Runtime Analysis of the (µ+1) EA on Simple Pseudo-Boolean Functions

Transcription

1 Rutime Aalysis of the (µ+1) EA o Simple Pseudo-Boolea Fuctios Carste Witt FB Iformatik, LS 2, Uiversität Dortmud, Dortmud, Germay carste.witt@cs.ui-dortmud.de November 14, 2005 Abstract Although Evolutioary Algorithms (EAs) have bee successfully applied to optimizatio i discrete search spaces, theoretical developmets remai weak, i particular for populatio-based EAs. This paper presets a first rigorous aalysis of the (µ+1) EA o pseudo-boolea fuctios. Usig three wellkow example fuctios from the aalysis of the (1+1) EA, we derive bouds o the expected rutime ad success probability. For two of these fuctios, upper ad lower bouds o the expected rutime are tight, ad o all three fuctios, the (µ+1) EA is ever more efficiet tha the (1+1) EA. Moreover, all lower bouds grow with µ. O a more complicated fuctio, however, a small icrease of µ provably decreases the expected rutime drastically. This paper develops a ew proof techique that bouds the rutime of the (µ+1) EA. It ivestigates the stochastic process for creatig family trees of idividuals; the depth of these trees is bouded. Thereby, the progress of the populatio towards the optimum is captured. This ew techique is geeral eough to be applied to other populatio-based EAs. 1 Itroductio Evolutioary Algorithms (EAs) are successfully applied to optimizatio tasks, but theoretical kowledge is still far behid practical experiece. I recet years, advaces have bee made i the theoretical aalysis of the computatioal time complexity of EAs, i particular for the optimizatio i discrete search spaces iduced by pseudo-boolea fuctios f : {0, 1} R. However, this kid of theory ofte cocetrates o simple sigle-idividual EAs such as the (1+1) EA (e. g. Garier et al., 1999; Droste et al., 2002; Wegeer, 2002) ad its coevolutioary variats (Jase ad Wiegad, 2004). Therefore, it does ot explai the utility of populatios employed i may real-world EAs. Theoretical aalyses of the impact of crossover operators i populatio-based EAs (e. g. Jase ad Wegeer, 2001c) do ot ecessarily explai why a large populatio might be beeficial (Storch ad Wegeer, 2003). Hece, i this paper, we cosider istead populatio-based EAs where mutatio is the oly search operator. Such EAs have bee studied by Jase ad Wegeer (2001b) ad Witt (2003), who prove that a populatio ca be beeficial with fitessproportioal selectio schemes. Recetly, Storch (2004) aalyzed steady-state EAs usig uiform selectio ad diversity-maitaiig operators. Moreover, He ad Yao This work was supported by the Deutsche Forschugsgemeischaft (DFG) as a part of the Collaborative Research Ceter Computatioal Itelligece (SFB 531). 1

2 (2002) studied some variats of (µ+µ) EAs. However, results o the time complexity of stadard (µ+λ) EAs with uiform selectio are ot available for µ > 1, i. e., o-trivial sizes of the paret populatio. Up to ow, oly aalyses of (1+λ) EAs (Jase et al., 2005) are kow. The aim of this paper is to cotribute to a theory of stadard (µ+λ) EAs with µ > 1 i discrete search spaces. Here, we start with the simple case λ = 1 ad cosider a (µ+1) EA that is a geeralizatio of the (1+1) EA for the search space {0, 1}. I particular, we study the behavior of the (µ+1) EA o example fuctios ad compare the obtaied results with those for the (1+1) EA. To this ed, a ew ad geeral proof techique for boudig the expected rutime of the (µ+1) EA is developed. A advatage of the ew techique is that it has ot bee desiged for a special mutatio operator. I particular, we are able to aalyze the (µ+1) EA with a global search operator that may flip may bits. I other cotexts, aalysis of EAs is much more difficult with a global tha with a local search operator (for such examples see, e. g., Wegeer ad Witt, 2005). The paper is structured as follows. I Sectio 2, we defie the (µ+1) EA ad the three example fuctios cosidered. Furthermore, we itroduce the otio of family trees which will be used later i the ew proof techiques. I Sectio 3, upper bouds o the expected rutime of the (µ+1) EA o the fuctios are preseted. I Sectio 4, we describe how family trees ca be applied to derive a geeral lower boud o the expected rutime ad o the success probability of the (µ+1) EA. This tool is used agai i Sectio 5 to prove more specific lower bouds. These bouds are tight for two examples ad show for all three examples that the (µ+1) EA is asymptotically o more efficiet tha the (1+1) EA. However, it is a commo belief that a populatio helps to better explore the search space, ad oe should fid a situatio where the (µ+1) EA with µ > 1 outperforms the (1+1) EA. Ideed, a fuctio where a small icrease of µ decreases the expected rutime drastically is idetified i Sectio 6. We the fiish with some coclusios. 2 Defiitios We obtai the (µ+1) EA for the maximizatio of fuctios f : {0, 1} R as a geeralizatio of the well-kow (1+1) EA (see Droste et al., 2002). As i cotiuous search spaces, a pure (µ+1) evolutio strategy should do without recombiatio ad should employ a uiform selectio for reproductio. As usual, a trucatio selectio is applied for replacemet. The mutatio operator should be able to search globally, i. e., to flip may bits i a step. Therefore, a stadard mutatio flippig each bit with probability 1/ seems sesible. These argumets lead to the followig defiitio. Defiitio 1 ((µ+1) EA) 1. Choose µ idividuals x (i) {0, 1}, 1 i µ, uiformly at radom. Let the multiset X (0) = {x (1),...,x (µ) } be the populatio at time 0. Let t := Repeat (a) Choose some x from the populatio X (t) at time t uiformly at radom. (b) Create x by flippig each bit of x idepedetly with probability 1/. Let X be the populatio obtaied by addig x to X (t). (c) Create the multiset X (t+1), the populatio at time t + 1, by deletig a idividual with lowest f-value from X uiformly at radom. (d) Set t := t

