Saddle Points in Random Matrices: Analysis of Knuth Search Algorithms

Transcription

1 Saddle Points in Rando Matrices: Analysis of Knuth Search Algoriths Micha Hofri Philippe Jacquet Dept. of Coputer Science INRIA, Doaine de Voluceau - Rice University Rocquencourt - B.P. 05 Houston TX Le Chesnay Cedex, France e-ail: hofri@cs.rice.edu e-ail: Philippe.Jacquet@inria.fr Abstract We present an analysis of algoriths for finding saddle points in a rando atrix, presented by Donald E. Knuth as exercise in The Art of Coputer Prograing. We estiate the average coputing costs of three saddle point search algoriths. Ausingly, the asyptotic results in this analysis about atrix saddle points uses the sae approach that leads to the celebrated saddle point ethod in coplex analysis.. Introduction A atrix saddle point is defined in [3] as an eleent of the atrix which is the sallest in its row, and the largest in its colun. Exercise there shows two algoriths to find such saddle points, and suggests a third. In this paper we respond to exercise there, and analyze the average cost of these three saddle point search algoriths. Soe of the results are only given explicit for asyptotically, as the nubers of both rows and coluns tend to infinity. Note: while we selected here a direct approach to derive the asyptotics, by a curious coincidence they also follow fro recent depoissonization results of the second author and Wociech Szpankowski [2], results which were obtained via the saddle point ethod of coplex analysis. This is shown in an Appendix. The calculations below concern the rando variables one needs in order to copute expected processing ties. In particular, unlike the custo in [3], we do not count the instructions and their execution ties in a MIX progra, to deterine the ultipliers of these variables. We consider atrices with distinct values. The scenario where values ay be repeated introduces additional difficulties, which are not addressed here, except in the last section. We do not quite follow the way the proble is presented a atrix with distinct eleents with all its perutations equally

2 likely since one can avoid looking at all possible rearrangeents of a fixed set of eleents by using the following result: Theore : Let an algorith operate on an n nuerical atrix in such a way that only orderrelations between the ters affect its operation (never the actual values). Then the sae probabilistic behavior is observed under the following two scenarios: () All the atrix eleents are distinct, and all (n)! possible rearrangeents of the eleents are equally likely. (2) The atrix eleents are drawn independently using the sae continuous, ato-less distribution. The proof is direct: since a atrix can be associated with a sequence in a consistent way (e.g. by catenating successive rows), and each sequence constructed by ethod (2) can be naturally related to a perutation fro the syetric group S n (using the relative order of its eleents), we only need to show inductively on the size of the sequence that each such eber is obtained in probability =(n)! an iediate calculation. 2. Notation and Assuptions We denote the nuber of rows and coluns by and n respectively. Rows are nubered through etc. The values of the iniu eleent of row i and the axiu eleent of colun are denoted by R i and C, respectively. In our odel they are rando variables. The operations of finding row-inia or colun-axia in each of the algoriths are independent of each other, and each takes a tie as discussed in [3, x.2.0], over n or entries. These durations are denoted by T n and T respectively. It is known that E[T n ] = α + βn + νlnn, where the Greek letters represent instruction counts and ties that depend on the ipleentation. We use the abbreviations pdf for probability distribution function and df for such a density function. Let the pdf fro which the eleents are drawn be F() with the corresponding density f (). Since any continuous ato-less pdf would do, we could pick a particular one, even the ever-so-convenient U(0; ). Keeping the sybol F akes soe of the calculations ore transparent. An arbitrary atrix ter is denoted by X. 2. The inax and axin inequality The following theore is shown in [3] by eleentary considerations: Theore 2: For an arbitrary nuerical atrix let inax be the iniu value of all coluns axia, and siilarly, let axin be the axiu value of all row inia. Then 2

