Exact L^2-distance from the limit for QuickSort key comparisons (extended abstract)

EXA CT L 2 -DIST ANCE FROM THE LIMIT F OR QUICKSOR T KEY COMP ARISONS (EXTENDED ABSTRA CT) P A TRICK BINDJEME JAMES ALLEN FILL Abstract Using a recursiv e approac h, we obtain a simple exact exp ression for the L 2 -distance from the limit in R´ egnier’s [5] classical limit theorem for the num b er of key comparisons requ ired by QuickSort . A previous study by Fill and Janson [1] using a similar approach found that th e d 2 -distance is of order b etw een n − 1 log n and n − 1 / 2 , and another by Neininger and Ruschendorf [4] found that the Zolotarev ζ 3 -distance is of exact order n − 1 log n . Our expression reveals that th e L 2 -distance is asymptotically equiva lent to (2 n − 1 ln n ) 1 / 2 . 1. Introduction, review of rela ted l itera ture, and summar y W e consider Hoare’s [3 ] QuickSo rt sorting a lgorithm applied to an infinite stream of iid (independent and identically distributed) uniform random v aria bles U 1 , U 2 , . . . . Q uickSo rt chooses t he first key U 1 as the “pivot”, compare s eac h of the other keys to it, and then pro cee ds recurs ively to sor t b oth the keys smaller than the pivot and those lar ger than it. If, for exa mple, the initial r o und of comparis ons finds U 2 < U 1 , then U 2 is used a s the piv ot in the recursive call to the algo r ithm that s orts the keys smaller than U 1 bec ause it is the first element in the sequence U 1 , U 2 , . . . which is smaller than U 1 . In a natural and o b vious w ay , a rea liz a tion (requiring infinite time) o f the alg o rithm pr o duces an infinite ro o ted binar y s earch tree whic h with probability one has the c ompleteness pr op e rty that each no de ha s t wo child-nodes . Essentially the same algor ithm ca n of co urse b e applied to the trunca ted sequence U 1 , U 2 , . . . , U n for any finite n , where the recur sion ends by declaring that a list of size 0 or 1 is already sorted. Let K n denote the n um b er of k ey c o mparisons require d by QuickS ort to sort U 1 , U 2 , . . . , U n . Then, with the wa y w e ha ve set things up, all the r andom v ar iables K n are defined on a commo n probability space, and K n is nondecreasing in n . Indeed, K n − K n − 1 is simply the cost of inser ting U n int o the usual (finite) binary search tree formed from U 1 , . . . , U n − 1 . In this fra mework, R ´ egnier [5 ] used martinga le techniques to establish the fol- lowing L p -limit theorem; she also prov ed a lmost sure con vergence. W e let µ n := E K n . Date : Jan uary 26, 2012 . Researc h supp orted by the Acheson J. Duncan F und for the Adv ancement of Researc h in Statistics. 