Generalization of the No-Free-Lunch Theorem

Transcription

1 Proceeings of the 9 IEEE International Conference on Sstes Man an Cbernetics San Antonio TX USA - October 9 Generalization of the No-Free-Lunch Theore Albert Y.S. La an Victor O.K. Li Departent of Electrical an Electronic Engineering The Universit of Hong Kong Hong Kong China {asla vli}@eee.hku.hk Abstract The No-Free-Lunch (NFL) Theore provies a funaental liit governing all optiization/search algoriths an has successfull rawn attention to theoretical founation of optiization an search. However we fin several liitations in the original NFL paper. In this work using results fro the nature of search algoriths we enhance several aspects of the original NFL Theore. We have ientifie the properties of eterinistic an probabilistic algoriths. We also provie an enueration proof of the theore. In aition we show that the NFL Theore is still vali for ore general perforance easures. This work serves as an application of the nature of search algoriths. Kewors No Free Lunch Nature of Search Algoriths Optiization. I. INTRODUCTION Search techniques have been wiel eploe to solve ifferent tpes of probles ranging fro sorting to optiization. The coon objective of such search algoriths is to fin an appropriate solution with esirable properties escribe b the proble. The general practice in solving search probles is: we have a proble an then tr to propose an algorith which can fin the esirable solution of the proble aong an other solutions. Most of the tie we keep on eveloping algoriths accoring to our experiences intuition an b trial-an-error. However an probles in science an engineering can be reucible to the stanar NPhar or NP-coplete probles []. Accoring to the general belief in the research counit P NP. In other wors attepting to fin an algorith which guarantees the esirable solution in polnoial tie for these probles is futile. Until the evelopent of stochastic search algoriths a few ecaes ago there were no eas an effective was to hanle NP-har an NP-coplete probles of large size. Stochastic search algoriths o not guarantee the best solutions in polnoial tie but the can norall generate goo solutions for a wier range of probles in a reasonable perio of tie when copare with eterinistic algoriths. Suppose we copare the perforance of a eterinistic algorith to that of a probabilistic algorith. Assuing the have the sae input an running tie the set of possible outputs generate b the latter ust be larger. Each output a correspon to goo perforance for a particular proble. Thus we conclue that the probabilit of a probabilistic algorith generating outputs with goo perforance for a larger range of probles is higher. A goo solution here eans that it is within certain tolerance of the best solution. In science an engineering goo solutions are generall sufficient to fulfill the requireent of various objectives. Most of the stochastic search algoriths are nature-inspire. Exaples inclue Genetic Algorith (GA) [ 3] Siulate Annealing [4] Ant Colon Optiization (ACO) [5 6] an Cheical Reaction Optiization [7]. There are also algoriths which are not aapte fro natural processes e.g. Tabu Search [8]. We generall classif the algoriths into heuristics an etaheuristics. The forer refers to those which are specialize to aress certain classes of probles but norall give unfavorable solutions to other probles. The reason is that proble-specific inforation has been ebee in the algorith evelopent process an thus the are tailore to the specific probles ver well. However when the are applie to other probles the ebee proble-specific inforation will not be useful anore an the result in ba perforance. On the other han the latter is not tailor-ae for an particular probles an oes not contain an problespecific inforation. When copare with heuristics etaheuristics can be applie to a wier range of probles with goo perforance. Thus there is an accurac-flexibilit traeoff between heuristics an etaheuristics. Most of the stochastic search algoriths are etaheuristics. Whenever a etaheuristic appears to work well on a particular proble gree approaches or heuristic coponents a be ae. This is tantaount to incluing ore proble-specific inforation to the etaheuristic an aking it ore heuristiclike. Accoring to the No-Free-Lunch Theore [9 ] the resulting etaheuristic is no longer able to solve other probles as well as before. In other wors we sacrifice flexibilit for accurac. Different etaheuristics are evelope base on istinct unerling echaniss or phenoena. For exaple GA is base on the iea of natural selection of living organiss while ACO akes use of the ecological behavior of ants in fining foo. Although etaheuristics can be applie to a wie range of probles the o have ifferent perforance when applie to ifferent classes of probles. Then it is natural to ask Is it possible to have a etaheuristic which is universall better than the others? Universall better can be interprete in the sense that the etaheuristic can solve ore probles with better solutions. Soe research is eicate to eveloping universall better etaheuristics an it is claie that soe search algoriths can beat others on average []. Most research focuses on the construction of search algoriths. There was harl an work on the theoretical /9/$5. 9 IEEE /9/$5. 9 IEEE 443

