A Comparison of Performance Measures for Online Algorithms


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1 A Comparison of Performance Measures for Online Algorithms Joan Boyar 1, Sany Irani 2, an Kim S. Larsen 1 1 Department of Mathematics an Computer Science, University of Southern Denmark, Campusvej 55, DK5230 Oense M, Denmark, 2 Department of Computer Science, University of California, Irvine, CA 92697, USA, Abstract. This paper provies a systematic stuy of several propose measures for online algorithms in the context of a specific problem, namely, the two server problem on three colinear points. Even though the problem is simple, it encapsulates a core challenge in online algorithms which is to balance greeiness an aaptability. We examine Competitive Analysis, the Max/Max Ratio, the Ranom Orer Ratio, Bijective Analysis an Relative Worst Orer Analysis, an etermine how these measures compare the Greey Algorithm an Lazy Double Coverage, commonly stuie algorithms in the context of server problems. We fin that by the Max/Max Ratio an Bijective Analysis, Greey is the better algorithm. Uner the other measures, Lazy Double Coverage is better, though Relative Worst Orer Analysis inicates that Greey is sometimes better. Our results also provie the first proof of optimality of an algorithm uner Relative Worst Orer Analysis. 1 Introuction Since its introuction by Sleator an Tarjan in 1985 [16], Competitive Analysis has been the most wiely use metho for evaluating online algorithms. A problem is sai to be online if the input to the problem is given a piece at a time, an the algorithm must commit to parts of the solution over time before the entire input is reveale to the algorithm. Competitive Analysis evaluates an online algorithm in comparison to the optimal offline algorithm which receives the input in its entirety in avance an has unlimite computational power in etermining a solution. Informally speaking, we look at the worstcase input which maximizes the ratio of the cost of the online algorithm for that input to the cost of the optimal offline algorithm on that same input. The maximum ratio achieve is calle the Competitive Ratio. Thus, we factor out the inherent ifficulty of a particular input (for which the offline algorithm is penalize along The work of Boyar an Larsen was supporte in part by the Danish Natural Science Research Council. Part of this work was carrie out while these authors were visiting the University of California, Irvine. The work of Irani was supporte in part by NSF Grant CCR
2 2 J. Boyar, S. Irani, K.S. Larsen with the online algorithm) an measure what is lost in making ecisions with partial information. Despite the popularity of Competitive Analysis, researchers have been well aware of its eficiencies an have been seeking better alternatives almost since the time that it came into wie use. (See [9] for a recent survey.) Many of the problems with Competitive Analysis stem from the fact that it is a worst case measure an fails to examine the performance of algorithms on instances that woul be expecte in a particular application. It has also been observe that Competitive Analysis sometimes fails to istinguish between algorithms which have very ifferent performance in practice an intuitively iffer in quality. Over the years, researchers have evise alternatives to Competitive Analysis, each esigne to aress one or all of its shortcomings. There are exceptions, but it is fair to say that many alternatives are applicationspecific, an very often, these papers only present a irect comparison between a new measure an Competitive Analysis. This paper is a stuy of several generallyapplicable alternative measures for evaluating online algorithms that have been suggeste in the literature. We perform this comparison in the context of a particular problem: the 2server problem on the line with three possible request points, nickname here the baby server problem. Investigating simple kservers problems to she light on new ieas has also been one in [2], for instance. We concentrate on two algorithms (Greey an Lazy Double Coverage (Lc) [8]) an four ifferent analysis techniques (measures): Bijective Analysis, the Max/Max Ratio, Ranom Orer Ratio an Relative Worst Orer Analysis. In investigating the baby server problem, we fin that accoring to some quality measures for online algorithms, Greey is better than Lc, whereas for others, Lc is better than Greey. The ones that conclue that Lc is best are focuse on a worstcase sequence for the ratio of an algorithm s cost compare to Opt. In the case of Greey an Lc, this conclusion makes use of the fact that there exists a family of sequences for which Greey s cost is unbounely larger than the cost of Opt, whereas Lc s cost is always at most a factor two larger than the cost of Opt. On the other han, the measures that conclue that Greey is best compare two algorithms base on the multiset of costs stemming from the set of all sequences of a fixe length. In the case of Greey an Lc, this makes use of the fact that for any fixe n, both the maximum as well as the average cost of Lc over all sequences of length n are greater than the corresponing values for Greey. Using Relative Worst Orer Analysis a more nuance result is obtaine, concluing that Lc can be a factor at most two worse than Greey, while Greey can be unbounely worse than Lc. All omitte proofs may be foun in the full version of the paper [7].
