Faster Searching by Elimination

Transcription

1 Faster Searchng by Elmnaton Theodore S. Norvell Electrcal and Computer Engneerng Memoral Unversty December 6, 010 Abstract The SIMPLE system, under development at Memoral Unversty, allows abstract problem descrptons to be refned by abstract algorthms. By data refnng both the problem and ts soluton, we can reuse verfed algorthms. We use bnary search as an example of ths method. 1 Introducton The SIMPLE system[1], currently under development by the author, s a programmng envronment for nteractvely dervng programs from ther specfcatons. One of the goals s to allow the programmer to leverage abstracton to the greatest extend possble. Abstracton allows one to concentrate on some aspects of a problem whle gnorng others. Ths paper llustrates how SIMPLE allows one to solve some aspects of algorthmc problems wthout solvng others. We wll show how the SIMPLE language allows one to separate the descrpton and proof of an algorthmc technque from ts applcaton to partcular problems and data representatons. Ths wll allow a lbrary of abstract problems and algorthmc technques to solve them to be bult, stored, and later appled to concrete problems. As an example, we wll look at the algorthmc problem of search and the algorthmc technque of bnary search. We wll then apply the technque to two concrete problems. Although SIMPLE s stll under development, ths paper s ntended to be a realstc exposton of the capabltes that the SIMPLE language and system are expected to have, as well as an exploraton of the capabltes t wll be requred to have n order to meet ts aspratons. Search and Bnary Search Thesearchproblemwewlllookatsthatoffndngavaluenasetthatsatsfesagvendescrpton Defnton Search:: requres G ensurex G Ths specfcaton consst of a precondton, S G, that ndcates that t s an assumpton that a soluton exsts wthn a search space S, and a postcondton that ndcates that a soluton stobeplacednvarablex. Thsspecfcatonsgenercoverananonymoustypeforthevarable x. Ifthetypeofxsα,thetypeofS andgsset(α). 1

2 As n Hehner s predcatve programmng approach[, 3], each specfcaton can be nterpreted as a boolean expresson relatng ntal and fnal states, specfyng whch behavours are acceptable. Ofcoursenotallsearchproblemsftthspattern;npartculartsoftennotknownapror whether a soluton exsts. The bnary search algorthm can be gven by Defnton BnarySearch:: whle S >1nvS G lets 0,S 1 SS 0 S 1 (S 0 G S:S 0 []S 1 G S:S 1 ) requre S 1ensureS{x } The algorthm uses nondetermnstc choce [] and guarded commands. A guarded command, A P,ensuresthattsguard,A,struebeforeexecutngtsbody,P. That ths algorthm refnes the search problem s expressed n SIMPLE as a theorem Theorem Search BnarySearch The proof of ths theorem wll also be expressble n SIMPLE and the SIMPLE envronment wllbeabletochecktheproof. Notethat expressesapartalcorrectnessrelaton,sothatthe theorem means that any behavour acceptable to BnarySearch s acceptable to Search. 3 Data Refnement Adatarefnement[4,5]ofaspecfcatonP underanabstractonnvaranti wthvarablesvto be removed s a predcate dr v;i P v I v I P Data refnement s a means of representng certan varables that occur n P wth other varables. For example, f we have a statement x : x but we represent the boolean varable x wth an ntegerundertheabstractonnvarant(x 0) ( x 1)thenthedatarefnements dr x;(x 0) ( x 1) (x: x)(:1 ) Nowletuslookattwoexamplesofapplcatonsofthebnarysearchproblem. 4 Applcaton 4.1 Searchng a sorted array The classc example of bnary search s that of searchng a sorted array. Wecandefnewhattsforanarraytobesorted DefntonSorted A::,j ndex(a) j A Aj Nowtheproblemwewshtosolvessearchngasortedarraywhenweknowthethngweare lookngfor,p,snthearray. Defnton ArraySearch A p::

3 requresorteda ( x ndex(a) Ap) ensurea[x ]p In order to understand the array search problem as a concretzaton of the abstract search problem, we need to relate ther varables wth an abstracton nvarant. We use: I(S{a,..b} G{ ndex(a) Ap}) where{a,..b}sthesetofallntegersgreaterorequaltoaandlessthanb. Nowdatatransformng Search, we get ArraySearch requresorteda vara,b:0,length(a) dr S,G;I (Search) Where represents refnement of specfcatons. That s P Q means that any behavour acceptedbyqsacceptedbyp. As data refnement s monotonc wth respect to refnement we have ArraySearch requresorteda vara,b:0,length(a) dr S,G;I (Search) requresorteda vara,b:0,length(a) dr S,G;I (BnarySearch) The bnary search under ths data refnement s further refned by whlea+1<bnv {a,...b} Ap lets 0,S 1 {a,..b}s 0 S 1 ( S 0 Ap changea,b ensure{a,..b }S 0 [] S 1 Ap changea,b ensure{a,..b }S 1 ) requre a+1bensure{a,..b}{x } NowsetsS 0 ands 1 canbechosensuchthats 0 {a,.. }ands1 {,..b}gvng algorthm whleb a>1nv {a,...b} Ap ( {a,.. } Ap b: [] {,..b} Ap a: ) x:a FromthefactthatAssortedwecanobtanthatfA >x andthatfa x {a,..b} Ap {a,.. } Ap {a,.. } Ap 3 {,..b} Ap