3 axin inax; The atrix has a saddle point if and only if inax = axin, and then this is the value of the saddle point. A atrix with distinct ters can have one saddle point at ost. 2.2 Saddle point probability The likelihood that a saddle point occurs in a distinct-values atrix is coputed in [3], and is shown there to be quite sall: P n Pr[An n atrix has a saddle point] = + n : () This quantity is n ties the probability of occurrence of a saddle point at any given position in the atrix. The last observation allows us to rederive the result in [3] fro the equivalent odel, by theore. Let x be the value of a specific eleent. The probability that the other n? ters of its row are greater than x is (? F(x)) n?. Siilarly, the probability that the other? ters in its colun are saller than x is F(x)?. Therefore the probability of occurrence of saddle point in the specified R position is? F(x)? (? F(x)) n? f (x)dx = n(n+)(n+?) (?)(?2). Multiplying by n reproduces the above result.? +n n 3. Knuth s saddle point search algoriths These algoriths coe with no clai of being optial or even particularly good. They are used as tutorial prograing exaples in The Art of Coputer Prograing. Correspondingly, the following should be considered an exercise in analysis of algoriths, in the sense that although the analyses point to various possible iproveents in the algoriths, we rarely follow these issues. Note: We discuss soe of these aspects briefly in the last section. A ore sophisticated approach to searching for saddle points is described in []. 3. Algorith I: row-oriented At the level of detail needed here, this algorith does the following:. For rows i = through, 2. Locate inia R i, (only one is found in the current scenario) 3

4 3. Copare R i against eleents in its colun: 4. If a larger eleent is found, continue to next row, 5. Else return the iniu and its location: a saddle point has been found. Exit. 3.2 Algorith II: two-phase algorith Phase one: scan coluns.. For coluns = through n, 2. Locate axia C, (only one is found in the current scenario) 3. Copare C against the sallest found so far, and keep the record in µ. At the end of this phase the variable µ holds the value of inax. Phase two: scan rows. 4. For row i = through, 5. Search the row for an eleent saller than µ. 6. If the search succeeds, continue to the next row. 7. If the search fails, µ is the value of the saddle point; by virtue of theore 2 µ itself is found in that row. Exit. 3.3 Algorith III: algorith II + theore 2 The third algorith has two phases like algorith II, and the first phase is nearly identical with the first one there: the only difference is that in addition to recording the sallest colun axiu found so far in µ, it also records the row in which each new record is found, denoted by ρ. By the coent at the end of the previous subsection, it suffices, during phase 2, to check that µ is indeed the sallest ter of row ρ. 4. Average case analysis of the search algoriths 4. Analysis of the first algorith Locating each row iniu R i is independent of all other operations and requires the duration T n. Now we exaine the nuber of coparisons of R i with eleents in its colun, denoted by D i. Consider row. Since R is a iniu of n iid F-rando variables, its pdf is? (? F (x)) n. Since the first coparison is of R with itself, D 2. We copute the distribution of D. This is easy to do exactly. We shall then ake an approxiation, and clai that up to a detail we treat explicitly, the sae distribution applies for all the other R i. The detail is that while each R i is copared with 4

5 the ters in its colun, if a larger value is not found in the first i? trials, the next coparison is duy when R i is copared with itself since it never terinates the search. Concerning the need for approxiation: If in any of the first i? rows the local row-iniu was found in the sae colun as R i, we know that it ight have been copared with R i before. This introduces a dependence between the coparisons. We rid ourselves of this coplicated dependence by a siple device: we odify the algorith. We assue that it does not terinate when a saddlepoint is found, as line 5 in x2. now says; instead, the algorith always exaines all row-inia. This accoplishes the goal, since now the fact that we reached row i provides no inforation about the outcoes of the coparisons done in the previous searches, aking the above clai that the D i have (essentially the sae) independent distributions precise. What is the effect of the odification? It only changes the operation of the algorith when a saddle point is found. This event is so rare, that the relative error it introduces, on the order of the probability given in (), is sufficiently sall that it can be safely disregarded. We illustrate this clai by an exaple below. No calculations are needed to see by syetry that u Pr[D = 2] = Pr[R < X] = n=(n + ): If R survives the coparison, in probability v? u = =(n + ), it is the second order-statistic aong n +, and has the density f (2) n+ (x) = n(n + )F(x) f (x) (? F(x))n?. We proceed in this way to find successive f n+ ( +) (x), the conditional df of R following + coparisons (including the duy one), given by f n+ ( +) n + (x) = f (x)f (x) (? F(x)) n? ; (2) ;;n? and v +, the probability that a variable with this distribution is larger than X, is given by With the natural definition u +? v + we have v + = + n + + ; (v 0 ): (3) p r? Pr[D = r] = v v 2 :::v r?2 u n n + r?? r? = ; 2 r : (4) r? n And our assuption is that for row i, 2 i the following holds: 8 >< p r r < i Pr[D i = r] = 0 r = i : (5) >: i < r p r? These probabilities do not su to one: if D i > we declare that R i is a saddle point We use this approach to copute once again the probability of such a atrix having a saddle point. It provides an exaple for our clai about the error our odification of the algorith introduces. We 5