3 Performance Measures for Online Algorithms 3 2 Preliminaries 2.1 The Server Problem Server problems [4] have been the objects of many stuies. In its full generality, one assumes that some number k of servers are available in some metric space. Then a sequence of requests must be treate. A request is simply a point in the metric space, an a kserver algorithm must move servers in response to the request to ensure that at least one server is place on the request point. A cost is associate with any move of a server (this is usually the istance move in the given metric space), an the objective is to minimize total cost. The initial configuration (location of servers) may or may not be a part of the problem formulation. In investigating the strengths an weaknesses of the various measures for the quality of online algorithms, we efine the simplest possible nontrivial server problem: Definition 1. The baby server problem is a 2server problem on the line with three possible request points A, B, an C, in that orer from left to right, with istance one between A an B an istance > 1 between B an C. The cost of moving a server is efine to be the istance it is move. We assume that initially the two servers are place on A an C. All results in the paper pertain to this problem. Even though the problem is simple, it contains a core kserver problem of balancing greeiness an aaptability, an this simple setup is sufficient to show the noncompetitiveness of Greey with respect to Competitive Analysis [4]. 2.2 Server Algorithms First, we efine some relevant properties of server algorithms: Definition 2. A server algorithm is calle noncrossing if servers never change their relative position on the line. lazy [15] if it never moves more than one server in response to a request an it oes not move any servers if the requeste point is alreay occupie by a server. A server algorithm fulfilling both these properties is calle compliant. Given an algorithm, A, we efine the algorithm lazy A, LA, as follows: LA will maintain a virtual set of servers an their locations as well as the real set of servers in the metric space. There is a onetoone corresponence between real servers an virtual servers. The virtual set will simulate the behavior of A. The initial server positions of the virtual an real servers are the same. Whenever a virtual server reaches a request point, the corresponing real server is also move to that point (unless both virtual servers reach the point simultaneously,
4 4 J. Boyar, S. Irani, K.S. Larsen in which case only the physically closest is move there). Otherwise the real servers o not move. In [8], it was observe that for any 2server algorithm, there exists a noncrossing algorithm with the same cost on all sequences. In [15], it was observe that for an algorithm A an its lazy version LA, for any sequence I of requests, A(I) LA(I) (we refer to this as the laziness obervation). Note that the laziness observation applies to the general kserver problem in metric spaces, so the results which epen on it can also be generalize beyon the baby server problem. We efine a number of algorithms by efining their behavior on the next request point, p. For all algorithms, no moves are mae if a server alreay occupies the request point (though internal state changes are sometimes mae in such a situation). Greey moves the closest server to p. Note that ue to the problem formulation, ties cannot occur (an the server on C is never move). If p is in between the two servers, Double Coverage (Dc), moves both servers at the same spee in the irection of p until at least one server reaches the point. If p is on the same sie of both servers, the nearest server moves to p. We efine adc to work in the same way as Dc, except that the rightmost server moves at a spee a times faster than the leftmost server. We refer to the lazy version of Dc as Lc an the lazy version of adc as alc. The balance algorithm [15], Bal, makes its ecisions base on the total istance travelle by each server. For each server, s, let s enote the total istance travelle by s from the initiation of the algorithm up to the current point in time. On a request, Bal moves a server, aiming to obtain the smallest possible max s s value after the move. In case of a tie, Bal moves the server which must move the furthest. If p is in between the two servers, Dummy moves the server that is furthest away to the request point. If p is on the same sie of both servers, the nearest server moves to p. Again, ue to the problem formulation, ties cannot occur (an the server on A is never move). 2.3 Quality Measures In analyzing algorithms for the baby server problem, we consier input sequences I of request points. An algorithm A, which treats such a sequence has some cost, which is the total istance move by the two servers. This cost is enote by A(I). Since I is of finite length, it is clear that there exists an offline algorithm with minimal cost. By Opt, we refer to such an algorithm an Opt(I) enotes the unique minimal cost of processing I. All of the measures escribe below can lea to a conclusion as to which algorithm of two is better. In contrast to the others, Bijective Analysis oes not inicate how much better the one algorithm might be; it oes not prouce a ratio, as the others o.
5 Performance Measures for Online Algorithms 5 Competitive Analysis: In Competitive Analysis [11, 16, 12], we efine an algorithm A to be ccompetitive if there exists a constant α such that for all input sequences I, A(I) c Opt(I) + α. The Max/Max Ratio: The Max/Max Ratio [3] compares an algorithm s worst cost for any sequence of length n to Opt s worst cost for any sequence of length n. The Max/Max Ratio of an algorithm A, w M (A), is M(A)/M(Opt), where M(A) = limsup max A(I)/t. t I =t The Ranom Orer Ratio: Kenyon [13] efines the Ranom Orer Ratio to be the worst ratio obtaine over all sequences, comparing the expecte value of an algorithm, A, with respect to a uniform istribution of all permutations of a given sequence, to the value of Opt of the given sequence: E σ [A(σ(I))] lim sup Opt(I) Opt(I) The original context for this efinition is Bin Packing for which the optimal packing is the same, regarless of the orer in which the items are presente. Therefore, it oes not make sense to take an average over all permutations for Opt. For server problems, however, the orer of requests in the sequence may very well change the cost of Opt. We choose to generalize the Ranom Orer Ratio as shown to the left, but for the results presente here, the efinition to the right woul give the same: E σ [A(σ(I))] lim sup E Opt(I) σ [Opt(σ(I))] [ ] A(σ(I)) lim sup E σ Opt(σ(I)) Opt(I) Bijective Analysis an Average Analysis: In [1], Bijective an Average Analysis are efine, as methos of comparing two online algorithms irectly. We aapt those efinitions to the notation use here. As with the Max/Max Ratio an Relative Worst Orer Analysis, the two algorithms are not necessarily compare on the same sequence. In Bijective Analysis, the sequences of a given length are mappe, using a bijection onto the same set of sequences. The performance of the first algorithm on a sequence, I, is compare to the performance of the secon algorithm on the sequence I is mappe to. If I n enotes the set of all input sequences of length n, then an online algorithm A is no worse than an online algorithm B accoring to Bijective Analysis if there exists an integer n 0 1 such that for each n n 0, there is a bijection f : I n I n satisfying A(I) B(f(I)) for each I I n. Average Analysis can be viewe as a relaxation of Bijective Analysis. An online algorithm A is no worse than an online algorithm B accoring to Average Analysis if there exists an integer n 0 1 such that for each n n 0, Σ I In A(I) Σ I In B(I).