2 founation until Wolpert an Macrea propose the No-Free- Lunch (NFL) Theore [9 ]. It states that all search algoriths perfor statisticall ienticall on solving coputation probles. In other wors it is theoreticall ipossible to have a best general-purpose universal search strateg []. The NFL Theore is controversial an raises an iscussions (e.g. []) on its ipact an correctness. The search for a best general-purpose etaheuristic will be futile if the NFL Theore hols. We believe that the theore is correct an the conservation of perforance for search exists. As the NFL results are iportant we exaine [9 ] carefull again. We enhance an eonstrate the theore with the nature of search algoriths [3]. The contributions of this work inclue clarifing the properties of NFL-governe algoriths proviing a verification etho of the theore an generalizing the perforance easures. The rest of this paper is organize as follows. Section II escribes the terinolog use in search an optiization an the NFL Theore an its extensions. In Section III we introuce the nature of search algoriths. We eonstrate soe enhanceents an an enueration proof of the theore in Section IV an conclue this paper in Section V. II. RELATED WORK A. Terinolog We escribe a search proble as a function f F with an arguent x X an a functional value Y i.e. f ( x). X an Y are iscrete an finite. We assue that their carinalities are equal to X an Y respectivel an thus F Y X. Let a be a eterinistic non-potentiall retracing search algorith [9 ]. At iteration {... X } a takes a histor [( x )( x )...( x )] as input an outputs x i.e. a ( ) x. (We also use to refer to a histor in general.) Let x an enote the x- an -coponents of. Deterinistic eans that a prouces the sae x with probabilit one whenever given the sae while nonpotentiall retracing inicates x. x B. An Overview of the NFL Theore The NFL Theore is firstl propose in [9 ] an consists of several results. The ost representable one which establishes the core iea is Theore in []. Matheaticall it states that for an pair of eterinistic algoriths a an a ( ) ( ) P f a P f a where P ( f a ) represents the probabilit of having given f an a. Note that each f is taken fro F with the sae probabilit. Equation () eans that for an possible the perforance of an algorith (which can be obtaine The iscreteness an finiteness are trivial if we eplo a igital coputer to solve the probles. X an Y represente in a coputer ust be iscrete an finite. fro P ( ) ) is inepenent of the algorith itself when ever f F is evaluate once. In other wors all algoriths (heuristics or etaheuristics) have equal perforance when average over all possible functions. However the theore oes not hol when we focus on a certain class of probles. Thus it is still possible for an algorith to outperfor another one in this situation. C. Extensions The original NFL Theore requires all possible functions to be evaluate. Soe researchers think that this requireent is far fro realit as soe functions are ore practical (i.e. have higher probabilit of occurrence) than the others. In [4] the authors consier soe special subsets of F an propose the sharpene NFL Theore. It states that the original NFL Theore stills hol in a subset of F provie that the subset is close uner perutation (c.u.p.). Igel an Toussaint stu the necessar an sufficient conitions such that the NFL result hols in non-unifor istributions of functions in a c.u.p. subset [5]. Steeter investigates classes of functions where the NFL Theore oes not hol [6] while an Alost No Free Lunch Theore is propose to eonstrate how the result can hol approxiatel [7]. More people have realize the iportance an ipact of the NFL Theore. The research counit has been getting ore intereste in it an relate papers are publishe ever ear. III. NATURE OF SEARCH ALGORITHMS Previousl we worke out the nature of search algoriths which is focuse on the algorithic perspective instea of the functional viewpoint [8]. We will highlight the ajor results below. For ore etails an proofs intereste reaers can refer to [3]. We aopt the sae working principle of search algoriths escribe in Subsection II-A. With the sets X an Y efine we