The Knuth-Yao Quadrangle-Inequality Speedup is a Consequence of Total-Monotonicity

Transcription

1 The Knuth-Yao Quadrangle-Ineuality Seedu is a Conseuence of Total-Monotonicity Wolfgang W. Bein Mordecai J. Golin Lawrence L. Larmore Yan Zhang Abstract There exist several general techniues in the literature for seeding u naive imlementations of dynamic rogramming. Two of the best known are the Knuth-Yao uadrangle ineuality seedu and the SMAWK algorithm for finding the row-minima of totally monotone matrices. Although both of these techniues use a uadrangle ineuality and seem similar they are actually uite different and have been used differently in the literature. In this aer we show that the Knuth-Yao techniue is actually a direct conseuence of total monotonicity. As well as roviding new derivations of the Knuth-Yao result, this also ermits showing how to solve the Knuth-Yao roblem directly using the SMAWK algorithm. Another conseuence of this aroach is a method for solving online versions of roblems with the Knuth-Yao roerty. The online algorithms given here are asymtotically as fast as the best reviously known static ones. For examle the Knuth-Yao techniue seeds u the standard dynamic rogram for finding the otimal binary search tree of n elements from Θ(n 3 ) down to O(n 2 ), and the results in this aer allow construction of an otimal binary search tree in an online fashion (adding a node to the left or right of the current nodes at each ste) in O(n) time er ste. We conclude by discussing how the general techniue described here is also alicable to later extensions of the Kuth-Yao result, such as those develoed Borchers and Guta. Deartment of Comuter Science, University of Nevada, Las Vegas, NV bein@cs.unlv.edu. Research suorted by NSF grant CCR Det. of Comuter Science, Hong Kong UST, Clear Water Bay, Kowloon, Hong Kong. golin@cs.ust.hk Research artially suorted by Hong Kong RGC CERG grant HKUST312/E. Deartment of Comuter Science, University of Nevada, Las Vegas, NV larmore@cs.unlv.edu. Research suorted by NSF grant CCR Det. of Comuter Science, Hong Kong UST, Clear Water Bay, Kowloon, Hong Kong. cszy@cs.ust.hk Research artially suorted by Hong Kong RGC CERG grant HKUST312/E. 1

2 1 Introduction 1.1 History The construction of otimal binary search trees is a classic otimization roblem. The inut is 2n+1 weights (robabilities) 1,..., n,, 1...., n ; i is the weight that a search is for Key i ; such a search is called successful. The value i is the weight that the search argument is unsuccessful and is for an argument between Key i and Key i+1 (where we set Key = and Key n+1 = ). Our roblem is to find an otimal binary search tree (OBST) with n internal nodes corresonding to successful searches and n + 1 leaves corresonding to unsuccessful searches that minimizes the average search time. Let d( i ) be the deth of internal node corresonding to i and d( i ) the deth of leaf corresonding to i. Then we want to find a tree that minimizes j (1 + d( j )) + k d( k ). 1 j n k n It is not hard to see that this roblem reduces to solving the following recurrence: { if i = j B i,j = j t=i+1 i + j t=i j + min i<t j {B i,t 1 + B t,j } if i < j (1) where the cost of the otimal OBST is B,n. The naive way of calculating B i,j reuires Θ(j i) time, so calculating all of the B i,j would seem to reuire Θ(n 3 ) time. In fact, this is what was done in by Gilbert and Moore in 19 [8]. More than a decade later, in 191, it was noticed by Knuth [9] that, using a comlicated amortization argument, the B i,j can all be comuted using only Θ(n 2 ) time. Around another decade later, in the early 198s, Yao [1, 1] simlified Knuth s roof and, in the rocess, showed that this dynamic rogramming seedu worked for a large class of roblems satisfying a uadrangle ineuality roerty. Many other authors then used the Knuth-Yao techniue, either imlicitly or exlicitly, to seed u different dynamic rogramming roblems. See e.g., [13, 3, ]. In the 198s a variety of researchers develoed various related techniues for exloiting roerties, such as convexity and concavity, to yield dynamic rogramming seedus; a good early survey is []. A high oint of this strand of research was the develoment in the late 198s of the linear time SMAWK algorithm [1] for finding the row-minima of totally monotone matrices. The work in [] rovides a good survey of the techniues mentioned as well as alications and later extensions. One articular extension we mention (since we will use it later) is the LARSCH algorithm of Larmore and Schieber [1] which, in some cases, ermits finding row-minima even when entries of the matrix can imlicitly deend uon other entries in the matrix (a case SMAWK cannot handle). As we shall soon see, both the Knuth-Yao (KY) and SMAWK techniues rely on an underlying uadrangle ineuality in their structure and have a similar feel. In site of this, they have until usually been thought of as being different aroaches. See, e.g., [12] which uses both KY and SMAWK to seed u different roblems. In [2] Aggarwal and Park demonstrated a relationshi between the KY roblem and totally-monotone matrices by building a 3-D monotone matrix based on the KY roblem and then using an algorithm due to Wilber [1] to find tube minima in that 3-D matrix. They left as an oen uestion the ossibility of using SMAWK directly to solve the KY roblem. The main theoretical contribution of this aer is to show that the KY techniue is really just a secial case of the use of totally monotone matrices. We first show a direct solution to the KY 2