6 6 J. Boyar, S. Irani, K.S. Larsen Relative Worst Orer Analysis: Relative Worst Orer Analysis was introuce in [5] an extene in [6]. It compares two online algorithms irectly. As with the Max/Max Ratio, it compares two algorithms on their worst sequence in the same part of a partition. The partition is base on the Ranom Orer Ratio, so that the algorithms are compare on sequences having the same content, but possibly in ifferent orers. Definition 3. Let I be any input sequence, an let n be the length of I. If σ is a permutation on n elements, then σ(i) enotes I permute by σ. Let A be any algorithm. Then, A(I) is the cost of running A on I, an A W (I) = max A(σ(I)). σ Definition 4. For any pair of algorithms A an B, we efine c l (A, B) = sup {c b: I : A W (I) c B W (I) b} an c u (A, B) = inf {c b: I : A W (I) c B W (I) + b}. If c l (A, B) 1 or c u (A, B) 1, the algorithms are sai to be comparable an the Relative WorstOrer Ratio WR A,B of algorithm A to algorithm B is efine. Otherwise, WR A,B is unefine. If c l (A, B) 1, then WR A,B = c u (A, B), an if c u (A, B) 1, then WR A,B = c l (A, B). If WR A,B < 1, algorithms A an B are sai to be comparable in A s favor. Similarly, if WR A,B > 1, the algorithms are sai to be comparable in B s favor. Definition 5. Let c u be efine as in Definition 4. If at least one of the ratios c u (A, B) an c u (B, A) is finite, the algorithms A an B are (c u (A, B),c u (B, A)) relate. Definition 6. Let c u (A, B) be efine as in Definition 4. Algorithms A an B are weakly comparable in A s favor, 1) if A an B are comparable in A s favor, 2) if c u (A, B) is finite an c u (B, A) is infinite, or 3) if c u (A, B) o(c u (B, A)). 3 Competitive Analysis The kserver problem has been stuie using Competitive Analysis starting in [14]. In [8], it is shown that the competitive ratios of Dc an Lc are k, which is optimal, an that Greey is not competitive. 4 The Max/Max Ratio In [3], a concrete example is given with two servers an three noncolinear points. It is observe that the Max/Max Ratio favors the greey algorithm over the balance algorithm, Bal.
7 Performance Measures for Online Algorithms 7 Bal behaves similarly to Lc an ientically on Lc s worst case sequences. The following theorem shows that the same conclusion is reache when the three points are on the line. Theorem 1. Greey is better than Lc on the baby server problem with respect to the Max/Max Ratio. It follows from the proof of this theorem that Greey is close to optimal with respect to the Max/Max Ratio, since the cost of Greey ivie by the cost of Opt tens towar one for large. Since Lc an Dc perform ientically on their worst sequences of any given length, they also have the same Max/Max Ratio. 5 The Ranom Orer Ratio The Ranom Orer Ratio correctly istinguishes between Dc an Lc, inicating that the latter is the better algorithm. Theorem 2. Lc is better than Dc accoring to the Ranom Orer Ratio. Proof. For any sequence I, E σ [Dc(σ(I))] E σ [Lc(σ(I))], by the laziness observation. Let I = (ABC) n. Whenever the subsequence CABC occurs in σ(i), Dc moves a server from C towars B an back again, while moving the other server from A to B. In contrast, Lc lets the server on C stay there, an has cost 2 less than Dc. The expecte number of occurrences of CABC in σ(i) is cn for some constant c. The expecte costs of both Opt an Lc on σ(i) are boune above an below by some other constants times n. Thus, Lc s ranom orer ratio will be less than Dc s. Theorem 3. Lc is better than Greey on the baby server problem with regars to the Ranom Orer Ratio. Proof. The Ranom Orer Ratio is the worst ratio obtaine over all sequences, comparing the expecte value of an algorithm over all permutations of a given sequence to the expecte value of Opt over all permutations of the given sequence. Since the competitive ratio of Lc is two, on any given sequence, Lc s cost is boune by two times the cost of Opt on that sequence, plus an aitive constant. Thus, the Ranom Orer Ratio is also at most two. Consier all permutations of the sequence (BA) n 2. We consier positions from 1 through n in these sequences. Refer to a maximal consecutive subsequence consisting entirely of either As or Bs as a run. Given a sequence containing h As an t Bs, the expecte number of runs is 1 + 2ht h+t. (A problem in [10] gives that the expecte number of runs of As is h(t+1) h+t, so the expecte number of runs of Bs is t(h+1) h+t. Aing these gives the result.) Thus, with h = t = n 2, we get n expecte number of runs.