can eterine the whole set of functions F. Algoriths work on functions an generate histories. The analsis starts b enuerating all possible histories which inclue all the results i.e. (x ) pairs generate b an possible algoriths evaluating ever f F (the wa to enuerate histories will be explaine soon). We group the histories in a table an nae it an enueration table of histories. Table I(a) shows the enueration table for the exaple with X = {3} an Y = {3} (ou can ignore the shaings boxes an the last colun about counting the nuber of shae entries in boxes at this oent). The construction of an enueration table is eas. Each row an colun correspon to a function an a perutation of X respectivel. For exaple there are 7 (= 3 3 ) rows an 6 (= 3!) coluns in Table I(a). A perutation of X is Ever algorith shoul be given an input which is equal to the initial point where the search shoul start. All algoriths shoul not be restricte to fixe starting points. Otherwise the are just pieces of coes to illustrate the pre-efine search paths instea of tools which we can utilize to solve ifferent probles. 4433

3 (a) (b) Table Entries are - coponents of the Histories Functions TABLE I. f f f 3 f 4 f 5 f 6 f 7 f 8 f 9 f f f f 3 f 4 f 5 f 6 f 7 f 8 f 9 f f f f 3 f 4 f 5 f 6 f 7 Table Entries are - coponents of the Histories Functions f f f 3 f 4 f 5 f 6 f 7 f 8 f 9 f f f f 3 f 4 f 5 f 6 f 7 f 8 f 9 f f f f 3 f 4 f 5 f 6 f 7 ENUMERATION TABLES OF HISTORIES FOR X={3} AND Y={3} x-coponents of the Histories x-coponents of the Histories Nuber of Histories both in Shaing an Box Nuber of Histories both in Shaing an Box 4434

4 x in fact one possible X. An entr gives the X of the x corresponing X an f. Note that an enueration table is a coplete escription of search probles. Although its size grows exponentiall with X an Y the entries can be easil generate b inputting all perutations of X into the functions. Iportant properties of search algoriths can be eterine fro the enueration table. We consier eterinistic algoriths first an obtain the following results: Lea. Consier a eterinistic algorith a which is a apping fro the set containing all possible to the set containing all possible x i.e. a() = x. If there are an n possibilities for an x respectivel there will be n possible ifferent a. Theore. Given a sste with the sets X an Y with X X i X Y X 3 an Y there are i possible ifferent eterinistic algoriths. Corollar. Given a sste with the sets X an Y with X 3 an Y the set of all eterinistic algoriths is finite. We can list all of the b inicating the choices of x x 3... an x X one-b-one for all the partial histories... an respectivel. X i TABLE II. LOGIC MATRICES OF TWO DETERMINISTIC ALGORITHMS Algorith Algorith Theore. For i X let X () i be the value of i specifie in X.Given a sste with sets X an Y with X 3an Y an specific [ X () X ()... X ( i)] can be eterine the sae nuber of ties in the enueration table b each possible eterinistic algorith. We fin that ever algorith can be represente b a logic atrix [3]. Table II gives two exaples of such atrices euce fro the enueration table shown in Table I(a). A logic atrix can be interprete like this: the st colun stans for x the n for the 3 r for x an so on. A row represents a histor generate b that algorith. We also call a row a priitive logic because it gives an if-then relationship between the past histor an the next point x for = X -. For exaple the first priitive logic of Algorith can be interprete as: if [( x )] [()] then x. The whole atrix in fact characterizes all possible histories which can be generate b that algorith. The analsis can be extene to probabilistic search algoriths [3]. A probabilistic algorith is efine siilarl as a eterinistic algorith but instea of apping to a single x it aps to ultiple x with ifferent probabilities fro. B consiering the logic atrices an their priitive logics we obtain the following results: Theore 3. Given a sste with the sets X an Y with X 3an Y the corresponing eterinistic algoriths for the basis of its algorithic space which consists of all eterinistic an probabilistic algoriths. All algoriths are fore b probabilisticall cobining soe of the eterinist- Figure. An illustration of a probabilistic algorith with a set of eterinistic algoriths. ic algoriths. Corollar. Given a sste with the sets X an Y with X 3 an Y an algorith both eterinistic an probabilistic can be represente b a probabilistic istribution on the set of the eterinistic algoriths. We can eonstrate the above results with the exaple shown in Fig. which epicts a scenario of solving a iniization proble 3. In the figure the rectangle represents the solution space which is copose of all feasible solutions x of the proble. Each thin line gives a