3 We have kept the defiitio of the (µ+1) EA as simple as possible ad refrai from employig diversity-maitaiig mechaisms. Therefore, the (µ+1) EA with µ = 1 is very similar to the (1+1) EA, yet differs i oe respect. If a idividual created by mutatio has the same f-value as its paret, either oe is retaied with equal probability. As usual i theoretical ivestigatios, we leave the stoppig criterio of the (µ+1) EA uspecified ad aalyze the umber of iteratios (also called steps) of the ifiite loop util for the first time the curret populatio cotais a optimal idividual, i. e., oe that maximizes f. We the say that the (µ+1) EA has reached the optimum. The sum of this umber of steps ad the populatio size µ is deoted as the rutime of the (µ+1) EA ad correspods to the umber of fuctio evaluatios executed so far (a commo approach i black-box optimizatio, cf. Droste et al., 2006). Throughout the paper, we cosider oly µ = poly(), i. e., values of µ bouded by a polyomial of. For super-polyomial values, eve iitializatio aloe would produce a super-polyomial rutime. As metioed i the itroductio, we study the (µ+1) EA o well-kow fuctios already cosidered with the (1+1) EA. These fuctios are well structured ad serve as a startig poit for the aalysis w. r. t. to more complicated problems. Sice the followig three fuctios are meat to exhibit typical behavior of EAs, they are ofte called example fuctios. The first oe is the famous fuctio OeMax(x) = x x, which couts the umber of oes of a search poit x {0, 1}. The secod, LeadigOes(x) = i x j, i=1 j=1 couts the umber of leadig oes. Fially, we ivestigate the fuctio + 1 if x = 1 i 0 i for i <, SPC(x) = 2 if x = 1, OeMax(x) otherwise (SPC stads for short path with costat fitess), which was itroduced by Jase ad Wegeer (2001a). The search poits that ca be writte as 1 i 0 i, i <, form a so-called plateau, i. e., a regio of the search space that is coected via 1-bit flips ad has the same fitess, where the uique optimum is coected to oe ed of the plateau. This fuctio is of particular iterest sice the (µ+1) EA will have to search withi the plateau of costat fitess without guidace by fitess values. A iterestig questio is whether or ot this exploratio of the plateau beefits from the populatio. To elucidate the utility of the (µ+1) EA s populatio, throughout the paper, we compare the (µ+1) EA with µ parallel rus of the (1+1) EA. The total cost (eglectig iitializatio cost) of t steps of the (µ+1) EA correspods to the cost of performig µ parallel rus of the (1+1) EA util time t/µ. Thus, if we cosider the (µ+1) EA at time t, we deote µ parallel rus of the (1+1) EA cosidered at time t/µ as the correspodig parallel ru. I particular, we are iterested i examples where the (µ+1) EA is sigificatly more efficiet tha its correspodig parallel ru. I order to derive rutime bouds for the (µ+1) EA, it is helpful to cosider the so-called family trees of the idividuals from the iitial populatio. This cocept has bee itroduced i a differet cotext by Rabai et al. (1998) ad has already bee studied for differet populatio-based EAs i discrete ad cotiuous search 3

4 spaces by Witt (2003) ad Jägersküpper ad Witt (2005), respectively. Let x 0 be a arbitrary idividual from the iitial populatio. We visualize the descedats of x 0 created by time t 0 i the family tree T t (x 0 ) whose odes deote time steps ad are labeled with idividuals geerated at these steps; here it comes i hady that at each time step, exactly oe idividual is created. The edges of the tree correspod to mutatio steps ad model direct paret-child relatios betwee the idividuals labelig the odes. I may of our cosideratios, the time steps where idividuals are created do ot matter, ad for coveiece, we will idetify odes of family trees with the idividuals labelig these odes. For istace, we will sometimes say that odes of family trees are the idividuals that label them. I this otio, T t (x 0 ) cotais all descedats of x 0 produced by direct mutatio ad idirect mutatio of x 0 util time t. Although formally, the tree is udirected, the poits of time where idividuals are created lead to a orderig of the correspodig odes. Hece, descedats of x 0 are called successors of the root (= x 0) Figure 1: I a family tree, odes deote the poits of time where idividuals appear (left picture) ad are labeled with the search poit of the idividuals (right picture). Formally, each T t (x 0 ) is a udirected graph (V t, E t ) with V t N 0 ad a mappig c t : V t {0, 1}. These compoets are obtaied iductively. For T 0 (x 0 ), it holds that V 0 = {0}, E 0 = ad c 0 (0) = x 0. If at time t 0, the (µ+1) EA chooses the idividual x 0 for mutatio ad creates y, V t+1 := V t {t + 1}, c t+1 is obtaied from c t by extedig it via c t+1 (t+1) := y, ad E t+1 := E t {{0, t+1}}. If at time t, the idividual created i the mutatio step at time t < t is chose ad t + 1 V t holds, V t+1 ad c t+1 are defied as before ad E t+1 := E t {{t + 1, t + 1}}. (Note that due to techical reasos, the idividual created at time t labels ode t + 1. Hece, the coditio t + 1 V t meas that the idividual created at time t is cotaied i the tree at time t.) Otherwise V t+1 := V t, E t+1 := E t ad c t+1 := c t ; hece, i this case, T t+1 (x 0 ) = T t (x 0 ) because at time t, the (µ+1) EA chooses a idividual cotaied i a differet family tree. At ay time t, there are µ differet family trees T t (x 0 ), amely oe tree for each x 0 from the iitial populatio. Family trees whose roots are idividuals ot belogig to the iitial populatio ca be idetified with subtrees of trees T t (x 0 ) for some idividual x 0 from the iitial populatio. Sice the (µ+1) EA describes a ifiite stochastic process, the process growig the family trees T t (x 0 ) for t 0 is ifiite as well. Note, however, that T t (x 0 ) ca cotai idividuals that have already bee deleted from the populatio at time t. It ca eve happe that a give tree oly cotais deleted idividuals. O the other had, sice at least oe x 0 from the iitial populatio always has alive descedats, the tree T t (x 0 ) of this x 0 cotais arbitrarily may odes for growig t. Fially, 4