8 8 J. Boyar, S. Irani, K.S. Larsen The cost of Greey is equal to the number of runs if the first run is a run of Bs. Otherwise, the cost is one smaller. Thus, Greey s expecte cost on a permutation of s is n The cost of Opt for any permutation of s is, since it simply moves the server from C to B on the first request to B an has no other cost after that. Thus, the Ranom Orer Ratio is n+1 2, which, as n tens to infinity, is unboune. 6 Bijective Analysis Bijective analysis correctly istinguishes between Dc an Lc, inicating that the latter is the better algorithm. This follows from the following general theorem about lazy algorithms, an the fact that there are some sequences where one of Dc s servers repeately moves from C towars B, but moves back to C before ever reaching B, while Lc s server stays on C. Theorem 4. The lazy version of any algorithm for the baby server problem is at least as goo as the original algorithm accoring to Bijective Analysis. Theorem 5. Greey is at least as goo as any other lazy algorithm Lazy (incluing Lc) for the baby server problem accoring to Bijective Analysis. Proof. Since Greey has cost zero for the sequences consisting of only the point A or only the point C an cost one for the point B, it is easy to efine a bijection f for sequences of length one, such that Greey(I) Lazy(f(I)). Suppose that for all sequences of length k that we have a bijection, f, from Greey s sequences to Lazy s sequences, such that for each sequence I of length k, Greey(I) Lazy(f(I)). To exten this to length k +1, consier the three sequences forme from a sequence I of length k by aing one of the three requests A, B, or C to the en of I, an the three sequences forme from f(i) by aing each of these points to the en of f(i). At the en of sequence I, Greey has its two servers on ifferent points, so two of these new sequences have the same cost for Greey as on I an one has cost exactly 1 more. Similarly, Lazy has its two servers on ifferent points at the en of f(i), so two of these new sequences have the same cost for Lazy as on f(i) an one has cost either 1 or more. This immeiately efines a bijection f for sequences of length k + 1 where Greey(I) Lazy(f (I)) for all I of length k + 1. If an algorithm is better than another algorithm with regars to Bijective Analysis, then it is also better with regars to Average Analysis [1]. Corollary 1. Greey is the unique optimal algorithm with regars to Bijective an Average Analysis. Proof. Note that the proof of Theorem 5 shows that Greey is strictly better than any lazy algorithm which ever moves the server away from C, so it is better than any other lazy algorithm with regars to Bijective Analysis. By Theorem 4, it is better than any algorithm. By the observation above, it also hols for Average Analysis.
9 Performance Measures for Online Algorithms 9 Theorem 6. Dummy is the unique worst algorithm among compliant server algorithms for the baby server problem accoring to Bijective Analysis. Lemma 1. If a b, then there exists a bijection σ n : {A,B,C} n {A,B,C} n such that alc(i) blc(σ n (I)) for all sequences I {A,B,C} n. Theorem 7. Accoring to Bijective Analysis an Average Analysis, slower variants of Lc are better than faster variants for the baby server problem. Proof. Follows immeiately from Lemma 1 an the efinition of the measures. Thus, the closer a variant of Lc is to Greey, the better Bijective an Average Analysis preict that it is. 7 Relative Worst Orer Analysis Similarly to the ranom orer ratio an bijective analysis, relative worst orer analysis correctly istinguishes between Dc an Lc, inicating that the latter is the better algorithm. This follows from the following general theorem about lazy algorithms, an the fact that there are some sequences where one of Dc s servers repeately moves from C towars B, but moves back to C before ever reaching B, while Lc s server stays on C. If is just marginally larger than some integer, even on Lc s worst orering of this sequence, it oes better than Dc. Let I A enote a worst orering of the sequence I for the algorithm A. Theorem 8. The lazy version of any algorithm for the baby server problem is at least as goo as the original algorithm accoring to Relative Worst Orer Analysis. Theorem 9. Greey an Lc are (,2)relate an are thus weakly comparable in Lc s favor for the baby server problem accoring to Relative Worst Orer Analysis. Proof. First we show that c u (Greey,Lc) is unboune. Consier the sequence (BA) n 2. As n tens to infinity, Greey s cost is unboune, whereas Lc s cost is at most 3 for any permutation. Next we turn to c u (Lc,Greey). Since the competitive ratio of Lc is 2, for any sequence I an some constant b, Lc(I Lc ) 2Greey(I Lc ) + b 2Greey(I Greey ) + b. Thus, c u (Lc,Greey) 2. For the lower boun of 2, consier a family of sequences I p = (BABA...BC) p, where the length of the alternating A/Bsequence before the C is Lc(I p ) = p(2 + 2). A worst orering for Greey alternates As an Bs. Since there is no cost for the Cs an the A/B sequences start an en with Bs, Greey(σ(I p )) p(2 ) + 1 for any permutation σ. p(4) p(2)+1. As p goes to infinity, this Then, c u (Lc,Greey) p(2 +2) p(2 )+1 approaches 2. Thus, Greey an Lc are weakly comparable in Lc s favor.