3 roblem by decomosing it into O(n) totally-monotone O(n) O(n) matrices, ermitting direct alication of the SMAWK algorithm to yield another O(n 2 ) solution. After that we describe how the Knuth-Yao techniue itself is actually a direct conseuence of total-monotonicity of certain related matrices. Finally, we show that roblems which can be solved by the KY techniue statically in O(n 2 ) time can actually be solved in an online manner using only O(n) worst case time er ste. This is done by using a new formulation of the roblem in terms of monotonematrices, along with the LARSCH algorithm Definitions Definition 1 A two dimensional uer triangular array a(i, j), 1 i j n satisfies a uadrangle ineuality (QI) if a(i, j) + a(i, j ) a(i, j) + a(i, j ) for i i j j. Note: In some alications we will write a i,j instead of a(i,j). Definition 2 A 2 2 matrix is monotone if the minimum of the uer row is not to the right of [ ] a b the minimum of the lower row. More formally, is monotone if b < a imlies that d < c c d and b = a imlies that d c. A 2-dimensional matrix M is totally monotone if every 2 2 submatrix of M is monotone. [ ] a b Definition 3 A 2 2 matrix is Monge if a + d b + c. c d A 2-dimensional matrix M is Monge if every 2 2 submatrix of M is Monge. The imortant observations (all of which can be found in []) are Observation 1 An m n matrix M is Monge if, for all 1 i < m and 1 j < n, M(i, j) + M(i + 1, j + 1) M(i, j + 1) + M(j + 1, i) (2) Observation 2 Every Monge matrix is totally monotone. Combining the above leads to the test that we will often use: Observation 3 Let M be an m n matrix. M is totally monotone, if for all 1 i < m and 1 j < n, M(i, j) + M(i + 1, j + 1) M(i, j + 1) + M(j + 1, i). (3) 1.3 Mathematical Framework Even though both the SMAWK algorithm [1] and the Knuth-Yao (KY) seedu [9, 1, 1] use an imlicit uadrangle ineuality in their associated matrices, on second glance, they seem uite different from each other. In the SMAWK techniue, the uadrangle ineuality is on the entries of a given m n inut matrix, which can be any totally monotone matrix. 2 It is not necessary for the inut matrix to 1 We should oint out that, as discussed in more detail at the end of Section 3, an alternative online algorithm to the one resented here could be derived by careful deconstruction of the static Aggarwal-Park [2] method; somehow, this never seems to have been remarked before in the literature. 2 Note that Monge Matrices satisfy a uadrangle ineuality, but in general, a totally monotone matrix may not. However, in ractice, most alications of the SMAWK algorithm make use of Monge matrices. If the inut matrix is triangular, the missing entries are assigned the default value, reserving total monotonicity. 3