eterinistic algorith 4 in which the orer of picking x has been eterinisticall efine when the algorith is built. The thick line escribes the resulting search path of a probabilistic algorith. Along the path fro the starting point to the global iniu there are intersection points where the algorith has ore than one alternative path to choose fro. This exactl fulfills the efinition of probabilistic algoriths. In fact Theore 3 an Corollar tell us that a probabilistic algorith is characterize b a cobination of eterinistic algoriths 3 The scenario shown in Fig. is not tpical. The function is ignore. The actual cases shoul be uch ore coplicate. For siplicit an without loss of generalit we use this siplifie scenario to illustrate the results. 4 Here we characterize a eterinistic algorith with a partial x- coponent of one of its possible coputable histories. In realit it shoul be represente b a set of histories which traverse all solution points an each histor has a ifferent starting point. 4435

5 i.e. the thin lines. The behavior of the probabilistic algorith is intrinsicall efine b the cobination of the unerling eterinistic algoriths. If we know which an how the unerling eterinistic algoriths are cobine we can infer the perforance of the probabilistic algorith. We will ake use of the above results to re-investigate several issues relate to the NFL Theore. IV. A RE-VISIT OF THE NFL THEOREM We stu the etails of the NFL Theore again ainl fro []. There are several possible enhanceents of the theore. We iscuss the aspects of optiization algoriths verification of the theore an perforance easures accoringl in the following subsections. A. Optiization Algoriths In [] a eterinistic optiization algorith is efine as a apping fro previousl visite sets of points to a single previousl unvisite point in X i.e. a: D { x x x }. Then the NFL result (e.g. ()) is evelope base on a a which are ifferent representations of the general ter a. This iplies all a a for the set of eterinistic algorith A i.e. a a A 5. But what oes A inee look like? Is the set finite? Fro Theore an Corollar in Section III we have alrea given the answers. A is finite an all the eleents can be liste in the for of logic atrices. Then the authors of [] efines a probabilistic (i.e. stochastic) optiization algorith as a apping fro D to a -epenent istribution over X with zero x probabilit for all x. Let A p be the set of all possible probabilistic algoriths. The NFL result for eterinistic algoriths is also vali for probabilistic algoriths b replacing a with throughout the proof as explaine in []. Matheaticall () becoes P( f ) P( f ) for an pair of algoriths an in result is legitiate for an algoriths in A p. In this wa the NFL A an A p respectivel. The ieiate follow-up question is whether the NFL result is still true when we consier an algoriths in both A an A p at the sae tie i.e. for an A Ap P ( f ) P ( f ) We will prove (3) in Subsection C. B. Verification of the NFL Theore As we can eterine the iportant properties of search fro the enueration table an the NFL Theore is one of the funaental results of search it is logical to euce the NFL Theore fro the enueration table. In the following we tr to look at the NFL result i.e. () in the ipleentation perspective instea of the original approach base on probabilit theor []. For an sste with X an Y specifie we can prouce the corresponing enueration table (as explaine in Section III). We now consier eterinistic algoriths first. B Theore an Corollar in Section III we can list all a A in the for of logic atrices. We can check the NFL result with the following proceures:. Select a specific for X ;. For each entr in the enueration table shae it if it atches ; 3. For each a A i. Put all possible -coponents of the histories which can be generate b a in boxes; ii. Count how an entries are both in shaings an in boxes for each function an calculate the total for all functions; iii. Delete all the boxes; 4. Theore in Section III guarantees all the totals (of all a A ) are the sae. The shae entries an those in boxes represent those - coponents of histories which atch an those which can be generate b a respectivel. The shae ones in boxes enote coputable histories with goo perforance b a. The total nuber of shae entries of a in boxes reflects its abilit to generate goo results. In Step 3(ii) the irect suation to get the total iplies that we give equal weight to ever function. After fining the total we have finishe the exaination of a particular a. Step 3(iii) restores the setting for onl an akes it rea for another a. The positive answer at Step 4 tells us that the NFL result i.e. all eterinistic search algoriths a A perfor statisticall ienticall on solving coputation probles