10 10 J. Boyar, S. Irani, K.S. Larsen Recalling the efinition of alc, a request for B is serve by the rightmost server if it is within a virtual istance of no more than a from B. Thus, when the leftmost server moves an its virtual move is over a istance of l, then the rightmost server virtually moves a istance al. When the rightmost server moves an its virtual move is over a istance of al, then the leftmost server virtually moves a istance of l. In the results that follow, we frequently look at the worst orering of an arbitrary sequence. Definition 7. The canonical worst orering of a sequence, I, for an algorithm A is the sequence prouce by allowing the cruel aversary (the one which always lets the next request be the unique point where A oes not currently have a server) to choose requests from the multiset efine from I. This process continues until there are no requests remaining in the multiset for the point where A oes not have a server. The remaining points from the multiset are concatenate to the en of this new request sequence in any orer. The canonical worst orering of a sequence for alc is as follows: Proposition 1. Consier an arbitrary sequence I containing n A As, n B Bs, an n C Cs. A canonical worst orering of I for alc is I a = (BABA...BC) pa X, where the length of the alternating A/Bsequence before the C is Here, X is a possibly empty sequence. The first part of X is an alternating sequence of As an Bs, starting with a B, until there are not both As an Bs left. Then we continue with all remaining As or Bs, followe by all remaining Cs. Finally, { } n A n p a = min B,,n C. + 1 Theorem 10. If a b, then alc an blc are ( +, ( b +) b + ( +) b )relate for the baby server problem accoring to Relative Worst Orer Analysis. We provie strong inication that Lc is better than blc for b 1. If b > 1, this is always the case, whereas if b < 1, it hols in many cases, incluing all integer values of. Theorem 11. Consier the baby server problem evaluate accoring to Relative Worst Orer Analysis. For b > 1, if Lc an blc behave ifferently, then they are (r,r b )relate, where 1 < r < r b. If a < 1, alc an Lc behave ifferently, an is a positive integer, then they are (r a,r)relate, where 1 < r a < r. The algorithms alc an 1 alc are in some sense of equal quality: Corollary 2. When a an b are integers, then alc an blc are (b,b) relate when b = 1 a. We now set out to prove that Lc is an optimal algorithm in the following sense: there is no other algorithm A such that Lc an A are comparable an A is strictly better or such that Lc an A are weakly comparable in A s favor. a
11 Performance Measures for Online Algorithms 11 Theorem 12. Lc is optimal for the baby server problem accoring to Relative Worst Orer Analysis. Similar proofs show that alc an Bal are also optimal algorithms. In the efinitions of Lc an Bal given in Sect. 2, ifferent ecisions are mae as to which server to use in cases of ties. In Lc the server which is physically closer is move in the case of a tie (equal virtual istances from the point requeste). The rationale behin this is that the server which woul have the least cost is move. In Bal the server which is further away is move to the point. The rationale behin this is that, since > 1, when there is a tie, the total cost for the closer server is alreay significantly higher than the total cost for the other, so moving the server which is further away evens out how much total cost they have, at least temporarily. With these tiebreaking ecisions, the two algorithms behave very similarly when is an integer. Theorem 13. Lc an Bal are not comparable on the baby server problem with respect to Relative Worst Orer Analysis, except when is an integer, in which case they are equivalent. 8 Concluing Remarks The purpose of quality measures is to give information for use in practice, to choose the best algorithm for a particular application. What properties shoul such quality measures have? First, it may be esirable that if one algorithm oes at least as well as another on every sequence, then the measure ecies in favor of the better algorithm. This is especially esirable if the better algorithm oes significantly better on important sequences. Bijective Analysis, Relative Worst Orer Analysis, an the Ranom Orer Ratio have this property, but Competitive Analysis an the Max/Max Ratio o not. This was seen in the lazy vs. nonlazy version of Double Coverage for the baby server problem (an the more general metric k server problem). Similar results have been presente previously for the paging problem LRU vs. FWF an lookahea vs. no lookahea. See [6] for these results uner Relative Worst Orer Analysis an [1] for Bijective Analysis. Seconly, it may be esirable that, if one algorithm oes unbounely worse than another on some important families of sequences, the quality measure reflects this. For the baby server problem, Greey is unbounely worse than Lc on all families of sequences which consist mainly of alternating requests to the closest two points. This is reflecte in Competitive Analysis, the Ranom Orer Ratio, an Relative Worst Orer Analysis, but not by the Max/Max Ratio or Bijective Analysis. Similarly, accoring to Bijective Analysis, LIFO an LRU are equivalent for paging, but LRU is often significantly better than LIFO, which keeps the first k 1 pages it sees in cache forever. In both of these cases, Relative Worst Orer Analysis says that the algorithms are weakly comparable in favor of the better algorithm.
12 12 J. Boyar, S. Irani, K.S. Larsen Another esirable property woul be ease of computation for many ifferent problems, as with Competitive Analysis an Relative Worst Orer Analysis. It is not clear that the other measures have this property. References 1. Spyros Angelopoulos, Reza Dorrigiv, an Alejanro LópezOrtiz. On the separation an equivalence of paging strategies. In 18th ACMSIAM Symposium on Discrete Algorithms, pages , Wolfgang W. Bein, Kazuo Iwama, an Jun Kawahara. Ranomize competitive analysis for twoserver problems. In 16th Annual European Symposium on Algorithms, volume 5193 of Lecture Notes in Computer Science, pages Springer, Shai BenDavi an Allan Boroin. A new measure for the stuy of online algorithms. Algorithmica, 11(1):73 91, Allan Boroin an Ran ElYaniv. Online Computation an Competitive Analysis. Cambrige University Press, Joan Boyar an Lene M. Favrholt. The relative worst orer ratio for online algorithms. ACM Transactions on Algorithms, 3(2), Article No Joan Boyar, Lene M. Favrholt, an Kim S. Larsen. The relative worst orer ratio applie to paging. Journal of Computer an System Sciences, 73(5): , Joan Boyar, Sany Irani, an Kim S. Larsen. A comparison of performance measures for online algorithms. Technical report, arxiv: v1, Marek Chrobak, Howar J. Karloff, T. H. Payne, an Sunar Vishwanathan. New results on server problems. SIAM Journal on Discrete Mathematics, 4(2): , Reza Dorrigiv an Alejanro LópezOrtiz. A survey of performance measures for online algorithms. SIGACT News, 36(3):67 81, William Feller. An Introuction to Probability Theory an Its Applications, volume 1. John Wiley & Sons, Inc., New York, 3r eition, Problem 28, Chapter 9, page R. L. Graham. Bouns for certain multiprocessing anomalies. Bell Systems Technical Journal, 45: , Anna R. Karlin, Mark S. Manasse, Larry Ruolph, an Daniel D. Sleator. Competitive snoopy caching. Algorithmica, 3:79 119, Claire Kenyon. Bestfit binpacking with ranom orer. In 7th Annual ACMSIAM Symposium on Discrete Algorithms, pages , Mark S. Manasse, Lyle A. McGeoch, an Daniel Dominic Sleator. Competitive algorithms for online problems. In 20th Annual ACM Symposium on the Theory of Computing, pages , Mark S. Manasse, Lyle A. McGeoch, an Daniel Dominic Sleator. Competitive algorithms for server problems. Journal of Algorithms, 11(2): , Daniel D. Sleator an Robert E. Tarjan. Amortize efficiency of list upate an paging rules. Communications of the ACM, 28(2): , 1985.