5 whe studyig family trees (ad subtrees thereof), we omit the time idex t whe iterested i the ifiite stochastic process rather tha a specific poit of time. Figure 1 shows oe exemplary family tree with its labelig. The left-had picture displays the ode umbers while the right-had picture cotais the labels from {0, 1} 3 belogig to the odes. While the ode umbers are mootoically icreasig o paths, this eed ot be the case for the value of the goal fuctio. If the goal fuctio for the picture is, e. g., OeMax, the idividuals labelig odes 2 ad 10 are deleted first. I the example, there are several poits of time where the (µ+1) EA chooses from the family tree of aother idividual from the iitial populatio. For example, this holds for t = 0 ad t = 3 sice odes 1 ad 4 are missig. I this paper, family trees are used to capture the progress of the populatio towards the optimum of the goal fuctio. Suppose that at some large time t, the depth of ay family tree (w. r. t. to its root) is still very small. This depth bouds the legth of every path from the root to a leaf i the tree, ad, as metioed above, such a path correspods to a sequece of mutatios that fially creates the idividual at the leaf, startig from the idividual at the root. Hece, if the depth is small, this meas that all leaves are idividuals that are with a high probability similar to the root. This makes the optimizatio of eve simple fuctios very ulikely, which may lead to lower bouds o the rutime. O the other had, lower bouds o the depth of family trees guaratee some progress ad may imply upper bouds o the rutime. We will lower boud the depth of family trees i the followig sectio i order to derive oe specific upper boud o the rutime. I cotrast, we will upper boud the depth of the trees i Sectio 4 to derive a very geeral lower boud o the rutime. More specific upper bouds o the depth of family trees ad lower bouds o the rutime are show i subsequet sectios. Hece, while the family trees are useful i both respects, applicatios for lower boudig the rutimes of populatio-based EAs seem to be more geeral ad commo. 3 Upper Bouds I this sectio, we will derive upper bouds o the expected rutime of the (µ+1) EA for the example fuctios. Oe may cojecture that these bouds are larger tha the correspodig bouds for the (1+1) EA by at most a factor µ. Although this turs out to be true for the three examples, this eed ot always be the case as will be show i Sectio 6. The first two of the followig bouds are ot too difficult to obtai. The proofs use potetial fuctios, a straightforward geeralizatio of the proof techique of artificial fitess layers (cf. Wegeer, 2002). We start with the problem for which we ca preset the easiest of the three proofs. Theorem 1 Let µ = poly(). The the expected rutime of the (µ+1) EA o LeadigOes is bouded above by µ + 3e max{µ l(e), } = O(µ log + 2 ). Proof: Note that the term µ reflects the cost of iitializatio. We measure the progress to the optimum by the potetial L, defied as the maximum Leadig- Oes value of the curret populatio s idividuals. To icrease L, it is sufficiet to select a idividual with value L (hereiafter called a best idividual) ad to flip oly the leftmost zero. The probability of the latter equals (1/)(1 1/) 1. Hece, if there are i best idividuals, the probability of the cosidered evet is at least i µ 1 ( 1 1 ) 1 i eµ, 5

6 ad the waitig time is at most eµ/i. The potetial has to icrease at most times. Estimatig i by 1 would lead to a upper boud of µ+eµ 2 o the expected rutime. However, the (µ+1) EA ca produce replicas of idividuals. If the umber of best idividuals is i, the probability of creatig a replica of a best idividual is bouded below by ( i µ 1 1 ) i 2eµ. Furthermore, if i < µ, this replica replaces a worse idividual ad icreases the umber of best oes. Assume pessimistically that L does ot icrease util we have at least mi{/l(e), µ} best idividuals. The expected time for this is at most /l(e) 1 i=1 2eµ i 2eµ ( l(/l(e)) + 1 ) 2eµ l(e) sice we ca boud the k-th Harmoic umber accordig to k i=1 1/i (l k) + 1. Now the expected time to icrease L is at most eµ/(mi{/l(e), µ}). Altogether, the expected rutime is at most ( ) eµ µ + 2eµ l(e) + µ + 3e max{µ l(e), } mi{/l(e), µ} as suggested. By the precedig proof, we have also show the followig corollary, which will tur out to be useful i Sectio 6. Corollary 1 Let µ = poly(). The the expected time util the (µ+1) EA o LeadigOes creates a idividual with k leadig oes is bouded above by µ + 3ek max{µ l(e), }. Now we study the well-kow OeMax problem. Iterestigly, we require somewhat more complicated argumets tha before to show upper bouds o the expected rutime. This was also the case whe Jase et al. (2005) showed correspodig upper bouds for the (1+λ) EA. Theorem 2 Let µ = poly(). The the expected rutime of the (µ+1) EA o OeMax is bouded above by µ + 5eµ + e l(e) = O(µ + log ). Proof: The idea of the proof is similar as i Theorem 1. Let L be the maximum OeMax value of the curret populatio. I cotrast to LeadigOes, the probability of icreasig L depeds o L itself. Sice each idividual has at least L zeros, the cosidered probability is bouded below by i µ L ( 1 1 ) 1 i( L) eµ if the populatio cotais at least i idividuals with value L. We are iterested i the expected time util the populatio cotais at least mi{/( L), µ} idividuals with value L. Usig the same elemetary calculatios as i the proof of Theorem 1, this time is bouded above by /( L) 1 i=1 2eµ i 2eµ l(e/( L)) 6

7 if L does ot icrease before. If we sum up these expected waitig times for all values of L, we obtai (usig Stirlig s formula to estimate! (/e) ) a total expected waitig time of at most 1 2eµ L=0 ( ) e l L ( e ) = 2eµ l 2eµ l(e 2 ) = 4eµ.! After the desired umber of idividuals with value L has bee obtaied, the expected time for icreasig L is at most eµ mi{µ, /( L)} ( L) = eµ mi{µ( L), }. Hece, the expected waitig time for all L-icreases is at most 1 ( eµ µ( L) + eµ ) L=0 e l(e) + eµ, ad the total expected rutime, therefore, at most µ + e l(e) + 5eµ. For SPC, we ca oly prove a (seemigly) trivial upper boud. Surprisigly, it will tur out later that this boud is at least almost tight. For the proof, the family trees come ito play for the first time. Theorem 3 Let µ = poly(). The the expected rutime of the (µ+1) EA o SPC is bouded by O(µ 3 ). Before the proof, we will first itroduce some otios. We call the members of the iitial populatio iitial idividuals. I the ru of the (µ+1) EA o SPC, there is a first poit of time where all idividuals of the curret populatio are so-called plateau poits, i. e., idividuals of shape 1 i 0 i, 0 i. (Note that we allow i = whereas the poit 1 is ot from the plateau of costat fitess. Oe could call these poits path poits sice 1 i 0 i forms a coected path; however, we stick to the otio of plateau poits to avoid cofusios with paths withi family trees.) Let t pla deote this poit of time. The radom variable t pla has fiite expectatio ad ca take eve extreme values such as 0, i. e., all iitial idividuals are plateau poits. With high probability, however, t pla occurs later, ad the optimum is ot reached util (ad icludig) time t pla. Steps occurrig after time t pla ad producig idividuals outside the plateau, i. e., ot of shape 1 i 0 i, immediately delete the created idividual sice its fitess must be worse tha that of a plateau poit. Let x 0 be a arbitrary idividual from a populatio at some time s t pla. We ow study the sequece of family trees T t (x 0 ), t s, whose commo root is labeled with x 0. If s > 0, x 0 ca have predecessors i the family tree of a iitial idividual, i.e., T t (x 0 ) is a subtree of its tree. Let y be a idividual i T t (x 0 ). If y has bee deleted by time t, we call the ode labeled with y ad the path from the root to y dead, ad alive otherwise. There is always at least oe alive path i some family tree. Now cosider a alive path at some time. The evet that the path is curretly alive imposes coditios o the mutatios creatig the odes o the path. After time t pla, e. g., a path caot be alive if it eds with a idividual outside the plateau. Our goal is to show that o alive paths i family trees, we ca rediscover a ru of the (1+1) EA o SPC. This is formalized by the followig lemma. Lemma 1 Cosider the (µ+1) EA o SPC. Let t pla s t, let x 0 be a arbitrary idividual from the populatio at time s, ad at time t, let p be a arbitrary path 7