1 2 P A TRICK BINDJEME JAMES ALLEN FILL Theorem 1.1 (R ´ egnier [5]) . Ther e exists a r andom variable T satisfying Y n := K n − µ n n + 1 L p − → T for every finite p . R¨ osler [6] characteriz e d the distribution of R´ e g nier’s limiting T as the unique fixed p oint of a certain distributional transformation, but he also described explic- itly how to construct a rando m v aria ble having the same distribution a s T . W e will describ e his ex plicit co nstruction in equiv a lent terms, but fir st we need t wo paragr aphs of notation. The no des of the complete infinite binary sea r ch tree a r e labeled in the natura l binary wa y: the r o o t g ets a n empty la bel wr itten ε her e, the left (resp ectively , right ) child is la beled 0 (resp., 1 ), the left child of no de 0 is labe le d 00, etc. W e write Θ := ∪ 0 ≤ k< ∞ { 0 , 1 } k for the set of all suc h labels. If V θ denotes the key inserted at no de θ ∈ Θ, let L θ (resp., R θ ) denote the largest key s maller than V θ (resp., sma llest key lar ger than V θ ) inserted a t any ancesto r of θ , with the exceptions L θ := 0 and R θ := 1 if the specified ancestor k eys do not exist. F urther, for each no de θ , define φ θ := R θ − L θ , U θ := φ θ 0 /φ θ , G θ := φ θ C ( U θ ) = φ θ − 2 φ θ ln φ θ + 2 φ θ 0 ln φ θ 0 + 2 φ θ 1 ln φ θ 1 , (1.1) where for 0 < x < 1 w e define (1.2) C ( x ) := 1 + 2 x ln x + 2 (1 − x ) ln (1 − x ) . Let 1 ≤ p < ∞ . The d p -metric is the metric on the space of all probability distributions with finite p th absolute moment defined by d p ( F 1 , F 2 ) := inf k X 1 − X 2 k p , where we ta ke the infimum of L p -distances over all pair s of random v ar iables X 1 and X 2 (defined on the same pro bability space) with res pective marginal dis tributions F 1 and F 2 . By the d p -distance b etw een tw o rando m v a riables w e mean the d p - distance betw een their distributions. W e ar e now prepared to state R¨ osler ’s ma in result. Note : Here and la ter results hav e b een adjusted slightly a s necessary to utilize the same deno minator n + 1 (rather than n ) that R ´ egnier us e d. Theorem 1. 2 (R¨ osler [6]) . F or any finite p , the infinite seri es Y = P ∞ j =0 P | θ | = j G θ c onver ges in L p , and the se quenc e Y n = ( K n − µ n ) / ( n + 1 ) c onver ges in the d p -metric to Y . Of cour se it follows from Theor e ms 1.1 – 1.2 that T and Y hav e the sa me distri- bution. The purp os e of the presen t extende d abstract is to show that in fact T = Y and to pro vide a simpl e explicit expres sion for the L 2 -distance b et w een Y n and Y v alid for every n ; this is done in Theorem 1.4 below. W e are aw are of only tw o previo us studies o f the r ate of conv ergenc e of Y n to Y , and b oth of those concern cer ta in dis ta nces be tw een distributions rather than b etw een r andom variables . The first study , by Fill a nd Ja nson [1], provides upper and low er b ounds o n d p ( Y n , Y ) for genera l p ; w e c ho ose to fo cus here on d 2 . Theorem 1.3 (Fill and Janson [1]) . Ther e is a c onstant c > 0 such that for any n ≥ 1 we have cn − 1 ln n ≤ d 2 ( Y n , Y ) < 2 n − 1 / 2 . EXACT L 2 -DIST ANCE FOR QUICKSOR T KEY COMP ARISONS 3 T o our knowledge, the gap betw een the rates (log n ) /n and n − 1 / 2 has not been nar- row ed. Neininger and Ruschendorff [4] use d the Zolotarev ζ 3 -metric and found that the co rrect rate in that metric is n − 1 log n , but their tec hniques ar e not sufficiently sharp to