6 copute the copleentary probability, of the event fall D i g, none overflows. For any row i we (?)! have Pr[D i ] =? v :::v? =?(n+?)!=n!, hence (? )!n! Pr[no saddle point]? =? P n ( + O(P n)); (6) (n +? )! where P n is the sall probability given in (). One can define a rando variable, R, the nuber of rows the unodified algorith uses, but again, unless the atrix is very sall, it equals with a probability so close to that it is not very useful. The expected values of the D i are directly available fro equations (4) and (5): i? E[D i ] = r + rp r? = r=rp r>i n n? +? n+i? i? : (7) Over the execution of the algorith i varies fro through, and the expected overall (odified) nuber of coparisons is then E[D] = i= E[D i ] = n( + ) n?? n n? Except for very sall and n, the second ter above is negligible.? +n? : (8) It is interesting to see if this analysis can suggest whether this algorith, when used on nonsquare atrices, should follow the above outline, or rotate the atrix and exchange the roles of rows and coluns, assuing no other considerations (such as virtual storage anageent) intervene. Surely the algorith logic is invariant under such a rotation (this holds, utatis utandis, for all three algoriths). The total ties in both cases can be written as T rows = T n + T (D rows ) and T coluns = nt + T (D coluns ). Adding ore detail for the expectations, with appropriate ultipliers (α 0 represents all the fixed overhead needed at each row/colun, but we disregard the setup costs that would be the sae in each orientation), copared with E[T rows ] = (α 0 + βn + νlnn) + δ E[T coluns ] = n(α 0 + β + νln) + δ n( + ) n?? n n? (n + )???? +n?? +n? n! ; (9)! : (0) The leading ter is the sae. The last ter is negligible, and otherwise the difference can be written as n? E[T rows ]? E[T coluns ] = (? n) α 0? δ + νln n (n? )(? ) n : () Typically α 0 > δ; also the ultiplier of δ is nearly, and that of ν is positive when n > ; since we also expect α 0 to exceed ν, it sees the algorith should proceed along the shorter diension (i.e. as above, if n > ) but actual estiates of these paraeters would be needed to ake a definite call. 6