8 startig at the root of T t (x 0 ). Let x 0,...,x l deote the idividuals alog p. If p is alive ad the optimum has ot bee reached yet, the sequece x 0,..., x l has the same distributio as the sequece of search poits of the (1+1) EA o SPC i the time steps 0,...,l provided that the (1+1) EA starts with x 0 ad that it util (ad icludig) time l oly creates search poits of shape 1 j 0 j where 0 j <. Proof: For otatioal coveiece, we without loss of geerality assume s = 0. Let 0, t 1,...,t l deote the odes alog p. We prove the claim by iductio o i {0,...,l}, where the base case i = 0 is trivial sice the (1+1) EA is assumed to start with x 0. For the iductio step, let 0 i < l ad assume that the ode t i, labeled with x i, has bee created ad is alive. The latter is ecessary for p to be alive at time t. Now the lemma imposes exactly the followig set of coditios o the evets creatig the alive path 0, t 1,..., t i+1 cosidered i the iductio step. 1. A step appeds to t i some t labeled with x = 1 k 0 k for k <. 2. If i + 1 < l, t stays alive at least util a step chooses t ad produces t i+2 ; otherwise, t stays alive at least util time t. 3. t = t i+1. The last two coditios follow immediately from the evet cosidered. The first coditio holds sice the optimum has ot yet bee reached ad sice we cosider a poit of time after t pla. If x was ot a plateau poit, x would be deleted immediately, cotradictig the secod coditio. Let A be the itersectio of the three coditios. If A occurs, x = x i+1. Let y 0,..., y l be the radom curret search poits of the (1+1) EA i the steps to time l; by assumptio, y 0 = x 0. Aalogously to the first coditio, the lemma requires all mutatios of the (1+1) EA util time l to create plateau poits differet from 1. By the defiitio of the (1+1) EA, all these mutats are accepted. Hece, both the (µ+1) EA s mutatio geeratig x i+1 ad the mutatio of the (1+1) EA geeratig y i+1 are coditioed o the evet of creatig search poits of shape 1 j 0 j for 0 j <. We ivestigate the distributio of x i+1 more carefully. Here it is crucial that the (µ+1) EA chooses the idividual to be deleted uiformly from the set of worst idividuals. Sice all idividuals of shape 1 j 0 j, j, have the same SPC-value, the evet that such a idividual is deleted is idepedet of the value of j. Hece, the secod of the above coditios is idepedet of k, the umber of oes i x. It follows that the distributio of k is the same uder oly the first coditio as uder Coditio A. The (1+1) EA uses the same mutatio operator as the (µ+1) EA ad, accordig to the last paragraph, it creates y i+1 uder that coditio holdig for x. If x i ad y i take the same value 1 j 0 j, their successors have the same distributio. Sice, by iductio hypothesis, x i is idetically distributed as y i, the iductio step follows. Havig established the correspodece to the (1+1) EA o SPC, we are ready to prove Theorem 3. I Sectio 5.2, Lemma 1 will be applied agai to derive lower bouds o the rutime. Proof of Theorem 3: Let x 0 be a arbitrary idividual from the populatio at time t pla ad let T(x 0 ) be its family tree. We wat to show that the followig property (P) holds: the expected time util at least oe path i T(x 0 ) reaches legth k or util all paths i T(x 0 ) are dead is bouded above by 4eµk for all k 1, or the optimum is reached before. This property will etail the theorem for the followig reasos. We will show that E(t pla ) = O(µ log ). Moreover, we will apply Lemma 1 i the followig way. We cosider the evet that i T(x 0 ), a path 8

9 is created where at least oe ode is labeled with 1. It follows that the probability of obtaiig 1 o a path of legth l + 1 withi T(x 0 ) is bouded below by the probability that the (1+1) EA with iitial search poit x 0 creates 1 withi l + 1 steps. (Here, eve usuccessful steps of the (1+1) EA do ot cout.) By the results of Jase ad Wegeer (2001a), this implies that the optimum is obtaied after a expected path legth of O( 3 ) i a family tree. By Markov s iequality, a path legth of O( 3 ) is sufficiet to reach the optimum with probability at least 1/2. Sice at least oe path remais alive forever, this implies accordig to (P) that with probability at least 1/2, the optimum is reached after O(µ 3 ) expected steps. I the case of a failure, we ca repeat the above argumetatio with a idividual from some time t > t pla istead of x 0. The expected umber of repetitios is bouded above by 2 such that the optimum is reached after a expected umber of O(µ 3 ) steps. To prove (P), we oly have to cosider the case that there is always at least oe alive path i T(x 0 ) util time t pla + 4eµk. (Otherwise, there is othig to show.) For ay poit of time t t pla, we defie the followig potetial L t. Let S t be the set of those curretly alive successors of x 0 that will always have a alive descedat util time t pla +4eµk. The L t is defied to deote the maximum depth of S t -idividuals i T(x 0 ), i. e., the legth of a logest path leadig from the root to a S t -idividual. A special property of the potetial L t is that it depeds o the future, but is a fuctio mappig each curret poit of time to a value. Moreover, by defiitio, L t caot shrik i the ru of the (µ+1) EA, ad there is the followig sufficiet coditio for icreasig the L-value. A idividual x defiig the curret L-value is chose for mutatio, a child beig a plateau poit is produced, ad x is deleted from the populatio before its child is deleted. The probability is 1/µ for the first evet, (1 1/) 1/(2e) for the secod evet sice it is sufficiet to produce a replica, ad 1/2 for the third oe sice the cosidered idividuals have the same SPC-value. Hece, the expected time to icrease the L-value is bouded above by 4eµ, implyig that a alive path reaches legth k a expected umber of at most 4eµk steps after time t pla. We still have to show that E(t pla ) = O(µ log ). If the populatio cotais at least oe plateau poit, there is the followig sufficiet coditio for icreasig the umber of plateau poits: the (µ+1) EA chooses a plateau poit ad produces a replica of it. Hece, by similar argumets as i the proof of Theorem 1, the populatio is filled up by plateau poits after a expected umber of O(µ log µ) = O(µ log ) steps. Otherwise, the SPC-value of ay idividual x of the populatio is give by OeMax(x). The the expected time util creatig a plateau poit is bouded above by the expected time util the umber of oes has bee miimized for at least oe idividual, i. e., by O(µ log ). 4 A Geeral Lower Boud Techique For lower bouds o the rutime of (µ+1) EA, we cosider the growth of the family tree T(x 0 ) for a arbitrary iitial idividual x 0. As metioed i Sectio 2, upper bouds o the depth of family trees ca imply lower bouds o the rutime. Such upper bouds follow from the selectio mechaism of the (µ+1) EA, which always selects the idividual to be mutated uiformly from the curret populatio. Recall that a path i a family tree has bee defied as a mootoically icreasig sequece 0, t 1,...,t l i which t i deotes the poit of time where the i-th ode is preset for the first time. The probability that the path 0, t 1,...,t l is created ad labeled with 9

