smple of n records from n infinite file with bucket size b nd storge utiliztion. By pplying the theorem mentioned bove, we immeditely find tht the probbility tht the longest probe sequence encountered in the smple is t most k ccesses equls QJJ. The expected length of the longest probe sequence is thus R n = I *(QZ - QZ-,) = i + I (l - QZ) The distribution QZ hs been plotted in Fig. for n = looo.oc = 0.8nd6 =,. We see tht with probbility greter thn 90%, the length of the longest probe sequence will not exceed 9 when b = nd 6 when b =. The length of the longest probe sequence is very stble; the probbility tht it will be either or is s high s 7. per cent when b =. Tble shows the expected length of the longest probe sequence for few prmeter combintions. The vlues re surprisingly low, especilly for lrger buckets. The longest probe sequence in file with b =, = 0.8 nd storing one million records is expected to be only.9 ccesses, for exmple. For expected serch lengths, see the Appendix i i r J b = S 6 8 k (ccesses) Figure. The probbility tht lips <, k for uniform hshing, n = 00, o=0.8. b =,. LINEAR PROBING ~ Liner probing is one of the erliest nd most widely used techniques for hndling overflow records. The serch pth of record hshing to bucket k is k, k +,..., m,0, k nd the record is inserted into the first non-full bucket encountered on this pth. The performnce of liner probing hs been nlysed by severl uthors, but either only the cse b = or only the expected performnce is considered, ' ' 6 ' 7 The first nlysis of" the cse b>, which includes the whole probbility distribution, not merely the expected vlue, of the serch length, ws published in 978 by Blke nd Konheim. 8 The nlysis below is bsed on their model, but the formule re rewritten in form better suited for numericl computtions. Assuming n infinite file, the probbility tht k ccesses will be required to retrieve record cn be computed s {<p b (x)exp(-xb) s=0 The integrl must be computed numericlly. The numbers {V bj+s },j > 0,0 <, s <, b, occurring in the formul do not depend on the storge utiliztion x. They cn be computed using the recurrence V hi+s = j i=l for s = 0 + *)){(/+l)v w+,_, for =,,..., b - Tble. Expected Ups for uniform hshing No. of records Bucket size, b = = = = THE COMPUTER JOURNAL, VOL., NO.,98 Heyden & Son Ltd, 98 Downloded from

3 EXPECTED WORST-CASE PERFORMANCE OF HASH FILES where the boundry vlues for./ = 0 re given by V s = Wef/sl, s = 0,,..., b -. The numbers {V bj+s } re relted to the numbers {T b>bj+s } defined in Ref. 8 by V bj + s = T 6, 6j+s (6/e) b -' + 7(fe/ + s)\. The chnge hs been mde purely for numericl resons; the numbers {T(,,bj+ S } grow extremely rpidly. The function q> b (x) is normliztion fctor which ensures tht the probbilities sum to one. It is given by cp b (x) = b(l - 7= where <o is primitivefethroot of unity; o> = cos(nlb) + isin(7i/fe), for exmple. The complex-vlued function 0(z), zec, is defined by the series r = 0 The series converges slowly, however, nd numericlly the function 6(z) is most esily computed from the functionl eqution 9{z) = exp (zd(z)) using Newton- Rphson itertion. Afirstpproximtion is redily found by computing few terms of the series. The rest is then strightforwrd. The probbility tht retrievl of record requires t most k ccesses is nd the probbility tht the longest probe sequence occurring in smple of size n is of length k or less is Q. The expected lips is then given by R n = + k =i0-qz)- Exctly s for uniform hshing, the nlysis of liner probing is lso pproximte nd slightly overestimtes the expected vlue. The error is thus 'on the sfe side'. Compred with uniform hshing the numericl computtions re considerbly more complex. Figure shows the cumultive probbility distribution QJj for two different bucket sizes. Compred with uniform k (ccesses) Figure. The probbility tht lips < k when using liner probing, n= 00, = 0.8, b =,. hshing, the probbilities grow very slowly; the risk of long probe sequences occurring is much greter. The expected vlues given in Tble re considerbly lrger thn the corresponding vlues for uniform hshing. This poor worst-cse