4 actually be given.all that the SMAWK algorithm reuires is that, when needed, individual entries can be calculated in O(1) (amortized) time. The outut of the SMAWK algorithm is a vector containing the row-minima of the inut matrix. If m n, the SMAWK algorithm oututs this vector in O(n) time, an order of magnitude seedu of the naive algorithm that scans all mn matrix entries. The KY techniue, by contrast, uses a uadrangle ineuality in the uer-triangular n n matrix B i,j. That is, it uses the QI roerty of its result matrix to seed u the evaluation, via dynamic rogramming, of the entries in the same result matrix. More secifically, Yao s result [1] was formulated as follows: For 1 i j n let w(i, j) be a given value and { if i = j B i,j = () w(i, j) + min i<t j {B i,t 1 + B t,j } if i < j Definition w(i, j) is monotone in the lattice of intervals if [i, j] [i, j] imlies w(i, j) w(i, j ). As an examle, it is not difficult to see that the w(i, j) = j t=i+1 i+ j t=i j of the BST recurrence (1) satisfies the uadrangle ineuality and is monotone in the lattice of intervals. Definition Let K B (i, j) = max{t : w(i, j) + B i,t 1 + B t,j = B i,j }, i.e., the largest index which achieves the minimum in (). Yao then roves two Lemmas (see Figure 1 for an examle): Lemma 1 (Lemma 2.1 in [1]) If w(i j) satisfies the uadrangle ineuality as defined in Definition 1, and is also monotone on the lattice of intervals, then the B i,j defined in () also satisfy the uadrangle ineuality. Lemma 2 (Lemma 2.2 in [1]) If the function defined in () satisfies the uadrangle ineuality then K B (i, j) K B (i, j + 1) K B (i + 1, j + 1) for i < j Lemma 1 roves that a QI in the w(i, j) imlies a QI in the B i,j. Suose then that we evaluate the values of the B i,j in the order d = 1, 2,...,n, where, for each fixed d, we evaluate all of B i,i+d, i =, 1, n d. Then Lemma 2 says that B i,i+d can be evaluated in time O(K B (i + 1, i + d) K B (i, i + d 1)). Note that n d i= (K B (i + 1, i + d) K B (i, i + d 1)) K B (n d + 1, n) n and thus all entries for fixed d can be calculated in O(n) time. Summing over all d, we see that all B i,j can be obtained in O(n 2 ) time. As mentioned, Lemma 2 and the resultant O(n 2 ) running time have usually been viewed as unrelated to the SMAWK algorithm. While they seem somewhat similar (a QI leading to an order of magnitude seedu) they aeared not to be directly connected. The main theoretical result of this aer is the observation that if the w(i, j) satisfy the QI and are monotone in the lattice of intervals, then the B i,j defined by () can be derived as the row-minima of a seuence of O(n) different totally monotone matrices, each of size O(n) O(n), where the entries in a matrix deend uon the row-minima of revious matrices in the seuence. In fact, we will show three totally different decomosition of the B i,j into O(n) totally monotone matrices. In articular, our first decomosition, will ermit the direct use of SMAWK.