10 the idividuals x 0, x 1,...,x l is upper bouded by l 1 i=0 mut(x i, x i+1 ) µ = ( ) l l 1 1 mut(x i, x i+1 ), µ where mut(x, y) deotes the probability of the mutatio operator creatig y {0, 1} from x {0, 1}. This holds sice both the selectio ad the mutatio operator i a step of the (µ+1) EA work idepedetly of previous steps ad sice the probability of choosig ode t i at time t i+1 1 is either 0 (if the idividual created at time t i 1 has bee deleted i the meatime) or 1/µ. Regardless of the labelig, this implies a upper boud o the depth of family trees, which holds with overwhelmig probability. Lemma 2 Let D(t) deote the depth of a family tree of the (µ+1) EA at time t. For all t 0, Prob(D(t) 3t/µ) = 2 Ω(t/µ). Proof: Let l := 3t/µ. We show the claim by cosiderig all possible paths of legth exactly l. The evet that at least oe such path emerges equals the evet D(t) 3t/µ. We already kow that the probability of a fixed path 0, t 1,..., t l is upper bouded by (1/µ) l. The umber of differet paths of legth l possible util time t is bouded above by ( t l) sice this equals the umber of choices such that 1 t 1 < < t l t. Hece, the cosidered probability is bouded above by ( ) ( ) l ( ) ( ) l t 1 lµ/3 1. l µ l µ Stirlig s formula implies ( ) k ( e k )k for 0 k. Hece, we ca fially upper boud the probability by ( ) l ( ) l lµe/3 1 ( e ) l = = 2 Ω(l) = 2 Ω(t/µ) l µ 3 sice e/3 < 1. Lemma 2 states that with overwhelmig probability, a family tree of the (µ+1) EA becomes asymptotically o deeper tha the total umber of mutatios performed i a sigle ru of the correspodig parallel ru (cf. Sectio 2). The tree ca become wide, but a flat tree meas that o ay path from the root, few mutatios occur. As metioed above, ituitively, this makes the optimizatio of eve simple fuctios very ulikely. The followig theorem makes this precise by eve coverig some fuctios whose set of global optima has expoetial size. Essetially, we assume the set of global optima to have size 2 o() or that all global optima have a liear Hammig distace to search poits with /2 oes. The rutime bouds derived are tight for some simple fuctios such as OeMax (if µ is ot too small). To formulate the theorem, we itroduce the abbreviatio x = x x for x {0, 1}. Theorem 4 Let µ = poly(), f : {0, 1} R, S opt := argmax{f(x) x {0, 1} }, o mi := mi{ x x S opt } ad o max := max{ x x S opt }. If at least oe of the three coditios i=0 S opt = 2 o(), o mi = /2 + Ω(), o max = /2 Ω() holds, the expected rutime of the (µ+1) EA o f is at least Ω(µ), ad the success probability i cµ steps is 2 Ω() if the costat c > 0 is chose appropriately small. If S opt = 1, the expected rutime o f is eve Ω(µ + log ). 10

11 Proof: The lower boud Ω( log ) i the case S opt = 1 eeds oly be show for µ c log where c > 0 is a arbitrary costat. Now let S opt = 1 ad µ log /2. We argue accordig to a geeralizatio of the Coupo Collector s Theorem that has bee described by Droste et al. (2002) for liear fuctios ad the (1+1) EA. To geeralize their argumetatio, we estimate the probability that bit i {1,...,} is i all iitial idividuals differet from the uique optimal assigmet. Sice µ log /2, this probability is at least 1/2. By Cheroff bouds (see Motwai ad Raghava, 1995), at least /2 bits are wrog i all iitial idividuals with probability 1 2 Ω( ). Therefore, assumig that there are /2 such bits, the probability that at least oe of these bits is ever flipped withi t := ( 1)(l )/2 steps is bouded below by 1 ( 1 ( 1 1 ) ) t /2 ( ) /2 1 e 1/2, which implies that t steps are required with probability at least 1 e 1/2 2 Ω( ) = Ω(1). Hece, the lower boud Ω( log ) o the expected rutime follows. For the lower rutime boud Ω(µ) uder oe of the three coditios, we set up a phase of legth s := cµ for some costat c > 0 ad show that the (µ+1) EA requires at least s steps with probability 1 2 Ω() if c is small eough. The idea for the proof is as follows. Withi s steps, each family tree created by the (µ+1) EA (with high probability) does ot cotai odes labeled with optimal idividuals. To prove this, we study the maximal Hammig distace betwee the idividuals at the odes i the tree ad the idividual at the root. Afterwards, we show that with high probability, the Hammig distace of the root idividual to all optimal idividuals is greater. Let x 0 be a arbitrary iitial idividual ad let T t (x 0 ) deote its radom family tree at time t. Accordig to Lemma 2, the probability that T s (x 0 ) reaches depth greater tha 3c is 2 Ω(). Now the aim is to prove that also with probability 2 Ω(), there is a ode i T s (x 0 ) at depth at most 3c ad labeled with a idividual whose Hammig distace to x 0 is bouded below by 8c. Let us cosider a sequece of 3c poits where each poit is the result of a mutatio of its predecessor by meas of the (µ+1) EA s mutatio operator. Each bit i each poit i this sequece is flipped idepedetly with probability 1/. Hece, the expected Hammig distace of ay two poits i this sequece is at most 3c, ad, by applyig Cheroff bouds w. r. t. to the upper boud 4c, it is at least 8c with probability bouded above by e 4c/3. We call a path bad if it starts at the root of T s (x 0 ), has legth l 3c ad cotais a label with Hammig distace at least 8c w. r.t. x 0. Together with our cosideratios from the begiig of this sectio, we obtai that the probability of creatig a specific bad path 0, t 1,...,t l is bouded above by (1/µ) l e 4c/3. We ow are able to estimate the probability of T s (x 0 ) cotaiig a bad path. Sice the umber of paths of legth at most l is bouded above by ( s l), the probability is at most 3c ( s l l=1 ) ( 1 µ ) l e 4c/3 3c 3c max l=1 4c/3 3c = 3ce { (ceµ ) ( ) } l l 1 e 4c/3 l µ ( ce ) l max. l=1 l Sice the expressio ((ce)/l) l is maximized for l = c, we obtai the upper boud 3ce c e 4c/3 = 3ce c/3 = 2 Ω(). 11