obtain ζ 3 ( Y n , Y ) ∼ ˜ cn − 1 ln n for some constant ˜ c . In our m ain Theorem 1.4 , pr ov ed using the s ame recursive approach as in Fill and Janso n [1], w e find not only the lead-order asymptotics for the L 2 -distance k Y n − Y k 2 , but in f act an exa ct expr ession for g eneral n . It is interesting to note that the r ate n − 1 / 2 (log n ) 1 / 2 for L 2 -conv ergence is la rger even than the upp er- bo und ra te of n − 1 / 2 for d 2 -conv ergence from Theorem 1.3. Theorem 1.4 ( main theorem). F or n ≥ 0 we have k Y n − Y k 2 2 = ( n + 1 ) − 1  2 H n + 1 + 6 n + 1  − 4 ∞ X k = n +1 k − 2 = 2 ln n n + O  1 n  , wher e H n := P n j =1 j − 1 is the n th harmonic numb er and the asymptotic expr ession holds as n → ∞ . The remainder of this e x tended a bstract is devoted to a pro o f o f Theorem 1.4, which is completed in Section 5. 2. Preliminaries In this section we provide rec ur sive re pr esentations of Y n (for genera l n ) and Y that w ill b e useful in proving Theorem 1.4. Our fir st prop osition concerns the limit Y and gives a sample-p oint wise extension of the v ery well known [6] distribu- tional identit y sa tisfied by Y . Recall the notation (1.1) and the definition of Y in Theorem 1.2 as the infinite series P ∞ j =0 P | θ | = j G θ in L 2 . Prop ositio n 2. 1. Ther e exist r andom variables F θ and H θ for θ ∈ Θ su ch that (i) the joi nt distributions of ( G θ : θ ∈ Θ) , of ( F θ : θ ∈ Θ) , and of ( H θ : θ ∈ Θ) agr e e; (ii) ( F θ : θ ∈ Θ) and ( H θ : θ ∈ Θ) ar e indep endent; (iii) the s eries (2.1) Y (0) := ∞ X j =0 X | θ | = j F θ and Y (1) := ∞ X j =0 X | θ | = j H θ c onver ge in L 2 ; (iv) the r andom variables Y (0) and Y (1) ar e indep endent, e ach with the same distribution as Y , and (2.2) Y = C ( U ) + U Y (0) + U Y (1) . Her e U := U 1 , with U := 1 − U 1 , and C is define d at (1.2) . Pr o of. Recall from (1.1) that G θ = φ θ − 2 φ θ ln φ θ + 2 φ θ 0 ln φ θ 0 + 2 φ θ 1 ln φ θ 1 . F or θ ∈ Θ, define the rando m v ariable ϕ θ (resp ectively , ψ θ ) b y ϕ θ := φ 0 θ /U (resp., ψ θ := φ 1 θ / U ) . 4 P A TRICK BINDJEME JAMES ALLEN FILL Then U and ϕ θ are independent (resp., U a nd ψ θ are independent), ϕ θ and ψ θ each hav e the sa me distribution as φ θ , and G 0 θ = U F θ and G 1 θ = U H θ , where F θ := ϕ θ − 2 ϕ θ ln ϕ θ + 2 ϕ θ 0 ln ϕ θ 0 + 2 ϕ θ 1 ln ϕ θ 1 , H θ := ψ θ − 2 ψ θ ln ψ θ + 2 ψ θ 0 ln ψ θ 0 + 2 ψ θ 1 ln ψ θ 1 . The prop osition follows eas ily from the clear equalit y L ( F θ : θ ∈ Θ) = L ( G θ : θ ∈ Θ) = L ( H θ : θ ∈ Θ) , of join t laws a nd the fact that Y = ∞ X j =0 X | θ | = j G θ = G ε + ∞ X j =0 X | θ | = j G 0 θ + ∞ X j =0 X | θ | = j G 1 θ = C ( U ) + U ∞ X j =0 X | θ | = j F θ + U ∞ X j =0 X | θ | = j H θ = C ( U ) + U Y (0) + U Y (1) .  