f F. Note that the perforance of an algorith is base on the histories obtaine uring execution an the histories are generate b the algorith itself. Thus the NFL results shoul be unerstoo fro the algorithic perspective. On the other han the above etho originates fro the enueration table whose generation is inepenent of the execution of the algoriths. Therefore we approach the NFL Theore fro a ifferent angle. In fact the above etho provies an enueration proof (also calle exhaustive proof) for the NFL Theore. Enueration proof 6 [] is vali when the nuber of test cases is finite. Fro Theore in Section III the set of eterinistic algoriths is finite an thus enueration proof is vali. 5 Although A is not explicitl inicate in [] it is trivial to see that all eterinistic algoriths will for the set A. 6 An exaple of using enueration proof to valiate atheatical stateents can be foun in [9]. 4436

6 To eonstrate the above etho we consier the sste with X = {3} an Y = {3} again. Let. We efine equal to [ ]. Table I shows the results of copleting up to Step 3(ii) for the two eterinistic algoriths shown in Table II (Table I(a) an I(b) correspon to Algoriths an respectivel). It is eas to check both the totals of Algorith an are equal to 9 b suing up the last coluns of Table I(a) an I(b). C. Perforance Measures In [] perforance easures are inicate b ( ) which can be interprete as an function with the -coponent of the histor as the input. The perforance is base on onl. For exaple ( ) in i{ ( i): i... } for iniizing f 7. For optiization we oubt the perforance easure shoul be base solel on. When we are optiizing f with an algorith a we want to evaluate the perforance of a with respect to the ost esirable function values of f. Without consiering f the actual values of copute b a are eaningless. We can illustrate the above iea with the following exaple: Suppose we tr to iniize two ifferent functions f an f (see Fig. ) with the sae optiization algorith a. We are onl intereste in evaluating the perforance of a after choosing the first three x in X i.e. = 3. If a generates the sae 3 [43] for both f an f can we sa that a has better perforance on either f or f b solel looking at 3? If we have soe ieas about all the -values of the functions (e.g. we have the curves of the functions as in Fig. ) we can efinitel conclue that a has better perforance on f than on f because the iniu of f has been obtaine in 3 but not for f. B onl looking at 3 however a has the sae perforance on both functions. Thus the perforance easure shoul be function-epenent an it shoul reflect the relative function values in to the ost esirable function values in question (e.g. for iniizing f with iniu * the perforance easure can check the ratio of to *). More general perforance easures can then be inicate b f ( ). Let D { : f( ) } where can be an reasonable conitions for perforance evaluation. For exaple can be specifie to require in{ Y} when = 3. Equation () i.e. the ke forula which constitutes the NFL Theore also inherits the above entione proble. The core of () is P ( f a ). In [] it is interprete as the conitional probabilit of obtaining a particular saple uner the conition that an algorith a iterates ties on a cost function f. Then the perforance of a is sai to be easure with P ( f a ) b stating that P( ( ) f a) can be easil erive fro P ( f a ). 7 This is the original exaple explaining perforance easures in []. Figure. Exaples to illustrate the effect of the perforance easure. If we aopt a ore general f ( ) as the perforance easure the NFL Theore shoul be re-state as: the average over all f of P( f( ) f a) is inepenent of a with ( ) being replace b f ( ). This result can in turn be erive b showing that for an pair of eterinistic algoriths a an a PD ( f a ) PD ( f a ) We are not tring to sa that () fails to coe up with the ore reasonable interpretation of perforance easures but we can show that () can be use to erive (4). We can obtain the connection b observing the enueration table of an sste with X an Y specifie. We tr to eonstrate the connection with the siple exaple shown in Table I(a). We first iscuss in (). The sste has X an Y with X = Y = 3. Thus there are 6 (= X! = 3!) coluns of 3 an each entr in a colun is the 3 of the corresponing f. As Y = 3 the axiu length of 3 is three. For an particular there are X exactl Y Y corresponing entries in each colun. Their locations a be ifferent in ifferent coluns i.e. the sae appears at rows corresponing to ifferent functions in ifferent coluns. For exaple [] [ ] an X X 3 [ 3] has exactl 9 ( = Y Y ) 3 (= Y Y ) an X 3 (= Y Y ) entries in each colun respectivel. We now coe to D ( ) in (4). Recall that D is a set of with the conition satisfie. When checking D fro colun to colun in an enueration table we fin that D appears exactl once in ever colun but D a correspon to ifferent f. In other wors we can consier each in D ( ) one-b-one an appl () to each of the. We have 4437