12 The last estimatio holds sice c is assumed to be a positive costat. We fially show that the Hammig distace of x 0 to all optimal idividuals with high probability is at least 8c if c is small eough. First we cosider the case S opt = 2 o(). For ay y S opt, it holds that x 0 has a expected Hammig distace /2 to y ad by Cheroff bouds, the Hammig distace is at least /3 with probability 1 2 Ω(). Sice S opt = 2 o(), the Hammig distace from x 0 to all y S opt is also at least /3 with probability 1 2 Ω(). I the case o mi = /2 + Ω(), we apply Cheroff bouds w. r. t. to the expectatio E( x 0 ) = /2. Hece, the Hammig distace from x 0 to ay optimal idividual is bouded below by Ω() with probability 1 2 Ω(). The case o max = /2 Ω() is symmetrical. Therefore, i all three cases, choosig c small eough results i 8c beig smaller tha the lower boud Ω(). All i all, for a fixed iitial idividual x 0, T s (x 0 ) with probability 1 2 Ω() does ot cotai odes labeled with optimal idividuals. Sice µ = poly(), this also holds for all iitial idividuals together. Theorem 4 covers the wide rage of uimodal fuctios. For certai subclasses of the class of uimodal fuctios (icludig liear fuctios), the (1+1) EA s expected rutime is at most O( log ). With respect to this class, Theorem 4 states that the (µ+1) EA is at most by a factor of O(log ) more efficiet tha the correspodig parallel ru. For more difficult fuctios (which meas that the (1+1) EA has a larger expected optimizatio time tha O( log )), the proof cocept of Theorem 4 ca be carried over to show larger lower bouds also for the (µ+1) EA. However, we have to derive better lower bouds o the depth that family trees eed for optimizatio. Thus, more structure of the fuctio f ad of the ecoutered search poits comes ito play. 5 Special Lower Bouds I this sectio, we show specialized lower bouds o the rutime of the (µ+1) EA. As i the proof of Theorem 4, the key idea here is to boud the depth of family trees. 5.1 Lower Boud for LeadigOes Theorem 5 Let µ = poly(). The the expected rutime of the (µ+1) EA o LeadigOes is Ω(µ log + 2 ). Moreover, the success probability withi some cµ log steps, c > 0, is 2 Ω(). Proof: The boud Ω( 2 ) follows by meas of the aalysis of the (1+1) EA o LeadigOes described by Droste et al. (2002, Theorem 17). This aalysis ca directly be applied to the potetial L from the proof of Theorem 1, i. e., the maximum LeadigOes value of the (µ+1) EA s curret populatio. The probability of icreasig L is at most 1/ ad at least the rightmost L 1 bits are uiformly distributed i each idividual of the populatio. This is agai a cosequece of the (µ+1) EA s deletio operator. Therefore, the estimatios for the umber of so-called free-riders carry over to the potetial L. The basic idea for the boud Ω(µ log ) is the same as i Theorem 4. We will show that for some small eough costat c > 0, the (µ+1) EA requires at least s := cµ log steps with probability 1 2 Ω(). To this ed, we cosider the family tree T s (x 0 ) obtaied after s steps for a arbitrary iitial idividual x 0. Usig Lemma 2 ad µ = poly(), it oly remais to show that with probability 1 2 Ω(), o ode at depth at most 3c log i T s (x 0 ) is optimal. 12