5 1. Online Algorithms Generally, an online roblem is defined to be a roblem where a stream of oututs must be generated in resonse to a stream of inuts, and where those resonses must be given under a rotocol which reuires some oututs be given before all inuts are known. The online versions of the roblems in which we are interested are given below. Our goal is to achieve the otimal result, while maintaining the same asymtotic time comlexity as the offline versions. Let L R be given along with values w(i, j) for all L i j R that satisfy the QI and the monotone on lattice of intervals roerty. Let { if i = j B i,j = w(i, j) + min i<t j {B i,t 1 + B t,j } if i < j and assume all B ij for L i j R have already been calculated and stored. The Right-online roblem is: Given new values w(i, R + 1) for L i R + 1, such that w(i, j) still satisfy the QI and lattice roerty, calculate all of the values B i,r+1 for L i R + 1. The Left-online roblem is: Given new values w(l 1, j) for L 1 j R, such that w(i, j) still satisfy the QI and lattice roerty, calculate all of the values B L 1,j for L 1 j R. These online roblems restricted to the otimal binary search tree would be to construct the OBST for items Key L,...,Key R, and, at each ste, add either Key R+1, a new key to the right, or Key L 1, a new key to the left. Every time a new element is added, we want to udate the B i,j (dynamic rogramming) table and thereby construct the otimal binary search tree of the new full set of elements. (See Figure 1.) To achieve this, it is certainly ossible to recomute the entire table; however this comes at the rice of O(n 2 ) time, where n = R L + 1 is the number of keys currently in the table. What we are interested in here is the uestion of how one can handle a new key where the extra comutational work is neutral to the overall comlexity of the roblem, i.e., a new key can be added in linear time. Our goal is an algorithm in which a seuence of n online key insertions will result in a worst case O(n) er ste to maintain an otimal tree, yielding an overall run time of O(n 2 ). Unfortunately, the KY seedu cannot be used to do this. The reason that the seedu fails is that the KY seedu is actually an amortization over the evaluation of all entries when done in a articular order. In the online case, adding a new item n to reviously existing items 1, 2,...,n 1, reuires using () to comute the n new entries B i,n, in the fixed order i = n, n 1,...,1, and it is not difficult to construct an examle in which calculating these new entries in this order using () reuires Θ(n 2 ) work. We will see later that the decomosition given in section 3 ermits a fully online algorithm with no enalty in erformance, i.e., after adding the n th new key, the new B i,j can be calculated in O(n) worst case time. Furthermore, this will be true for both the left-online and right-online case.

6 Figure 1: An examle of the online case for otimal binary search trees where (,,, ) = (2, 9, 38, 8) and ( 3,,,, ) = (2, 9, 31,, 1). The leftmost table contains the B i,j values; the rightmost one, the K B (i, j) values. The unshaded entries in the table are for the roblem restricted to only keys,. The dark gray cells are the entries added to the table when key is added to the right. The light gray cells are the entries added when key is added to the left. The corresonding otimal binary search trees are also given, where circles corresond to successful searches and suares to unsuccessful ones. The values in the nodes are the weights of the nodes (not their keys). 2 The First Decomosition Definition For 1 d < n define the (n d + 1) (n + 1) matrix D d by Di,j d = { w(i, i + d) + Bi,j 1 + B j,i+d if i < j i + d n, otherwise. () Figure 2 illustrates the first decomosition. Note that () immediately imlies B i,i+d = min j n Dd i,j () so finding the row-minima of D d yields B i,i+d, i =,...,n d. Put another way, the B i,j entries on diagonal j i = d are exactly the row-minima of matrix D d. Lemma 3 If the function B i,j defined in () satisfies the QI then, for each d n, D d is a totally monotone matrix Proof : From Observation 3 it suffices to rove that Note that if i < j j + d, then from Lemma 1, D d i,j + D d i+1,j+1 D d i+1,j + D d i,j+1 () B i,j 1 + B i+1,j B i+1,j 1 + B i,j (8)