performnce is not unexpected if the tendency of liner probing to crete long clusters of full buckets is borne in mind. For expected serch lengths, see the Appendix. SEPARATE CHAINING When using seprte chining, overflow records re stored by linking one or more secondry pges from seprte storge re to the overflowing (primry) pge. ' 9 Ech secondry pge holds records from only one chin, i.e. chins re not llowed to colesce. The secondry pge size, denoted by c, c >, my differ from the primry pge size b. Consider file where the expected number of records per bucket is zb, z > 0. The prmeter z denotes the lod fctor. Note tht the lod fctor is not the sme s storge utiliztion nd tht the lod fctor my be lrger thn one. For lrge file the probbility tht j,j = 0, records hsh to bucket cn be pproximted by the Poisson probbility exp(-zb)(zb) J /j\. Hence the probbility tht chin of pges will be of length k or less, k >, (the primry pge plus k secondry pges) is = exp(-z6) If file consisting of N primry pges is considered to hve been creted by smpling from n infinite file, the probbility tht the longest chin will be of length k or less is Q k (z) N. This is t the sme time the probbility tht the longest probe sequence will be of length k or fewer ccesses. Observe tht we now hve the number of primry pges in the exponent, not the number of records s in the two previous cses. Seprte chining is totlly different type of overflow hndling scheme from uniform hshing nd liner probing. Performnce figures for seprte chining cnnot be compred with those of the two other techniques unless the following two conditions re fulfilled. () Storge utiliztion is the sme for ll three methods, nd () files of the sme size, i.e. the sme number of records, re considered. These conditions cn be fulfilled by djusting the prmeters z nd N respectively. In order to be ble to compute the storge utiliztion, we must mke two simplifying ssumptions. The spce occupied by pointers, (one per pge) nd overflow spce Tble. Expected Ups for liner probing No. of records Bucket size, b = =»» = " ti = 0.8 " = " Heyden & Son Ltd, 98 THE COMPUTER JOURNAL, VOL., NO.,98 9 Downloded from

4 P.-A. LARSON llocted but not in use, i.e. empty overflow pges, is ignored. They both depend on the detils of the implementtion. Under these ssumptions the lod fctor z required to chieve storge utiliztion cn be computed numericlly from Eqn (). OO zb/ = b + cexp(-zb) (k + ) fc = O t (zbf +kc+i /(b + kc + 0! () This eqution defines z s function of nd this function will be denoted by z(). The expected number of records per bucket is z{d)b. To store n records in thefile, the number of buckets must be pproximtely = n/(z()b). () Finlly the expected length of the longest probe sequence cn be computed s fc=l To sum up: given storge utiliztion nd pge size b nd c, the required lod fctor z() is computed numericlly from Eqn (). Therefter the number of pges N() is given by Eqn () nd finlly the expected vlue cn be computed. Figure shows the cumultive probbility distribution of the lips for = 0.8, n = 00, c = nd b =,,. The computed lod fctor is z(0.8) =. for b=\, z(0.8) = for b = nd z(0.8) = 0.8 for b =. Observe tht seprte chining behves differently from the two other techniques: the probbility of long probe sequences increses with incresing primry pge size. The sme behviour cn be observed in Tble for Figure. chining,/;. Ai o r =H 6 8 k (ccesses) tht llpss k when using seprte r h c in / The probbility n D ft higher storge utiliztions. For expected serch lengths, see the Appendix. The tbles lists numericl results for the cse c = only, the secondry pge size most often used in prctice. CONCLUSIONS The expected worst-cse retrievl performnce of three hshing schemes, uniform hshing (uh), liner probing (lp) nd seprte chining (sc) hs been nlysed. Figure is n ttempt to summrize the mjor results of this nlysis. The nlysis revels tht the expected worst-cse performnce is not very bd. With the exception of liner probing using smll buckets, the longest probe sequence occurring in file is expected to be of quite resonble length. At lest to the uthor, this ws surprise. The expected length of the longest probe sequence increses with the file size. For uniform hshing the growth is logrithmic, nd for the two others nerly logrithmic. It is not esily seen from Fig., but the growth is (/I 8u < 0 Figure. uh, ( ) (b) - - s' /' Number of records /' _s' s' s' s s /' i i i i i i i i i i J Number of record: > J Expected length of the longest probe sequence s Fnr h ' IM Fr-- <> \D) rr > Tble. Expected Ups for seprte chining withc = l No. of records Bucket size, b = = 6 ij = 0.8 " = THE COMPUTER JOURNAL, VOL., NO.,98 Heyden & Son Ltd, 98 Downloded from