13 For otatioal coveiece, let f := LeadigOes. Sice all iitial idividuals are uiform over {0, 1}, f(x 0 ) /2 with probability 1 2 Ω(). As metioed above (show by Droste et al., 2002), it holds that i each idividual of ay populatio, the bits after the leftmost zero of the idividual are uiformly distributed. This implies that with probability 1 2 Ω(), alog ay fixed path i T s (x 0 ), it is ot eough to icrease the f-value at most /6 times to obtai a total icrease i f-value of at least /2. We call a path bad if the f-value icreases at least /6 times o the path. Hece, we have to estimate the probability that a specific bad path 0, t 1,..., t l is created. The probability of icreasig the f-value is bouded above by 1/ sice the leftmost zero has to flip. Hece, usig the estimatio from the begiig of Sectio 4, the probability of creatig such a path is bouded above by ( 1 µ ) l ( ) /6 1. Fially, we estimate the probability of T s (x 0 ) cotaiig a bad path by the same method as i the proof of Theorem 4. Sice the umber of paths of legth at most l is bouded above by ( s l), the probability is at most 3clog l=1 ( s l ) ( 1 µ ) l ( ) /6 1 3c(log) ( 1 ) /6 3clog max l=1 ( ) l ce log. l Sice the expressio ((ce log )/l) l is maximized for l = c log, we obtai the upper boud 3c(log) e clog e ((l2)/6)log = 3c(log) e (c (l2)/6) log. We choose c small eough such that the last expressio is 2 Ω(log ). Altogether, T s (x 0 ) does ot cotai a optimal ode with probability 1 2 Ω(). Fially, sice µ = poly(), this also holds for all iitial idividuals together. We have see that the waitig time Θ() required by the (1+1) EA for a icrease of the LeadigOes-value traslates ito a factor Θ(µ log ) withi the rutime of the (µ+1) EA provided that µ = Ω(/log ). A similar correspodece seems to hold for a geeralizatio of LeadigOes called LOB b (leadig oes blocks) studied by Jase ad Wiegad (2004). For costat b N, let LOB b (x) := bi j=1 x j OeMax(x). The OeMax part of LOB b (x) implies that after /b i=1 a short time ad with high probability, each curret idividual of the (1+1) EA ad the (µ+1) EA is of shape 1 bi 0 bi for some i. To create a search poit with higher LOB b -value, a mutatio of probability Θ( b ) is ecessary ad sufficiet. It seems that the argumets from the proofs of Theorem 1 ad Theorem 5 ca be geeralized i a straightforward maer. We cojecture that the expected rutime of the (µ+1) EA o LOB b equals Θ( b + µ log ). 5.2 Lower Boud for SPC It is iterestig to study i how far the exploratio of the plateau of costat fitess posed by the fuctio SPC beefits from the populatio of the (µ+1) EA. Therefore, i this sectio, we will derive a lower boud o the expected rutime of the (µ+1) EA o SPC. I fact, as i the upper boud of Theorem 3, the mai idea will be to aalyze situatios where the whole populatio cosists of idividuals from the plateau of costat fitess. Ufortuately, the followig lower rutime boud does ot match the upper boud from the above-metioed theorem. Note, however, that the lower boud is Ω( 3 ), implyig that the (µ+1) EA is asymptotically ever more efficiet tha the (1+1) EA o SPC. 13

14 Theorem 6 Let µ = poly(). The the expected rutime of the (µ+1) EA o SPC is lower bouded by Ω(µ 3 /log µ). Moreover, the success probability withi some cµ 3 /log µ steps, c > 0, is 2 Ω(log2 µ). The mai proof idea is the same as i the proof of Theorem 5. By Lemma 2, the probability of a family tree s depth reachig at least 3c 3 /log µ withi a phase of cµ 3 /log µ steps is 2 Ω(3 /log µ). However, we will prove that a depth of at least 3c 3 /logµ is ecessary for optimizatio with probability 1 2 Ω(log2 µ) i every family tree. To this ed, we fix a arbitrary iitial idividual x 0 ad cosider the followig evet. I the family tree T(x 0 ), startig from the root, a path p = (0, t 1,..., t l ) is created that is alive at time t l. Some odes of p might be labeled with the optimal search poit 1 ; however, we will show that this is ulikely for well-chose values l = Θ( 3 /log µ). Note that i cotrast to the proofs of Theorem 4 ad Theorem 5, we restrict our attetio to alive paths i family trees. There are two reasos for this. First, we will be able to reapply Lemma 1. Secod, the umber of alive paths i a family tree always is at most µ, i. e., is bouded by a polyomial rather tha expoetial value. For techical reasos, we are iterested i odes of p that are far away from both 1 ad 0 ad study oly a subpath p of p with the followig property: all search poits o p are of shape 1 i 0 i with /4 i 3/4. Of course, there is a positive probability that o such subpath p exists. However, our goal is to show that with high probability, such a p exists ad has legth at least Ω( 3 /log µ). The latter is prove i two steps. First, we show that with high probability, the labels of the odes chage Ω( 2 /logµ) times alog p. Secod, the distace of two odes with differet labels is Ω() with probability Ω(1). To be able to ivestigate p, we cocetrate o poits of time where the populatio cotais plateau poits, i. e., idividuals of shape 1 i 0 i, 0 i. Let t pl1 deote the first poit of time such that at least oe plateau poit exists. Moreover, let t pla still deote the first poit of time such that µ plateau poits are i the populatio. Recall that the optimum may have bee reached before. We prove that this is very ulikely. Lemma 3 With probability 1 2 Ω( ), all populatios util (ad icludig) time t pla cotai oly idividuals with at most 3/5 oes. Proof: First of all, we prove that with probability 1 2 Ω(), all idividuals util time t pl1 have at most 4/7 oes. To show this, the followig argumets are used. Sice the iitial idividuals are draw uiformly at radom ad µ = poly(), the probability that there exists a iitial idividual with more tha 5/9 oes is bouded above by 2 Ω() accordig to Cheroff bouds. Before time t pl1, the goal fuctio SPC(x) equals i=1 x i, also called ZeroMax. Hece, the selectio mechaism of the (µ+1) EA implies that the miimum umber of zeros i the populatio caot decrease util time t pl1 1. By the same argumets as i the proof of Theorem 3, we have E(t pl1 ) = O(µ log ) ad, by Markov s iequality ad the fact that idepedet phases may be repeated, t pl1 = O(µ 2 log ) with probability 1 2 Ω(). Sice flippig a liear umber of bits i at least oe out of at most O(µ 2 log ) steps of the (µ+1) EA has probability 2 Ω(log ) due to µ = poly(), the plateau poit created at time t pl1 has at most 4/7 oes, ad the other idividuals have at most 5/9 oes with probability 1 2 Ω(). I the followig, we assume this evet to have happeed. By similar argumets as i the proof of Theorem 1, the expected umber of steps from time t pl1 util time t pla is O(µ log µ). Moreover, this time is bouded by O(µ log µ) with probability 1 2 Ω( ) sice we ca apply Markov s iequality ad repeat idepedet phases. Accordig to Lemma 2, o family tree becomes 14