7 i j D 12 D 11 D 1 D 9 D 8 D D D D D 3 D 2 D 1 i j Row-Min B,8 B 1,9 B 2,1 B 3,11 B,12 Figure 2: The lefthand figure shows the B i,j matrix for n = 12. Each diagonal, d = i j, in the matrix will corresond to a totally monotone matrix D d. The minimal item of row i in D d will be the value B i,i+d. The righthand figure shows D 8. and Thus, B j,i+d + B j+1,i+1+d B j,i+1+d + B j+1,i+d. (9) D d i,j + D d i+1,j+1 = [w(i, i + d) + B i,j 1 + B j,i+d ] + [w(i + 1, i + d + 1) + B i+1,j + B j+1,i+1+d ] = w(i, i + d) + w(i + 1, i + d + 1) + [B i,j 1 + B i+1,j ] + [B j,i+d + B j+1,i+1+d ] w(i, i + d) + w(i + 1, i + d + 1) + [B i+1,j 1 + B i,j ] + [B j,i+d+1 + B j+1,i+d ] = [w(i + 1, i + d + 1) + B i+1,j 1 + B j,i+1+d ] + [w(i, i + d) + B i,j + B j+1,i+d ] = D d i+1,j + D d i,j+1 and () is correct (where we note that the right hand side is if i j or j i + d). Lemma Assuming that all of the row-minima of D 1, D 2,...,D d 1 have already been calculated, all of the row-minima of D d can be calculated using the SMAWK algorithm in O(n) time. Proof : From the revious lemma, D d is a totally monotone matrix. Also, by definition, its entries can be calculated in O(1) time, using the reviously calculated row-minima of D d where d < d. Thus SMAWK can be alied. Combined with () this immediately gives a new O(n 2 ) algorithm for solving the KY roblem; just run SMAWK on the D d in the order d = 1, 2,...,n 1 and reort all of the row-minima. We oint out that this techniue cannot hel us solve the online roblem as defined in subsection 1., though. To see why, suose that items 1,...,n have reviously been given, new item n + 1 has just been added, and we need to calculate the values B i,n+1 for i =,...,n. In our formulation this would corresond to adding a new bottom row to every matrix D d and creating a new matrix D n+1. In our formulation, we would need to find the row-minima of all of the n new bottom rows. Unfortunately, the SMAWK algorithm only works on the rows of matrices all at once and cannot hel to find the row-minima of a single new row.

8 i j R 1 R R 3 1 R R 1 R 1 R R 8 1 R 9 1 R 1 R 11 i j Row-Min B,8 B 1,8 B 2,8 B 3,8 B,8 B,8 B,8 B,8 R 12 Figure 3: The lefthand figure shows the B i,j matrix for n = 12. Each column in the B i,j matrix will corresond to a totally monotone matrix R m. The minimal element of row i in R m will be the value B i,m. The righthand figure shows R 8. 3 The Second & Third Decomositions So far we have seen that it is ossible to derive the KY running time via reeated calls to the SMAWK algorithm. We now see two more decomositions into totally-monotone matrices. These decomositions will trivially imly Lemma 2 (Lemma 2.1 in [1]), which is the basis of the KY seedu. Thus, the KY seedu is just a conseuence of total-monotonicity. These new decomositions will also ermit us to efficiently solve the online roblem given in subsection 1.. The second decomosition is indexed by the rightmost element seen so far. See Figure 3. Definition For 1 m n define the (m + 1) (m + 1) matrix R m by { Ri,j m w(i, m) + Bi,j 1 + B = j,m if i < j m, otherwise. (1) Note that () immediately imlies B i,m = min j m Rm i,j (11) so finding the row-minima of R m yields B i,m for i =,...,m. Put another way, the B i,j entries in column m are exactly the row minima of R m. The third decomosition is similar to the second excet that it is indexed by the leftmost element seen so far. See Figure. Definition 8 For m < n define the (n m) (n m) matrix L m by { L m w(m, j) + Bm,i 1 + B j,i = i,j if m < i j n, otherwise. (12) (For convenience, we set the row and column indices to run from (m+1)...n and not 1...(n m).) Note that () immediately imlies B m,j = min m<i j n Lm j,i (13) 8