7 4.2 Analysis of the second algorith The first phase requires tie of nt + T n and it finds the inax value, µ. The T n accounts for selecting µ fro the successively deterined C. Since this is not the sae as a direct search over n entries, the instruction counts in this T n would differ, but the sae variables appear. In the second phase the scan of any given row stops as soon as a ter strictly saller than µ is found. We now copute the distributions of lengths of these scans. The analysis requires the distributions of the relevant rando variables: the inax with value µ, the colun axia C with values (x ; : : :;x n ), which we condition on µ, and row eleents. The sae algorithic odification as before, aking the algorith search rows even when a saddle point is discovered, akes the distribution of the inspected row eleents independent of their row index, but once we condition the on (C ; : : :;C n ) = (x ; : : :;x n ), the distributions depend on their colun indices. The unconditional pdf of any C is Pr[C x] = F (x), and that of µ is Pr[µ x] =? (? F (x)) n. The distribution function of C conditioned on the value of inax being µ, and on its occurring in another colun, equals c (x µ) Pr[C x C > µ] = F (x)? F (µ)? F ; (x µ): (2) (µ) Finally, the conditional distribution of a atrix ter in position (i; ) is given by Pr[A i x C = x ] = F(x)=F(x ): We consider now the scan of an arbitrary row. Let q i (µ) be the probability that the scan overflows colun i, i.e., at least the first i scanned ters there are all greater than or equal to µ; clearly q 0 (µ) =. Let us look at q (µ); it equals Pr[A: µ]. Coputing it is easy using the above conditional distribution of A: given C = x, and the distribution of C given µ. There are additional considerations due to the following possibilities: (a) in probability = the first ter in the row is C, then A: = C µ and the scan continues, and (b) in probability =n the inax falls in the first colun, then C = µ > A: and the scan ends, except in the case covered in part (a) (and then C = µ = A:). Hence q (µ) = = + (? =n)(? =)Pr[A: µv], where V denotes the event finax is not in the first colun, and C is not in the scanned rowg. This last probability equals Z x µ? F(µ) F? (x ) F(x )? F f (x )dx ; (µ) and we find q (µ) = + (?? F (µ) n )? F? : (3) (µ) 7

8 For q (µ), n, we have a generalization of this result by noticing that the only way coputations for different coluns interact is via the above consideration (b). We use in (4) below the unusual notation (? n ), where its power (? n )k stands for (? n )(? n? )(? n?k ) =? k n. This is the probability that a given subset of k coluns does not contain inax. A consideration of the possible configurations of events that lead the scan to go beyond the first coluns allows us to write, even without the need to integrate over the axia x ; : : :;x explicitly, q (µ) = =? F(µ) +? F? (µ)? F (µ)? F (µ)? n + n (? n ) (4)? F (µ) : (5)? F(µ) The average nuber of coluns inspected in a row, after reoving the conditioning on the inax, is R µ=? (q 0(µ) ++q n (µ))nf? (µ)(?f (µ)) n? f (µ)dµ. It is possible to obtain a closed for for the su under the integral, but there is no advantage in doing so. The expressions siplify when we ake the change of variable under the integral of x and we can then write E[q ] = Z Z µ=? x=0 F(µ), q (µ)nf? (µ)(? F (µ)) n? f (µ)dµ (6)? x? x? n + n? x nx? (? x ) n? dx:? x We deterine an asyptotic expansion of this ean. In the process we shall discover that we need only few of these, since E[q + ] = o(e[q ]). R The basic expression we consider 2 is I k;n 0 (? x)k x? (? x ) n? dx: Distributing the first binoial and using the substitution t x produces an iediate integral: I k;n = r k r (?) r Γ( r + ) Γ(n) Γ(n + + r (7) ): The dependence of this value on the atrix size paraeters is not obvious, and we intend to clarify the issue via an asyptotic estiate. We assue that both n and!. In addition, we assue that their rates of growth are coparable; specifically, whenever we truncate expansions below we assue that = o(n lnn)and n = o( ln) (but not necessarily as close as = Θ(n)). The values of k above, and hence r, ay be assued sall (the largest value we actually use below is 2). We replace Γ( + r=) by its Taylor developent at, noting that it is at the sae tie a power series in r and and an asyptotic series in? : Γ( + r=) = 0 γ (r=), where γ = Γ ( ) ()=!. This expression ay also be written as (? n )k = (+ v )(? v n )k v=0. 2 We provide here a direct asyptotic evaluation of this integral. The appendix shows how to obtain for it an expansion using depoissonization, according to the results in [2], which are based on the coplex saddle-point ethod. 8