W e next pro ceed to provide an analog ue [namely , (2.4)] of (2.2) for ea c h Y n , rather than Y , but firs t w e need a little more notation. Given 0 ≤ x < y ≤ 1, let ( U xy n ) n ≥ 1 be the subsequence o f ( U n ) n ≥ 1 that falls in ( x, y ). The ra ndom v aria ble K n ( x, y ) is defined to b e the (random) num b er of key co mparisons used to sort U xy 1 , . . . , U xy n using QuickSort . The distribution o f K n ( x, y ) o f course does not dep end on ( x, y ). W e now define the random v aria ble Y n,θ := [ K ν θ ( n ) ( L θ , R θ ) − µ ν θ ( n ) ] / [ ν θ ( n ) + 1] , (2.3) with the cent ering here motiv ated by the fact that µ ν θ ( n ) is the conditional exp ec- tation of K ν θ ( n ) ( L θ , R θ ) given ( ν θ ( n ) , L θ , R θ ). Then for n ≥ 1 we hav e (2.4) Y n = n n + 1 C n ( ν 0 ( n ) + 1) + ν 0 ( n ) + 1 n + 1 Y n, 0 + ν 1 ( n ) + 1 n + 1 Y n, 1 , where, as in [2], for 1 ≤ i ≤ n w e define C n ( i ) := 1 n ( n − 1 + µ i − 1 + µ n − i − µ n ) . W e note for future reference that the classical divide-and-conquer recurre nc e for µ n asserts precisely that (2.5) n X i =1 C n ( i ) = 0 for n ≥ 1. EXACT L 2 -DIST ANCE FOR QUICKSOR T KEY COMP ARISONS 5 It follows fro m (2.2) and (2.4) that for n ≥ 1 we have Y n − Y =  ν 0 ( n ) + 1 n + 1 Y n, 0 − U Y (0)  +  ν 1 ( n ) + 1 n + 1 Y n, 1 − U Y (1)  +  n n + 1 C n ( ν 0 ( n ) + 1) − C ( U )  =: W 1 + W 2 + W 3 . (2.6) Conditionally given U and ν 0 ( n ), the ra ndom v ar iables W 1 and W 2 are indepe ndent, each with v anishing mean, and W 3 is constant. Hence E [( Y n − Y ) 2 | U, ν 0 ( n )] = E [ W 2 1 | U, ν 0 ( n )] + E [ W 2 2 | U, ν 0 ( n )] + W 2 3 and th us, taking exp ectations and using symmetry , for n ≥ 1 we ha ve (2.7) a 2 n := E ( Y n − Y ) 2 = E W 2 1 + E W 2 2 + E W 2 3 = 2 E W 2 1 + E W 2 3 . Note that (2.8) a 2 0 = E Y 2 = σ 2 := 7 − 2 3 π 2 (for example, [2]). 3. Anal ysis of E W 2 1 In this sectio n we analyze E W 2 1 , pro ducing the following result. Recall the definition of σ 2 at (2.8). Prop ositio n 3. 1. L et n ≥ 1 . F or W 1 define d as at (2.6) , we have E W 2 1 = 1 n ( n + 1) 2 n − 1 X k =0 ( k + 1 ) 2 a 2 k + σ 2 6( n + 1) . F or that, we first prov e the following t w o lemmas. Lemma 3.2. F or any n ≥ 1 , we have E "  ν 0 ( n ) + 1 n + 1  2  Y n, 0 − Y (0)  2 # = 1 n n − 1 X k =0  k + 1 n + 1  2 a 2 k . Lemma 3.3. F or any n ≥ 1 , we have E "  ν 0 ( n ) + 1 n + 1 − U  2  Y (0)  2 # = σ 2 6( n + 1) . Pr o of of L emma 3.2 . There is a probabilistic co p y Y ∗ = ( Y ∗ n ) of the sto chastic pro cess ( Y n ) such that Y n, 0 ≡ Y ∗ ν 0 ( n ) and Y ∗ and Y (0) are indep e nden t of ( U, ν 0 ( n )). This implies E "  ν 0 ( n ) + 1 n + 1  2  Y n, 0 − Y (0)  2 # = E "  ν 0 ( n ) + 1 n + 1  2  Y ∗ ν 0 ( n ) − Y (0)  2 # . 6 P A TRICK BINDJEME JAMES ALLEN FILL By conditioning on ν 0 ( n ), which is uniformly distr ibuted on { 0 , . . . , n − 1 } , w e get E "  ν 0 ( n ) + 1 n + 1  2  Y n, 0 − Y (0)  2 # = E "  ν 0 ( n ) + 1 n + 1  2 a 2 ν 0 ( n ) # = 1 n n − 1 X k =0  k + 1 n + 1  2 a 2 k .  Pr o of of L emma 3.3 . Conditionally g iven ν 0 ( n ) and Y (0) , we have that U is distributed as the order statistic of rank ν 0 ( n ) + 1 from a sample o f size n from the uniform(0 , 1) distribution, namely , Beta( ν 0 ( n ) + 1 , n − ν 0 ( n )), with ex pecta tion [ ν 0 ( n ) + 1] / ( n + 1) and v ariance [( ν 0 ( n ) + 1)( n − ν 0 ( n ))] / [( n + 1) 2 ( n + 2)]. So, using also the independence o f ν 0 ( n ) and Y (0) , we find E "  ν 0 ( n ) + 1 n + 1 − U  2  Y (0)  2 # = E  ( ν 0 ( n ) + 1)( n − ν 0 ( n )) ( n + 1) 2 ( n + 2)  Y (0)  2  = σ 2 E  ( ν 0 ( n ) + 1)( n − ν 0 ( n )) ( n + 1) 2 ( n + 2)  = σ 2 ( n + 1) 2 ( n + 2) × 1 n n − 1 X k =0 ( k + 1 )( n − k ) = σ 2 n ( n + 1 ) 2 ( n + 2) × 1 6 n ( n + 1)( n + 2) = σ 2 6( n + 1) .  