9 i j L L 1 L 2 L 3 L L L L L 8 L 9 L 1 L 11 j i Row-Min B 8,9 B 8,1 B 8,11 B 8,12 Figure : The lefthand figure shows the B i,j matrix for n = 12. Each row in the B i,j matrix will corresond to a totally monotone matrix L m. The minimal element of row j in L m will be the value B m,j. The righthand figure shows L 8. so finding the row-minima of L m yields B m,j for j = m+1,...,n. Put another way, the B i,j entries in row m are exactly the row minima of matrix L m. Lemma If the function defined in () satisfies the QI then R m and L m are totally monotone matrices. Proof : The roofs are very similar to that of Lemma 3. Note that if i < j m, we can again use (8); writing the entries from (8) in boldface gives R m i,j + R m i+1,j+1 = [w(i, m) + B i,j 1 + B j,m ] + [w(i + 1, m) + B i+1,j + B j+1,m ] [w(i + 1, m) + B i+1,j 1 + B j,m ] + [w(i, m) + B i,j + B j+1,m ] = R m i+1,j + R m i,j+1 and thus R m is Monge (where we note that the right hand side is if i j) and thus totally monotone. If m < i < j then we again use (8) (with j relaced by j + 1) to get L m j,i + L m j+1,i+1 = [w(m, j) + B m,i 1 + B i,j ] + [w(m, j + 1) + B m,i + B i+1,j+1 ] [w(m, j + 1) + B m,i 1 + B i,j+1 ] + [w(m, j) + B m,i + B i+1,j ] = L m j+1,i + L m j,i+1 and thus L m is Monge (where we note that the right hand side is if i j) and thus totally monotone. We oint out these two decomositions immediately imly a new roof of Lemma 2 (Lemma 2.1 in [1]) which states that K B (i, j) K B (i, j + 1) K B (i + 1, j + 1). (1) To see this note that K B (i, j +1) is the location of the rightmost row-minimum of row i in matrix R j+1, while K B (i + 1, j + 1) is the location of the rightmost row-minimum of row i + 1 in matrix R j+1. Thus, the definition of total monotonicity (Definition 2) immediately gives K B (i, j + 1) K B (i + 1, j + 1). (1) 9

10 Similarly, K B (i, j) is the rightmost row-minimum of row j in L i while K B (i, j + 1) is the location of the rightmost row-minimum of row j + 1 in L i. Thus K B (i, j) K B (i, j + 1). (1) Combining (1) and (1) yields (1), which is what we want. Since the actual seedu in the KY techniue comes from an amortization argument based on (1), we have just seen that the original KY-seedu itself is also a conseuence of total monotonicity. We have still not seen how to actually calculate the B i,j using the R m and L m. Before continuing, we oint out that even though the R m are totally monotone, their row minima cannot be calculated using the SMAWK algorithm. This is because, for < i < j m, the value of entry R m i,j = w(i, m) + B i,j 1 + B j,m, which is deendent uon B j,m which is itself the row-minimum of row j in the same matrix R m. Thus, the values of the entries of R m deend uon the other entries in R m which is something that SMAWK does not allow. The same roblem occurs with the L m. We will now see that, desite this deendence, we can still use the LARSCH algorithm to find the row-minima of the R m. This will have the added advantage of solving the online roblem as well. At this oint we should note that our decomositions L m could also be derived by careful cutting of the 3-D monotone matrices of Aggarwal and Park [2] along articular lanes. Aggarwal and Park used an algorithm of Wilber [1] (derived for finding the maxima of certain concaveseuences) to find various tube maxima of their matrices, leading to another O(n 2 ) algorithm for solving the KY-roblem. In fact, even though their algorithm was resented as a static algorithm, careful decomosition of what they do ermits using it to solve what we call the left-online KYroblem. A symmetry argument could then yield a right-online algorithm. This never seems to have been noted in the literature, though. In the next section, we resent a different online algorithm, based on our decomositions and the LARSCH algorithm. Online Algorithms Without Losing the KY Seedu To execute the LARSCH algorithm, as defined in Section 3 of [1] we need only that X satisfy the following conditions: 1. X is a totally monotone n m matrix. 2. For each row index i of X, there is a column index C i such that for j > C i, X i,j = Furthermore, C i C i Let C =. If C i < j C i+1 then, for any k > i, X k,j can be evaluated in O(1) time rovided that the row minima of the first i rows are already known. If these conditions are satisfied, the LARSCH algorithm then calculates all of the row minima of X in O(n + m) time. We can now use this algorithm to derive Lemma Given that all values B i,j, m < i j n have already been calculated, all of the row-minima of L m can be calculated in O(n m) time. Given that all values B i,j, i j < m have already been calculated, all of the row-minima of R m can be calculated in O(m) time. Proof : For the first art, it is easy to see that L m satisfies the first two conditions reuired by the LARSCH algorithm with C j = j. The third condition is euivalent to saying that if C i < j C i+1, 1