13 Performance Measures for Online Algorithms 13 A Appenix: omitte proofs Proof of Theorem 1. Given a sequence of length n, Greey s maximum cost is n, so M(Greey) = 1. Opt s cost is at most n (since it is at least as goo as Greey). Opt s maximal cost over all sequences of length n = k(2( + 1) + 1), for some k, is 2( +1) at least 2( +1)+1 n. since its cost on ((AB) +1 C) k is this much. Thus, we can boun M(Opt) by ( +1) 2( +1)+1 M(Opt) 1. We now prove a lower boun on M(Lc). Consier the sequence s = (BABA...BC) p X, where the length of the alternating A/Bsequence before the C is 2 +1, X is a possibly empty alternating sequence of As an Bs starting with a B, X = n mo (2 +2), an p = n X If X 2 + 1, then Lc(s) = p(2 + 2) + X = n + ( 1)(n X ) +1. Otherwise, Lc(s) = p(2 + 2) + X = n + ( 1)(n X ) Since we are taking the supremum, we restrict our attention to sequences where X = 0. Thus, M(Lc) Finally, while > 1. Since M(Opt) is boune, n+ ( 1)n +1 n w M (Greey) = M(Greey) M(Opt) = = M(Opt), w M (Lc) = M(Lc) M(Opt) 2 ( + 1)M(Opt). w M(Lc) 2 w M(Greey) +1, which is greater than 1 for Proof of Theorem 4. By the laziness observation, the ientity function, i, is a bijection such that LA(I) A(i(I)) for all sequences I. Proof of Theorem 8. By the laziness observation, for any request sequence I, LA(I LA ) A(I LA ) A(I A ). Proof of Theorem 6. Similar to the proof of Theorem 5, but now with cost for every actual move. Proof of Lemma 1. We use the bijection from the proof of Theorem 5, showing that Greey is the unique best algorithm, but specify the bijection completely, as oppose to allowing some freeom in eciing the mapping in the cases where we are extening by a request where the algorithms alreay have a server. Suppose that σ n is alreay efine. Consier a sequence I n of length n an the three possible ways, I n A, I n B an I n C, of extening it to length n + 1. Suppose that alc has servers on points X a,y a {A,B,C} after hanling the sequence I n, an blc has servers on points X b,y b {A,B,C} after hanling σ n (I n ). Let Z a be the point where alc oes not have a server an Z b the point
14 14 J. Boyar, S. Irani, K.S. Larsen where blc oes not. Then σ n+1 (I n Z a ) is efine to be σ n (I n )Z b. In aition, since the algorithms are lazy, both algorithms have their servers on two ifferent points of the three possible, so there must be at least one point P where both algorithms have a server. Thus, P {X a,y a } {X b,y b }. Let U a be the point in {X a,y a } \ {P } an U b be the point in {X b,y b } \ {P }. Then, σ n+1 (I n P) is efine to be σ n (I n )P an σ n+1 (I n U a ) to be σ n (I n )U b. Consier running alc on a sequence I n an blc on σ n (I n ) simultaneously. The sequences are clearly constructe so that, at any point uring this simultaneous execution, both algorithms have servers moving or neither oes. Clearly, the result follows if we can show that blc moves away from an back to C at least as often as alc oes. By construction, the two sequences, I n an σ n (I n ), will be ientical up to the point where blc (an possibly alc) moves away from C for the first time. In the remaining part of the proof, we argue that if alc moves away from an back to C, then blc will also o so before alc can o it again. Thus, the total cost of blc will be at least that of alc. Consier a request causing the slower algorithm, alc, to move a server away from C. If blc also moves a server away from C at this point, both algorithms have their servers on A an B, an the two sequences continue ientically until the faster algorithm again moves a server away from C (before or at the same time as the slower algorithm oes). If blc oes not move a server away from C at this point, since, by construction, it oes make a move, it moves a server from A to B. Thus, the next time both algorithms move a server, alc moves from B to C an blc moves from B to A. Then both algorithms have servers on A an C. Since alc has just move a server to C, whereas blc must have mae at least one move from A to B since it place a server at C, blc must, as the faster algorithm, make its next move away from C strictly before alc oes so. In conclusion, the sequences will be ientical until the faster algorithm, blc, moves a server away from C. Lemma 2. Let I a be the canonical worst orering of I for alc. I a is a worst permutation of I for alc, an the cost for alc on I a is c a, where p a (2 + 2) c a p a (2 + 2) Proof. Consier a request sequence, I. Between any two moves from B to C, there must have been a move from C to B. Consier one such move. Between the last request to C an this move, the other server must move from A to B + 1 times, which requires some first request to B in this subsequence, followe by at least occurrences of requests to A, each followe by a request to B. (Clearly, extra requests to A or B coul also occur, either causing moves or not.) Thus, for every move from B to C, there must be at least + 1 Bs, As an one C. Thus, the number of moves from B to C is boune from above by p a. There can be at most one more move from C to B than from B to C. If such a move occurs, there are no more Cs after that in the