9 The ratio Γ(n)=Γ(n + + r=) is developed asyptotically as in [4, x4.5]. It is a direct calculation fro the standard representation of the gaa function at x by the integral R t0 e?t t x? dt, and to third order in =n and = produces Γ(n) Γ(n + + r=) = n?? r? r 2n? r2 2 2 n + r 2n 2 + O(?2 n?2 ) The salient feature of this is again that in the parenthesized factor all powers of r ust appear with like powers of?. In addition, except the first, they all coe with additional powers of n? sprinkled along. For n?r= we write the natural Taylor developent i0 (?r lnn=) i =i!.? Note that k r r (?) r r s with s a non-negative integer equals k!(?) k s k (where s k is the subset coefficient, or Stirling nuber of the second kind). Therefore it vanishes, unless s k. Hence only ters of order?k and higher (i.e., saller) survive the suation in (7). In particular, ters with r k produce the leading ter. Inserting all these into equation (7), and only keeping the leading ter, we find I k;n = n r = n = k t=0 k r n k t (?) r γ t t (?)k?t k t?r i ln i n r + O i! n r γ i lnn k?t (k?t)! k!(?)k + O b t ln k?t n lnn n + O lnn n (8) ; (9) where we use the notation b t (?) t t!γ t = (?) t t!γ ( ) (). The next order ter coes fro aditting the next two ters in the parenthesized factor of equation (8), as well as including those fro the collection [r k+ ]fγ( + r=)n?r= g, which we do not use below. In ters of these integrals, E[q ] = (n? )I ;n? + I?;n? +. The second ter is here of lower order than the first, and we find E[q ] = lnn t lnn b t ln k?t n + O t n : (20) The above clai, that E[q + ] = o(e[q ]), clearly holds. The replaceent of 0 E[q ] by the first three ters is then adequate at the order we aintain. We evaluate the and find E[Q] + lnn + γ + O lnn n The total expected coputation tie for the second algorith would then be, for soe suitable coefficient δ, given by nt + T n + δ( + (γ + ln n)=). Note that the doinant ter is in the nt part, siilar to what we had for the first algorith. (2) 9

10 4.3 Analysis of the third algorith The first phase of the third algorith is nearly identical to that of the second algorith. The only difference is that during its operation we now record the rows where the successive candidates for inax reside. This adds n? coparisons and, on the average, lnn replaceents of the current record value of inax. The second phase consists of the scan of a single selected row; the scan stops when a atrix eleent strictly saller than µ is found. This is the only algorith of the present trio in which we need ake no change to provide an accurate analysis. We use the ethodology developed for the analysis of the second algorith. Let r i (µ) be the probability that the row scan overflows colun i, with the convention that r 0 (µ) =. Here again we start with an evaluation of r (µ), for an equivalent of equation (3). The ain difference with the second algorith is that here, when the scan reaches the colun whose axiu is the inax, the atrix eleent equals µ and the scan is never terinated there, while in the second algorith this was the case with probability = only. For the coputation of r (µ), a convenient way to proceed is to separate the case where C is the inax, in probability =n, and the scan continues; otherwise, with a (conditional) probability of =; C is in the sae row and again the scan continues. If it is not, we need to evaluate an integral, identical to the one that led to equation (3). Writing it all in ters of the transforation we used for equation (6), we find r (x) = n +?? n + Z x!? xx x? dx? x : (22) Hence r (x) = n +(??x n )(?x ). With, siilar considerations to those used to produce equation (4) lead to the generalization: r (x) = =? x? x? x (? n )? x?? x n +?? x n? x : (23) The expected nuber of coparisons in phase 2 of the algorith is, as before, R 0 (r 0(x) + + r n (x))nx? (? x ) n? dx. Expressing E[r ] in ters of the integrals of type I k;n we see that E[r ] = I?;n? + + (n? )I ;n? ; 0