7 P ( f a ) P ( f a ) D ff D ff which can further iel (4). Fro this we are confient that the NFL Theore hols for an reasonable perforance easure. Up to now we have verifie (4) (an () which is a special case of (4)). We are going to valiate (3). Consier probabilistic algoriths Ap. B Theore 3 in Section III all probabilistic algoriths can be expresse as a probabilit istribution over the set of efine eterinistic algoriths. Let the saple space equal A. We can oel each probabilistic algorith b a iscrete rano variable H with probabilit ass function ph ( ) where h is an possible outcoe in i.e. hp( h) Ap. Thus for an a A an Ap h P ( f ) P ( f hph ) ( ) ff h ff P ( f h ) ph ( ) ff h P ( f ). (5) ff This eans that the perforance of an eterinistic algorith an that of an probabilistic algorith are the sae when average over all f. B cobining () () an (5) we get (3). Siilar to the wa of getting (4) fro () we have an ieiate corollar that for an A Ap P( D f ) P( D f ) We know that eterinistic an probabilistic algoriths constitute all optiization algoriths. Therefore an reasonable perforance easure of an algoriths (eterinistic an/or probabilistic) is statisticall ientical when average over all f. V. CONCLUSION Before the NFL Theore was publishe ost research on optiization focuse on solving specific probles with propose algoriths. The theoretical founation was largel ignore. The NFL Theore raws our attention to the theoretical founation an eonstrates a funaental liit of all search algoriths. However we fin liitations of NFL stateents in the original NFL papers. B using results fro the nature of search algoriths we enhance several aspects of the original NFL Theore. We have ientifie the properties of eterinistic an probabilistic algoriths. We provie an enueration proof. In aition we generalize the NFL results with ore general perforance easures an reove the liitations. REFERENCES [] M.R. Gare an D.S. Johnson Coputers an Intractabilit: A Guie to the Theor of NP-Copleteness. New York NY: W. H. Freean & Co Lt 979. [] D.E. Golberg Genetic Algoriths in Search Optiization an Machine Learning. Reaing MA: Aison-Wesle 989. [3] J.H. Hollan Aaptation in Natural an Artificial Sstes. Cabrige MA: MIT Press 99. [4] S. Kirkpatrick C.D. Gelatt Jr. an M.P. Vecchi Optiization b Siulate Annealing Science vol. no pp Ma 983. [5] E. Bonabeau M. Dorigo an G. Theraulaz Inspiration for Optiization fro Social Insect Behaviour Nature vol. 46 no. 679 pp Jul. [6] M. Dorigo an T. Stützle Ant Colon Optiization. Cabrige MA: MIT Press 4. [7] A.Y.S. La an V.O.K. Li Cheical-Reaction-Inspire Metaheuristic for Optiization subitte for publication. [8] F. Glover an F. Laguna Tabu Search. Norwell MA: Kluwer Acaeic Publishers 997. [9] D.H. Wolpert an W.G. Macrea No Free Lunch Theores for Search Santa Fe Institute Santa Fe NM Tech. Rep [] D.H. Wolpert an W.G. Macrea No Free Lunch Theores for Optiization IEEE Transactions on Evolutionar Coputation vol. no. pp April 997. [] Y.C. Ho an D.L. Pepne Siple Explanation of the No-Free-Lunch Theore an Its Iplications Journal of Optiization Theor an Applications vol. 5 no. 3 pp Deceber. [] J.C. Culberson On the futilit of blin search: An algorithic view of "No Free Lunch Evolutionar Coputation vol. 6 no. pp [3] A.Y.S. La an V.O.K. Li Nature of Search Algoriths Departent of Electrical an Electronic Engineering The Universit of Hong Kong Hong Kong Tech. Rep. TR-9- Januar 9. [4] C. Schuacher M.D. Vose an L.D. Whitle The No Free Lunch an Proble Description Length In Proceeings of the Genetic an Evolutionar Coputation Conference. San Francisco CA Morgan Kaufann pp [5] C. Igel an M. Toussaint No-Free-Lunch theore for non-unifor istributions of target functions Journal of Matheatical Moelling an Algoriths vol. 3 no. 4 pp [6] M.J. Streeter Two broa classes of functions for which a no free lunch result oes not hol Lecture Notes in Coputer Science (incluing subseries Lecture Notes in Artificial Intelligence an Lecture Notes in Bioinforatics) pp [7] S. Droste T. Jansen an I. Wegener Optiization with ranoize search heuristics - The (A)NFL theore realistic scenarios an ifficult functions Theoretical Coputer Science vol. 87 no. pp Septeber. [8] T. English No ore lunch: Analsis of sequential search in Proceeings of the 4 Congress on Evolutionar Coputation Portlan OR pp [9] J. Sloan Two Rectangles are Constructible with Tangras: An Enueartion Proof using Matheatica Matheatica in Eucation vol. no. 4 pp [] J. Sloan Discrete Matheatics. Upper Sale River NJ: Prentice Hall