15 Performance Measures for Online Algorithms 15 sequence. Therefore, the sequences efine above give the maximal number of moves of istance possible. More As or Bs in any alternating A/Bsequence woul not cause aitional moves (of either istance 1 or ), since each extra point requeste woul alreay have a server. Fewer As or Bs between two Cs woul eliminate the move away from C before it was requeste again. Thus, the canonical worst orering is a worst orering of I. Within each of the p a repetitions of (BABA...BC), each of the requests for A an all but the last request for B cause a move of istance 1, an the last two requests each cause a move of istance, giving the lower boun on c a. Within X, each of the first 2 requests coul possibly cause a move of istance 1, an this coul be followe by a move of istance. After that, no more moves occur. Thus, aing costs to the lower boun gives the upper boun on c a. Proof of Theorem 10. By Lemma 2, in consiering alc s performance in comparison with blc s, the asymptotic ratio epens only on the values p a an p b efine for the canonical worst orerings I a an I b for alcan blc, respectively. Since a > b, the largest value of pa p b occurs when p a = n C, since more Cs woul allow more moves of istance by blc. Since the contribution of X to alc s cost can be consiere to be a constant, we may assume that n A = n C an nb = n C ( + 1). When consiering blc s canonical worst orering of this sequence, all the Cs will be use in the initial part. By Lemma 2, we obtain the following ratio, for some constant c: (2 + 2)nC (2 + 2)nC + c Similarly, a sequence giving the largest value of p b p a will have p b = na b, since more As woul allow alc to have a larger p a. Since the contribution of X to blc can be consiere to be a constant, we may assume that n A = n C, n B = n C ( + 1), an pb = n C. Now, when consiering alc s worst permutation of this sequence, the number of perios, p a, is restricte by the number of As. Since each perio has a n As, p = A n C = b. After this, there are a constant number of As remaining, giving rise to a constant aitional cost c. Thus, the ratio is the following: (2 a (2 + 2)nC + 2) n b C + c Consiering the above ratios asymptotically as n C goes to infinity, we obtain that alc an alc are ( +, ( b +) )relate. b + ( +) b
16 16 J. Boyar, S. Irani, K.S. Larsen Proof of Theorem 11. By Theorem 10, alc an blc are ( + b +, ( b +) ( +) b )relate Subtracting these two values an comparing to zero, we get + b + ( b +) ( +) b < 0 E = ( + )( + ) ( + )( + ) < 0 We can rewrite E = ( )( 2 ). Observe that the first term is positive when b > a an the algorithms behave ifferently. We now analyze Lc compare with variants of other spees. First, assume that a = 1 an b > 1. Since the algorithms are ifferent, we have that 1. Thus, the secon factor, 2, is boune from above by ( 1) 2 < 0. Hence, if Lc an blc are (c 1,c 2 )relate, then c 1 < c 2. Now, assume that a < 1 an b = 1. Again, the algorithms are ifferent, so 1. Note that if > 2, then alc an Lc are (c 1,c 2 )relate, where c 2 > c 1. This hols when is a positive integer. Proof of Theorem 12. Since the competitive ratio of Lc is 2, for any algorithm A an any sequence I, there is a constant b such that Lc(I Lc ) 2A(I Lc )+b 2A(I A ) + b. Thus, c u (Lc, A) 2, so A is not weakly comparable to Lc in A s favor. For A to be a better algorithm than Lc accoring to Relative Worst Orer Analysis, it has to be comparable to Lc an perform more than an aitive constant better on some sequences (which then necessarily must be an infinite set). Assume that there exists a family of sequences S 1,S 2,... such that for any constant c there exists an i such that Lc W (S i ) A W (S i ) + c. Then we prove that there exists another family of sequences S 1,S 2,... such that for any constant c there exists an i such that A W (S i ) Lc W(S i ) + c. This establishes that if A performs more than a constant better on its worst permutations of some family of sequences than Lc oes on its worst permutations, then there exists a family where Lc has a similar avantage over A which implies that the algorithms are not comparable. Now assume that we are given a constant c. Since we must fin a value greater than any c to establish the result, we may assume without loss of generality that c is large enough that 3c ( + 2) + 3. Consier a sequence S from the family S 1,S 2,... such that Lc W (S) A W (S) + 3c. From S we create a member S of the family S 1,S 2,... such that A W (S ) Lc W (S ) + c.