4 {a,..b} Ap {a,.. } Ap {,..b} Ap {,..b} Ap HencewecanstrengthentheguardstoA >xanda xrespectvely, afterwhch thebodyoftheloopsmplfesto fa >xthenb: elsea: 4. Calculatng a square root Asasecondexamplewewllderveanalgorthmforfndngthentegerpartofthesquarerootof a natural number. The specfcaton s Defnton Root y:: requrey N ensurex y We wll determne the square root bt-by-bt, startng wth the left-most bt. We represent the set S usng two natural numbers: x represents the bts calculated so far, whle represents the number of bts yet to be calculated. Usng an abstracton nvarant I ( S{x,..(x+1) } G{ y} ) Weget Rooty var:n x:0;dr S,G;I Search var:n x:0;dr S,G;I BnarySearch wheren log y,.e. N senoughbtstoholdtheanswer. The data refnement of the bnary search algorthm gves whle>0nvx y <(x+1) lets 0,S 1 {x,..(x+1) }S 0 S 1 ( y S0 changex, ensure{x,..(x +1) }S 0 [] y S1 changex, ensure{x,..(x +1) }S 1 ) WecanpckS 0 ands 1 as{x,..(x+1) 1 }and{(x+1) 1,..(x+1) }. Thsgves whle>0nvx y <(x+1) ( y <(x+1) 1 x,:x, 1 [] y (x+1) 1 x,:x+1, 1) 4

5 Nowweneedtorefnetheguards Thsgvesaloopbodyof y<(x+1) 1 Property of floor. y<(x+1) 1 Property of square root y< ( (x+1) 1) Algebra y< x + x+ fy< x + x+ thenx,:x, 1elsex,:x+1, 1 Nowthsnvolvesanumberofmultplcatons,whchwewouldlketogetrdof. Wecando ths by means of another data refnement, ths tme to ntroduce varables to track the three terms n the guard. J ( p x q x r ) The ntalzaton for these varables s varp,q,r:0,0, N Thetestbecomesy<p+q+r. Theassgnmentx,:x, 1stransformedto x x p q : 1 ( 1) (x) ( 1) x r ( 1) whch smplfes, by the abstracton nvarant to x q : r x 1 q/ r/4 Theassgnmentx,:x+1, 1stransformedto x x p q : 1 x ( 1) x r ( 1) whch smplfes to x p q r : 5 x 1 p+q+r q/+r r/4

6 Insummarywehave var:n x:0; whle>0nvx y <(x+1) fy<p+q+r x,,q,r:x, 1,q/,r/4 else x,,p,q,r:x, 1,p+q+r,q/+r,r/4 All the dvsons can be mplemented as shfts. Ths algorthm s sutable for mplementaton n ether hardware or software. 5 Concluson and related work The methods presented here can be extended to other problems. For example, dvson of ntegers, wth the goal of a hardware sutable mplementaton. The orgns of ths paper came wth the realzaton that the prncpled strength reducton methodn[6]appledtoasquarerootalgorthmcouldbeseenasadatarefnementandthenthe realzaton that the orgnal algorthm could be seen as a data refnement of a more abstract bnarysearch algorthm. The resultng abstract bnary search algorthm turns out to be an abstracton also of the Searchng by Elmnaton algorthm presented n[7]. Ther algorthm only elmnates a sngle member of the search space n each loop, yeldng algorthms wth tme complextes lnear n thesearchspaceszeatbest;myalgorthmallowsanysubsetofthesearchspacetobeelmnated, whchsfaster;hencethettleofthspaper. References [1] Theodore Norvell and Zhka Dng, An envronment for provng and programmng, n Newfoundland Electrcal and Computer Engneerng Conference, October [] Erc C. R. Hehner, Predcatve programmng, Communcatons of the ACM, vol. 7, no., pp , [3] ErcC.R.Hehner, Abstractonsoftme, naclasscalmnd,a.w.roscoe,ed.,chapter1, pp Prentce-Hall Internatonal, [4] C. A. R. Hoare, Proof of correctness of data representatons, Acta Informatca, vol. 1, pp , 197. [5] Davd Gres and Jan Prns, A new noton of encapsulaton, n SIGPLAN 85 Symposum on Language Issues n Programmng, 1985, pp [6] Yanhong A. Lu, Prncpled strength reducton, n Algorthmc Languages and Calcul, Rchard Brd and Lambert Meertens, Eds. IFIP, February 1997, pp , Chapman and Hall. [7] Anne Kaldewaj and Berry Schoenmakers, Searchng by elmnaton, Scence of Computer Programmng, vol. 14, pp ,