15 deeper tha O( log µ) i this umber of steps with probability 1 2 Ω( ), therefore we ca estimate the umber of oes i the idividuals (which are plateau poits) i the populatio at time t pla usig the argumets from the proof of Theorem 4. Replacig the value cµ used there for s with some value O(µ log µ) ad cosiderig the family trees for the populatio at time t pl1, we obtai that o idividual at time t pla has more tha 3/5 oes with probability 1 2 Ω( ). Now we ca study p, a well-structured subpath of the above-metioed path p. I the followig lemma, we may assume p to be as log as eeded for the evets cosidered there. Oly if p is log eough, the evets ca occur. To cope with the case that µ does ot grow with, we have to be careful whe applyig O-otatio with respect to µ. Lemma 4 Cosider a arbitrary alive path p = (0, t 1,..., t l ) i T(x 0 ) ad suppose that the optimum is ot reached by time t l. With probability at least max{1 2 Ω(log2 µ), Ω(1)}, there is a subpath p of p such that all odes of p are created at time t pla or later, p cotais oly labels 1 i 0 i for /4 i 3/4, ad the labels alog p chage Ω( 2 /log µ) times. Proof: Let v be the label of the last ode of p that is created util (ad icludig) time t pla. Accordig to Lemma 3, it cotais at most 3/5 oes with probability 1 2 Ω( ). From time t pla o, a idividual that is ot a plateau poit is deleted immediately after its creatio. We idetify odes i family trees with their labels. Hece, v ad all its successors o p are plateau poits sice p could ot be alive at time t l otherwise. Sice flippig a liear umber of bits i oe step out of polyomially may steps has probability 2 Ω(log ), it follows that with probability 1 2 Ω(), there is a successor v of v o p that has at least /3 ad still at most 2/3 oes. Hece, with probability 1 2 Ω( ), we have idetified a start poit v for p that is created at time t pla or later. The path p eds at the first ode that is ot of shape 1 i 0 i with /4 i 3/4. Now we have to show that with high probability, at least Ω( 2 /log µ) odes o p are differet plateau poits tha their parets. The start ode v is a search poit 1 i 0 i, /3 i 2/3, with Hammig distace Ω() to the optimal search poit 1. We wat to show that the radom walk describig the differet plateau poits o p is similar to a fair radom walk o the lie of legth Θ(), i. e., where the probability of movig to the left o the lie equals the probability of movig to the right (Feller, 1968). This is also a key idea i the proof of the upper boud for the (1+1) EA o SPC (Jase ad Wegeer, 2001a). For the described radom walk, it is well kow that distace Ω() to its start poit is ot overcome withi Θ( 2 /logµ) steps with probability 1 2 Ω(log2 µ). Sice we study paths that are alive at at least oe poit of time ad the optimum is assumed to be reached after time t l, we ca apply Lemma 1 o the sequece of family trees rooted at v ad argue about v s successors. We cosider the first at most s := 2 /(50(logµ + 1)) odes o p that are differet search poits tha their parets. Each cosidered search poit is the result of a mutatio of some 1 i 0 i ito some 1 j 0 j, j i. We also cosider these at most s mutatios. I cotrast to the above-metioed radom walk, it may be the case that j i > 1, i. e., steps of size larger tha 1 occur. Therefore, we ivestigate the evet that the (µ+1) EA flips the i-th bit of a search poit 1 i 0 i. The it is most likely to obtai 1 i 1 0 i+1 by the mutatio. To create a plateau poit 1 i 1 k 0 i+1+k for some k 1, the bits i k,...,i 1, so-called additioally flippig bits, have to be flipped simultaeously. The correspodig probability is bouded above by 1/ k for the (µ+1) EA s mutatio operator. A aalogous statemet holds for the case that the (i + 1)-st bit is flipped. The expected umber of additioally flippig 15

16 bits i s steps is, therefore, bouded above by i=1 s/i 2/(50(logµ+1)), ad the probability of /24 additioally flippig bits is 2 Ω() accordig to Cheroff bouds. This will be used i the ext paragraph. I the followig, we assume o mutatio step withi s steps to flip at least /4 bits, which holds with probability 1 2 Ω(log ). Now, give the evet that some 1 i 0 i, where /4 i 3/4, is mutated to a differet plateau poit, the probability of flippig bit i ad bit i + 1 is the same. Hece, the probability of obtaiig a poit with more tha i oes is the same as the probability of obtaiig less oes, ad we have established the correspodece to the fair radom walk. I s steps, we expect s/2 steps icreasig ad s/2 steps decreasig the umber of oes provided o search poit with less tha /4 or more tha 3/4 oes is ecoutered i these steps. We call the latter evet a failure. Accordig to Cheroff bouds, the probability that there is a surplus of /25 icreasig or decreasig steps is bouded above both by 2 Ω(log2 µ) ad by 1 Ω(1) if o failure occurs. Recall that v has at least /3 ad at most 2/3 oes. Usig the above aalysis of additioally flippig bits, we obtai that the probability of creatig 1 i 0 i, where i < /4 or i > 3/4, withi the s mutatios is bouded above by max{2 Ω(log2 µ), 1 Ω(1)}, which justifies the assumptio that o failure occurs. Now we oly eed to estimate the distace of two odes o p that are differet poits from their parets. We call such odes actio odes. To boud the distace of two actio odes, we agai make use of Lemma 1. Lemma 5 With probability Ω(1), the distace of ay two cosecutive actio odes o p is Ω(). This holds idepedetly of previous actio odes o p. Proof: First of all, we cosider the superpath p, which, by our assumptios, is alive before reachig the optimum. Let v be a arbitrary ode o p created at time t pla or later ad let v be its successor o the cosidered part of p. The path from v to v is alive at least oe poit of time such we ca apply Lemma 1 with x 0 := w. Agai, we idetify odes with their labels. Hece, we kow that v is the result of a mutatio uder the coditio that its predecessor, a plateau poit, is chaged agai ito a plateau poit differet from 1. The ucoditioal probability of chagig a plateau poit 1 i 0 i ito a differet plateau poit is bouded above by 2/ sice the i-th or (i + 1)-st bit has to flip. The probability of chagig a plateau poit agai ito a plateau poit is bouded below by (1 1/) 1/(2e) sice a replica is sufficiet. Hece, the probability that v differs from v is bouded above by 4e/. Cosequetly, the first /(8e) successors of v o p are the same plateau poit as v with probability at least 1/2. Actually, rather tha p, we cosider odes o the subpath p. By defiitio, these are created at time t pla or later as desired. However, also by defiitio, oly odes of shape 1 i 0 i, where /4 i 3/4, are created o p, ad the search poits alog p chage Ω( 2 / log µ) times. The first coditio oly icreases the probability of creatig a replica, ad the secod oe does ot ifluece the distace of actio odes o p. Fially, the above statemet o the successors of v holds also if v is a actio ode sice the defiitio of a actio ode oly implies a coditio o its predecessor. Therefore, the distace of a actio ode ad the ext actio ode o p is at least /(8e) with probability at least 1/2. This estimatio holds idepedetly of previous actio odes sice the mutatio operator of the (µ+1) EA works idepedetly of earlier steps ad creatig a replica does ot violate the properties of p. Fially, we ca put the precedig three lemmas together. 16