11 i.e., if j = i+1, then k > i, L m k,j = Lm k,i+1 = w(m, k)+b m,i+b i+1,k can be evaluated in O(1) time rovided that the row minima of the first i rows are already known. But, w(m, k) and B i+1,k are already known and can be retrieved in O(1) time and B m,i is the row-minimum of row i of L m so this condition is trivially satisfied. Thus, all of the row-minima of the (n m) (n m) matrix L m can be calculated in O(n m) time. For the second art, set X to be the (m+1) (m+1) matrix defined by X i,j = Rm i,m j m. Then X satisfies the first two LARSCH conditions with C i = i 1. The third condition is euivalent to saying that if C i < j C i+1, i.e., if j = i, then k > i, X k,j = Rm k,m i m = w(m k, m) + B m k,m i 1 + B m i,m can be evaluated in O(1) time rovided that the row minima of the first i rows are already known. But w(m k, m) and B m k,m i 1 are already known and can be retrieved in O(1) time and B m i,m is the row-minimum of row i of X so this condition is also trivially satisfied. Thus, all of the row-minima of X and euivalently of R m can be calculated in O(m) time. Note that Lemma immediately solves the right-online and left-online roblems described in subsection 1.; Given the new values w(i, R + 1) for L i R + 1, simly find the row minima of R R+1 in time O(R L). Given the new values w(l 1, j) for L 1 j R, simly find the row minima of L L 1. We have therefore just shown that any dynamic rogramming roblem for which the KY seedu can statically imrove run time from Θ(n 3 ) to O(n 2 ) time can be solved in an online fashion in O(n) time er ste. That is, online rocessing incurs no enalty comared to static rocessing. In articular, the otimum binary search tree (as illustrated in 1.), can be maintained in O(n) time er ste as nodes are added to both its left and right. A further alication In [], Borchers and Guta extend the Knuth-Yao uadrangle ineuality. One of their major alications is finding an otimal Rectilinear Steiner Minimal Arborescence (RSMA) of a slide. A slide is a set of oints (x i, y i ) such that, if i < j, then x i < x j and y i > y j. A Rectilinear Steiner Arborescence is a directed tree in which each edge either goes u or to the right. In [11] it was shown that the minimum cost Rectilinear Steiner Arborescence connecting slide-oints (x i, y i ), (x i+1, y i+1 ),...(x j, y j ) satisfies L(i, j) = min i s<j (L i,s + L s+1,j + x s+1 x 1 + y s y j ). (1) [11] solved this recurrence in O(n 3 ) time. Even though (1) is not in the KY form () Borchers and Guta [] were still able to show that L i,j satisfies a uadrangle ineuality. This sufficed to show that Lemma 2 still holds for L i.j and thus ermitted deriving an O(n 2 ) algorithm for calculating all of the L i,j. In fact, they showed that if L(i, j) is given by a DP recurrence of the form L(i, j) = min i s<j (w(i, s, j) + L i,s + L s+1,j ). (18) where w(i, s, j) satisfies generalized versions of the uadrangle ineuality and monotonicity on integer lattices roerty, then Yao s result could alsways be generalized to aly. Note that the major difference between () and (18) is that (18) allows w() to deend uon the slitting index s and be brought inside the min. We oint out that even though we derived our results for roblems that satisfy the KY form () it is uite straightforward to show all of the results in this aer can be extended to work for all L i,j that satisfy (18) with the Borchers and Guta seedu conditions. In articular, this yields an O(n) er-ste worst-case online algorithm for constructing the new RSMA when adding oints to the left and right of a slide. 11

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