Pr o of of Pr op osition 3.1 . W e hav e E W 2 1 = E  ν 0 ( n ) + 1 n + 1 Y n, 0 − U Y (0)  2 = E  ν 0 ( n ) + 1 n + 1  Y n, 0 − Y (0)  +  ν 0 ( n ) + 1 n + 1 − U  Y (0)  2 = E "  ν 0 ( n ) + 1 n + 1  2  Y n, 0 − Y (0)  2 # + E "  ν 0 ( n ) + 1 n + 1 − U  2 ( Y (0) ) 2 # + 2 E  ν 0 ( n ) + 1 n + 1  Y n, 0 − Y (0)   ν 0 ( n ) + 1 n + 1 − U  Y (0)  . The res ult follows fro m Lemmas 3.2 – 3.3, and the fact that, conditionally given ( ν 0 ( n ) , Y n, 0 , Y (0) ), the random v ariable U is distr ibuted Beta( ν 0 ( n ) + 1 , n − ν 0 ( n )), so that the last expec ta tion in the preceding equation v anishes.  4. Anal ysis of E W 2 3 In this section we analy z e E W 2 3 , producing the following result. EXACT L 2 -DIST ANCE FOR QUICKSOR T KEY COMP ARISONS 7 Prop ositio n 4. 1. F or any n ≥ 1 we have b 2 n := E W 2 3 = σ 2 − 7 3 + 4 3  n + 2 n + 1  H (2) n + 4 3 n ( n + 1) 2 H n , wher e H n = P n j =1 j − 1 is the n th harmonic nu mb er and H (2) n := P n j =1 j − 2 is the n th harmonic num b er of the se c ond or der. F or that, we first prov e the following t w o lemmas. Lemma 4.2. F or any 1 ≤ k ≤ n we have D ( n, k ) := 1 B ( k , n − k + 1 ) Z 1 0 t k − 1 (1 − t ) n − k (ln t ) dt = H k − 1 − H n , wher e B is the b eta function. Lemma 4.3. F or any n ≥ 1 we have E [ C n ( ν 0 ( n ) + 1) C ( U )] = n n + 1 E [ C n ( ν 0 ( n ) + 1)] 2 . Pr o of of L emma 4.2 . The r esult can be prov ed for each fixed n ≥ 1 by bac k- wards induction o n k and in tegration by parts, but w e give a simpler pro o f. Recall the defining expression B ( α, β ) = Z 1 0 t α − 1 (1 − t ) β − 1 dt for the beta function when α, β > 0. Differentiating with respe c t to α giv es Z 1 0 t α − 1 (1 − t ) β − 1 (ln t ) dt = B ( α, β )[ ψ ( α ) − ψ ( α + β )] , where ψ is the classic a l digamma function, i.e., the logarithmic deriv ativ e of the gamma function. But it is well kno wn that ψ ( j ) = H j − 1 for po sitive integers j , s o the lemma follows by s etting α = k and β = n − k + 1.  Pr o of of L emma 4.3 . W e know that ν 0 ( n ) + 1 ∼ unif { 1 , 2 , . . . , n } a nd that, con- ditionally giv en ν 0 ( n ), the random v ariable U has the Beta( ν 0 ( n ) + 1 , n − ν 0 ( n )) distribution. So from Lemma 4.2, repea ted us e o f (2.5), and the very well known and easily derived explicit express ion µ n = 2( n + 1) H n − 4 n, n ≥ 0 , 8 P A TRICK BINDJEME JAMES ALLEN FILL we hav e E [ C n ( ν 0 ( n ) + 1) C ( U )] = 1 n n X j =1 C n ( j ) 1 B ( j, n − j + 1) Z 1 0 t j − 1 (1 − t ) n − j C ( t ) dt = 1 n n X j =1 C n ( j )[1 + 2 j n + 1 ( H j − H n +1 ) + 2 n − j + 1 n + 1 ( H n − j +1 − H n +1 )] = 1 n ( n + 1 ) n X j =1 C n ( j )[2 j H j + 2( n − j + 1 ) H n − j +1 ] = 1 n ( n + 1 ) n X j =1 C n ( j )[2 j H j − 1 − 4( j − 1) + 2 ( n − j + 1 ) H n − j − 4( n − j )] = 1 n ( n + 1 ) n X j =1 C n ( j )[ µ j − 1 + µ n − j ] = 1 n ( n + 1 ) n X j =1 C n ( j )[ µ j − 1 + µ n − j − µ n ] = n n + 1 × 1 n n X j =1 C n ( j ) 2 = n n + 1 E [ C n ( ν 0 ( n ) + 1)] 2 , as desired.  