17 Performance Measures for Online Algorithms 17 Assume that S contains n A As, n B Bs, an n C Cs. The iea behin the construction is to allow the cruel aversary to choose points from the multiset efine from S as in the efinition of canonical worst orerings. This process continues until the cruel aversary has use all of either the As, Bs, or Cs in the multiset, resulting in a sequence S. If the remaining points from the multiset are concatenate to S in any orer, this creates a permutation of S. The performance of A on this permutation must be at least as goo as its performance on its worst orering of S. We now consier the performance of Lc an A on S an show that Lc is strictly better. Let n A, n B, an n C enote the number of As, Bs, an Cs in S, respectively. { n Let p = min A, n B +1,n C }. By Lemma 2, the cost of Lc on its canonical worst orering of S is at most p(2 + 2) + 3. The cost of A is 2n C + n A + n B n C, since every time there is a request for C, this is because a server in the step before move away from C. These two moves combine have a cost of 2. Every request to an A or a B has cost one, except for the request to B immeiately followe by a request to C, which has alreay been counte in the 2n C term. A similar argument shows that Lc s cost is boune from above by the same term. Assume first that n A = n B +1 = n C. Then S can be permute so that it is a prefix of Lc s canonical worst orering on S (see Lemma 2 with a = 1). Since, by construction, we have run out of either As, Bs, or Cs (that is, one type is missing from S minus S as multisets), Lc s cost on its worst orering of S is at most its cost on its worst orering on S plus 3. Thus, Lc W (S i ) A W (S i )+c oes not hol in this case, so we may assume that these values are not all equal. Case n A > p: Lc s cost on S is at most the cost of A minus (n A p) plus 3. Case n B > ( + 1)p: Lc s cost on S is at most the cost of A minus (n B ( + 1)p) plus 3. Case n C > p: Lc s cost on S is at most the cost of A minus (2 1)(n C p) plus 1. We compare Lc s canonical worst orerings of S an S. In both cases, the form is as in Lemma 2, with a = 1. Thus, for S the form is (BA) BC) p X, an for S, it is ((BA) BC) p+l Y for some positive integer l. The string X must contain all of the As, all of the Bs an/or all of the Cs containe in ((BA) BC) l, since after this the cruel aversary has run out of something. Thus, it must contain at least l As, l( + 1) Bs or l Cs. The extra cost that Lc has over A on S is at most its cost on ((BA) BC) l Y minus cost l for the As, Bs or Cs containe in X, so at most l(2 + 2) l + 3 = l( + 2) + 3. Thus, Lc W (S) A W (S) l( + 2) + 3 while A W (S ) Lc W (S ) l 3.
18 18 J. Boyar, S. Irani, K.S. Larsen From the choice of c an the boun just erive, we fin that ( + 2) + 3 3c Lc W(S) A W (S) l( + 2) + 3 Thus, l 3+1 1, which implies the following: Now, l l( 1) l 3 l + 1 l 3 l( +2)+3 3 A W (S ) Lc W (S ) l 3 l( + 2) c 3 = c. The proof of optimality only epens on the form of the canonical worst orerings for the optimal algorithm an the costs associate with these. More specifically, the canonical worst orerings prouce using the cruel aversary are worst orers, an these sequences have the form R Q 1 Q 2...Q k X, where R has constant cost Q i has the form (BA) ri BC with l r i l + 1 for some integer l, an X cannot inclue enough As, Bs an Cs so that the sequence coul instea have been written R Q 1 Q 2...Q k Q k+1 X for some Q k+1 having the same form as the others an for some X. Since the cruel aversary is use, the patterns Q i are uniquely efine by the algorithm an every request in each of them has a cost. In the above proof, all the Q i were ientical. As an example of where they are not all ientical, consier = 2 1 2, where the canonical worst orering for Bal has the following prefix: BABC(BABABABCBABABC) Note that this implies that alc an Bal are also optimal algorithms. Proof of Theorem 13. One can see that a canonical worst orering of a sequence I for Bal has the following prefix: R Q 1 Q 2...Q k, where for all i, Q i = (BA) ri BC, where r i an R contains at most two Cs. Suppose that is an integer. Consier any request sequence I. Lc s canonical worst orering has the prefix ((BA) BC) k, while Bal s canonical worst orering has the prefix BABC(BA) BC((BA) BC) k. These prefixes of Lc s an Bal s canonical worst orerings of I are ientical, except that R is empty for Lc, while Bal has R = BABC(BA) BC.
19 Performance Measures for Online Algorithms 19 This leas to some small ifference at the en, where Bal might have at most two fewer of the (BA) BC repetitions than Lc. Thus, their performance on their respective worst orerings will be ientical up to an aitive constant ue to the ifferences at the start an en of the prefixes. Now consier the case where is not an integer. Following the proof of Theorem 12, consier a sequence, I, where the canonical worst orering for Bal consists entirely of the prefix form escribe above. Since is not an integer, not all of the Q i s will have the same length. The ifference from Lc s canonical worst orering is that there are some Q i of the form (BA) BC, so in their respective prefixes, Bal has more As an Bs, compare to the number of Cs, than Lc oes. This means that Lc will have a lower cost on its worst orering. However, the sequence, I, prouce by letting the cruel aversary for Lc choose points from the multiset efine by this sequence, I, has all Q i = (BA) BC. Bal s canonical worst orering of I will have fewer Q i s because there will not be enough As to make the longer repetitions require. Therefore, the two algorithms are incomparable when is not an integer.
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