5 EXPECTED WORST-CASE PERFORMANCE OF HASH FILES sublogrithmic for seprte chining nd slightly fster thn logrithmic for liner probing. The worst-cse performnce of uniform hshing is considerbly better thn tht of liner probing. This ws to be expected becuse liner probing tends to crete long clusters of full buckets. For smll bucket sizes the performnce of seprte chining is better thn tht of uniform hshing, but for lrge buckets the order is reversed. The min result of ll the mthemtics bove is perhps simply this: it offers theoreticl explntion for the empiricl observtion tht very long probe sequences re unlikely to occur in well-designed hsh file. A few comments on the results listed in the ppendix re in order. The expected serch lengths of uniform hshing is lwys lower thn those of liner probing. This comes s no surprise. However, for lrge buckets the serch lengths for both uniform hshing nd liner probing re lower thn those of seprte chining with c =. The explntion for this somewht surprising fct is quite simple: When the primry pge size is incresed, keeping the totl size of the primry storge re constnt, the totl number of overflow records decreses,, but so does the number of overflow chins. So even though there re, in totl, fewer overflow records, there my be more of themper chin, resulting in longer chins. This problem cn be overcome by using either lrger overflow pges or severl overflow chins per primry pge, selecting one of the chins by hshing. Acknowledgement The help of B. Qvist in solving some of the computtionl problems ws impertive nd is grtefully cknowledged. REFERENCES. G. H. Gonnet, Expected length of the longest probe sequence in hsh code serching. Journl of the ACM 8 (No. ) (98).. P.-A. Lrson, Performnce nlysis of liner hshing with prtil expnsions. ACM Trnsctions on Dtbse Systems (to pper).. D. E. Knuth, The Art of Computer Progrmming, Vol. : Sorting nd Serching. Addison Wesley, Reding, Msschusetts (97).. P.-A. Lrson, Anlysis of uniform hshing. Abo Akdemi, Reports on Computer Science & Mthemtics Ser. A, No. (978).. P.-A. Lrson, Frequency loding nd liner probing, BIT, Bind 9 (No. ), -8(979). 6. G. Schy nd W. G. Spruth, Anlysis of file ddressing method. Communictions of the ACM (No. ), 9-6 (96). 7. G. Tiniter, Addressing for rndom-ccess storge with multiple bucket cpcities. Journl of the ACM (No. ), 07- (96). 8. I. F. Blke nd A. G. Konheim, Big buckets re (re not) better I Journl of the ACM (No. ), (977). 9. J. A. vn der Pool, Optimum storge lloction for initil loding of file. IBM Journl of Reserch nd Development 6 (No. 6), 79-86(97). Received June 98 Heyden & Son Ltd, 98 APPENDIX Expected length of successful nd unsuccessful serches The tbles below list the expected length of n rbitrry (not the longest) successful nd unsuccessful serch for uniform hshing, liner probing nd seprte chining. The results re symptotic, i.e. for n infinite file, nd slightly overestimte the serch lengths for correspondingfinitefile. Tble Al. Expected serch lengths for uniform hshing Bucket size Successful serch Unsuccessful serch ; Ct Heyden & Son Ltd, 98 THE COMPUTER JOURNAL, VOL., NO.,98 Downloded from

6 P.-A. LARSON Tble A. Expected serch lengths for liner probing ' ' 8 Bucket size SSuccessful serch Unsuccessful serch Tble A. Expected serch lengths for seprting chining (c = I) - 6 Bucket size Successful serch Unsuccessful serch ' denotes the storge utiliztion. The corresponding lod fctor is computed from Eqn (). THE COMPUTER JOURNAL, VOL, NO.,98 Heyden & Son Ltd, 98 Downloded from