11 the sae integrals we needed for E[q ], albeit with different ultipliers. A siilar calculation produces E[r ln n ] = t bt t ln t + n n t? + O lnn lnn : Since these contributions to the expected nuber of coparisons decrease geoetrically in we only consider the first ones and find E[r ] = 0 lnn + + γ? + O n n + : (24) We use no further ters, since this value is two orders (in or n) below the cost of the first phase and unlike algorith II, it does not get ultiplied by. 5. Conclusion We have carried out probabilistic analyses of three soewhat siilar algoriths, and should not be surprised that their costs are siilar as well. Let us write for coparison the costs of all three, inserting Greek letters wherever needed to represent instruction and tie ultipliers, and using a letter and its pried version to denote siilar values. In C I we exchanged and n (rotated the atrix) for coparison purposes. C I = nt + δ(n + )? = n(α + β + νln + δ) + δ; C II = nt + T n + δ 0 ( + lnn + γ) = n(α + β + ν 0 ln + δ 0 (=n + lnn=n + γ=n)); C III = nt + T n + κlnn + δ 0 ( + ln n ) = n(α + β + ν 0 ln) + α 0 + βn + (ν + κ)ln n + δ 0 : Several coents are in order: () In each algorith cost, the first phase (in II and III, or the deterination of row inia in I) doinates. Moreover, the ain ter is the sae in all βn and since β and δ are siilar in agnitude, the costs of the first and third algorith are as close as it sees to atter, with a shade favoring algorith III, whereas algorith II is slightly ore expensive. When we consider that careful prograing (as deonstrated in the ipleentations of the first two algoriths presented in [3]) can result in nontrivial savings, by cobining various coponents of each algorith, we realize that uch of these differences ay be illusory. (2) The conclusions in coent () depend soewhat on our assuption that and n are of siilar order (see the discussion following equation (7)). If this were not the case, we would need to odify the to soe extent, but not drastically, since the leading ter in all the cost functions βn is syetrical in the atrix diensions. (3) While we spent uch effort on the probabilistic analysis of the second-phase costs, ost of the

12 cost there is deterinistic. Indeed, in all three algoriths, the difference between shortest and longest possible coputations ( best case and worst case ), is quite odest. (4) The algoriths we describe assue unlike the versions presented in [3] that all ters are distinct. At the level of our analysis the differences do not play a role. (5) The perforance of saddle point search algoriths when the ters ay not be all distinct can be quite different, as soe extree experients reported in the solution to exercise in [3] show. Clearly it is possible to design specialized algoriths for soe scenarios. An obvious case is when the atrix can have two distinct ters only; then we need only search for a row of high value ters, or a colun of low values. An algorith that does ust this would require on the average only 2( + n) coparisons. Interestingly, here the worst case ay exceed substantially the average one. With ore possible values, special algoriths will not bring nearly as uch savings, and are probably not cost effective. The next point is relevant here too. (6) As noted in [], not uch can be done for atrices with repeated values (when known to have ore than two levels) to obtain a ore efficient algorith. In particular, there is no escaping the ter βn. However, for atrices with distinct ters, it is possible to design an algorith with running tie bound by O(n lg3 ), where lg3 :585 : : : (this expression assues the atrix is square, and hides aterially higher nuerical coefficients than we have seen above). (7) The assuption of coparable growth rates for and n sees natural in this context, but does not affect the techniques needed for developing the asyptotic results only the ters that need to be collected, to get all that belong to a desired order. Acknowledgeent We would like to thank Donald E. Knuth who pointed out this unsolved exercise in his Art of Coputer Prograing, and gave valuable coents and iproveents to the solution we proposed. Appendix Depoissonization Asyptotics This is a different approach to coputing an asyptotic expansion of the integral I k;n. It is convenient to change here slightly the notation, and look instead at Z k I k;n+ g ;n = k (? x) k x? (? x ) n dx; 0 for a fixed k, and n and both tending to infinity. We eploy the Poisson generating function G (z) = n z n g ;n n! e?z and use the depoissonization leas presented by Jacquet and Szpankowski in 2