Pr o of of Pr op osition 4.1 . It follows from Lemma 4.3 that b 2 n = E  n n + 1 C n ( ν 0 ( n ) + 1) − C ( U )  2 =  n n + 1  2 E [ C n ( ν 0 ( n ) + 1)] 2 − 2  n n + 1  E [ C n ( ν 0 ( n ) + 1) C ( U )] + E C ( U ) 2 = E C ( U ) 2 −  n n + 1  2 E [ C n ( ν 0 ( n ) + 1)] 2 . Knowing that E C ( U ) 2 = σ 2 / 3, and from the pro of of Lemma A.5 in [2] that E [ C n ( ν 0 ( n ) + 1)] 2 = 7 3  1 + 1 n  2 − 4 3  1 + 2 n   1 + 1 n  H (2) n − 4 3 n − 3 H n , we hav e b 2 n = σ 2 − 7 3 + 4 3  n + 2 n + 1  H (2) n + 4 3 n ( n + 1) 2 H n , as claimed.  5. A closed f orm for a 2 n In this final section w e complete the pro o f of Theorem 1.4, for which we need one more lemma. Lemma 5.1. F or H (2) n = P n j =1 j − 2 , the n th harmonic nu mb er of the se c ond or der, we have n X j =1 H (2) j = ( n + 1) H (2) n − H n for any nonne gative inte ger n . EXACT L 2 -DIST ANCE FOR QUICKSOR T KEY COMP ARISONS 9 The lemma is w ell known a nd easily proved. Pr o of of main The or em 1.4 . F o r n ≥ 1 we hav e from the decomp osition (2.7) and Prop ositions 3.1 and 4.1 that a 2 n = 2 n ( n + 1 ) 2 n − 1 X k =0 ( k + 1) 2 a 2 k + σ 2 3  n + 2 n + 1  − 7 3 + 4 3  n + 2 n + 1  H (2) n + 4 3 n ( n + 1) 2 H n , and we recall from (2.8) that a 2 0 = σ 2 . Setting x n := ( n + 1 ) 2 a 2 n , we ha ve x 0 = σ 2 and x n = 2 n n − 1 X k =0 x k + c n for n ≥ 1 , with c n := σ 2 3 ( n + 2)( n + 1) − 7 3 ( n + 1) 2 + 4 3 ( n + 2)( n + 1) H (2) n + 4 3 n H n . This is a standard divide-and-conquer recurrence relation for x n , with solution x n = ( n + 1 ) " σ 2 + n X k =1 k c k − ( k − 1) c k − 1 k ( k + 1) # , n ≥ 0 . After straightforward computation in volving the identit y in Lemma 5.1, one finds a 2 n = ( n + 1 ) − 1  2 H n + 1 + 6 n + 1  + σ 2 − 7 + 4 H (2) n = ( n + 1 ) − 1  2 H n + 1 + 6 n + 1  − 4 ∞ X k = n +1 k − 2 = 2 ln n n + O  1 n  , as claimed.  References [1] James Allen Fill and Sv an te Janson. Quicksort asymptot ics. J. Algo rithms , 44(1):4–28, 2002 . Analysis of algorithms. [2] James Allen Fill and Sv ant e Janso n. Quicksort Asymptotics : App endix . Unpublished, Av ail- able from http: //www.ams.j hu.edu/~fill/ , 2004. [3] C. A. R. Hoare. Quic ksort. Com put. J. , 5:10–1 5, 1962. [4] Ralph Neininger a nd Ludger R ¨ uschen dorf. Rates of con vergen ce f or Quicksort. J . Algorithms , 44(1):51–6 2, 2002 . A nalysis of al gorithms. [5] M. R ´ egnier. A limiting dis tribution of Quic ksort. RAIRO Informatique Th ´ eorique et App lic a- tions , 23:335–343, 1989. [6] U. R¨ osler. A limit theorem f or Quicksort. RAIRO Informatique Th ´ eorique et Applic ations , 25:85–100, 1991. Dep a r tmen t of Applied Ma thema tics and St at istics, The Johns Hopkins University, 34th and Charles Streets, Bal timore, MD 21218-268 2 USA E-mail addr e ss : bindjeme@ams.jhu .edu and jimfill@jhu .edu