13 [2]. Corollary there states that if a sequence of Poisson generating functions G (z) are all entire, and satisfy the following two properties, uniforly in :. For all z in a linear cone including the positive real axis 3, the functions G (z) are uniforly in O(z β L(z)), with L(z) a slowly varying function For all z outside the linear cone, G (z)e z = P(z)e z, where P(z) decreases faster than the reciprocal of any polynoial. Then g ;n = G (n) + O(n β? L(n)). Note: The expansion can be further continued, to any order: i=k g ;n = i=0 i+k? b i; n i G (i) (n) + O(n β?k? L(n)) ; =0 where G (i) (n) is the ith derivative of G (z) at z = n. The coefficients b i; are obtained fro the Taylor power series developent: i; b i; x i y = exp(xln(+y)?xy): They can coputed recursively, starting with b 0;0 = through: b i; + =?( b i; +b i?;? )=( +). The recursion iplies that b i; = 0, for all < 2i. We do not use this extension below. We show first that the Poisson generation functions of the double sequence g ;n satisfy the above two conditions. The suation to obtain the functions is iediate: G (z) = k R 0 (? x)k x? e?xz dx, and they are clearly entire. For this function, G (z) G (R(z)). Hence, when we prove condition for a z which is real and positive, we have shown that it holds for an entire linear cone S θ (in the right half coplex plane i.e., 0 < θ < π=2). Since z R(z), condition 2 is satisfied trivially for this function. Let z be real and positive. We prove that G (z) (lnz)k z by transforing it. The change of variable x = e?y leads to: G R (z) = k 0 (? e?y= ) k e?y exp[?ze?y ]dy. Since?e?x x for all positive x, we find G (z) Z 0 y k e?y exp[?ze?y ]dy: (25) We only add O(e?z ) to the integral when we extend the liit of integration to? (and this is shown below to be negligible). Hence, G (z) R? yk e?y exp[?ze?y ]dy + O(e?z ). 3 Such a cone is a region in the coplex plane defined by S θ = fz : argz θ; 0 < θ < π=2g. 4 L(z) is a slowly varying function (at infinity) in the cone S θ if li u! L(aue iρ )=L(u)=, for all a > 0 and ρ < θ. 3

14 By obvious change R of variable we get G R (z) z? (x + lnz)k e?x exp[?e?x ]dx. The integral? (?x) e?x exp[?e?x ]dx equals the th derivative of the gaa function at (using the sae representation that leads to (8)), which is bounded by!, hence as z increases the leading ter in the above bound is (lnz) k =z. Therefore condition is proved with β =? and L(z) = (lnz) k. To obtain our expansion we evaluate G (z) as above, up to the expression leading to equation (27), but avoid siplifying. Instead we consider G (z) = z Z? k (? e?(y+lnz)= ) k e?y exp[?e?y ]dy + O(e?z ) ; and expand the exponential in k (? e?(y+lnz)= ) k by powers of y: G (z) = z Z? (y + lnz) " k (?) i+ i0 (i + )! # y + ln z i k e?y exp[?e?y ]dy + O(e?z ) : Keeping the first ter (i = 0) only leads to Z G lnz (z) = (y + lnz) k e?y exp[?e?y ]dy + O z? z = z k ln k? (z)(?) Γ ( ) lnz () + O 0 z Note that we have lost, as proised, the contribution of O(e?z ). The depoissonization lea then gives g ;n = n 0 k lnn b ln k? n + O ; (26) n where b = (?) Γ ( ) (). This is the exact analog of the result obtained by direct calculation. We expect that for higher-order ters this approach is likely to prove ore anageable. : References [] D. Bienstock, F. Chung, M. Fredan, A. Schäffer, P. Shor, and S. Suri, A note on finding a strict saddle-point, Aerican Math. Monthly 98, (99). [2] P. Jacquet, W. Szpankowski, Analytical depoissonization and its applications, subitted for Theor. Cop. Science, as a Fundaental Study, 997. [3] D. Knuth, The Art of Coputer Prograing, Addison-Wesley, volue : Fundaental Algoriths, rd. Ed [4] F. W. J. Olver, Asyptotics and Special Functions, Acadeic Press, New York,