A Linear-Time Approximation Algorithm for Rotation Distance

A Linear-Time Appro ximation Algorithm for Rotation Distance Sean Cleary ∗ Katherine St. John † Septem b er 2, 2021 Abstract Rotation distance b et w een ro oted binary trees measures the num- b er of simple op erations it takes to transform one tree in to another. There are no known polynomial-time algorithms for computing rota- tion distance. W e giv e an efficient, linear-time approximation algo- rithm, whic h estimates the rotation distance, within a prov able factor of 2, b et w een ordered ro oted binary trees. 1 In tro duction Binary searc h trees are a fundamental data structure for storing and re- trieving information [4]. Roughly , a binary searc h tree is a ro oted binary tree where the no des are ordered “left to right.” The potential efficiency of storing and retrieving information in binary searc h trees dep ends on their heigh t and balance. Rotations pro vide a simple mechanism for “balancing” binary search trees while preserving their underlying order (see Figure 1). There has been a great deal of w ork on estimating, b ounding and computing rotation distances. By rotating to right caterpillar trees, Culik and W o od [5] ga v e an immediate upp er bound of 2 n − 2 for the distance b et w een t wo trees with n interior no des. In elegan t work using metho ds of h yp erb olic v olume, Sleator, T arjan, and Thurston [12] sho wed not only that 2 n − 6 is an upper b ound for n ≥ 11, but furthermore that for all very large n , that ∗ Departmen t of Mathematics, City College of New Y ork, City Univ ersity of New Y ork, New Y ork, NY 10031, cleary@sci.ccny.cuny.edu . Partial funding provided by NSF #0811002. † Departmen t of Mathematics & Computer Science, Lehman College & the Graduate Cen ter, City Univ ersity of New Y ork, Bronx, NY 10468, stjohn@lehman.cuny.edu . Partial funding pro vided by NSF #0513660. 1 right left Figure 1: A (right) rotation at a no de consists of rotating the righ t child of the left c hild of the node to the right c hild of the node. A left rotation is defined similarly b y mo ving the left child of the righ t c hild of the no de to the left child of the no de. The circled no de in the middle tree has b een rotated righ t to yield the tree on the righ t, and similarly rotated left to yield the tree on the left. b ound is realized. In remark able recent w ork, Dehorno y [7] gav e concrete examples illustrating that the low er b ound is at least 2 n − O ( √ n ) for all n . There are no kno wn p olynomial-time algorithms for computing rotation dis- tance, though there are p olynomial-time estimation algorithms of P allo [10], P allo and Baril [1], and Rogers [11]. Baril and P allo [1] use computational exp erimen tal evidence to sho w that a large fraction of their estimates are within a factor of 2 of the rotation distance. The problem has b een recently sho wn to b e fixed-parameter tractable in the parameter, k , the distance [3]. Li and Zhang [9] giv e a polynomial time appro ximation algorithm for the equiv alent diagonal flip distance with approximation ratio of almost 1.97. 1 In this short note, w e giv e a linear time appro ximation algorithm with an appro ximation ratio of 2, impro ving the running time at the v ery modest ex- p ense of approximation ratio. This is accomplished b y showing the distance b et w een the trees is b ounded b elow by n − e − 1 and ab o v e by 2( n − e − 1) where n is the num b er of internal no des and e is the num ber of edges in common in the reduced trees. The num b er of common edges is equiv alen t to Robinson-F oulds distance, widely used in phylogenetic settings, which Da y [6] calculates in linear time. 1 The exact ratio is b ounded b y the maxim um n umber of diagonals, d , allow ed at an y v ertex, and is 2 − 2 4( d − 1)( d +6)+1 . 2 2 Bac kground W e consider ordered, ro oted binary trees with n interior no des and where eac h in terior no de has t w o children. Such trees are commonly called extende d binary tr e es [8]. In the following, tr e e refers to such a tree with an ordering on the lea ves, no de refers to an in terior node, and le af refers to a non-in terior no de. Our trees will hav e n + 1 lea ves n um b ered in left-to-right order from 1 to n + 1. The size of a tree will be the n umber of internal no des it contains. Eac h in ternal edge in a tree separates the lea ves into tw o connected sets up on remo v al, and a pair of edges e 1 in S and e 2 in T form a c ommon e dge p air if their remov al in their resp ectiv e trees giv es the same partitions on the leav es. In that case, w e say that S and T ha ve a c ommon edge. Righ t rotation at a node of a ro oted binary tree is defined as a simple c hange to T as in Figure 1, taking the middle tree to the righ t-hand one. Left rotation at a no de is the natural in verse op eration. The r otation distanc e d R ( S, T ) b et ween tw o ro oted binary trees S and T with the same n umber of leav es is the minim um n um b er of rotations needed to transform S to T . The sp ecific instance of the rotation distance problem we address is: R ot a tion Dist ance: Input: Tw o ro oted ordered trees, S and T on n in ternal no des, Question: Calculate the rotation distance b et ween them, d R ( S, T ). Finding a sequence of rotations which accomplish the transformation giv es only an upp er b ound. The general difficulty of computing rotation distance comes from the low er b ound. 3 Appro ximation Algorithm W e first sho w that the rotation distance is b ounded b y the num b er of edges that differ b et w een the trees. F rom this, the appro ximation result follows easily . Theorem 1 L et S and T b e two distinct or der e d r o ote d tr e es with the same numb er of le aves. L et n b e the numb er of internal no des and e the numb er of c ommon e dges for S and T . Then, n − e − 1 ≤ d R ( S, T ) ≤ 2( n − e − 1) 3 Pr o of: The lo wer bound follows from t wo simple observ ations. First, if w e use a single rotation to transform T 1 to T 2 , all but one of the in ternal edges in eac h tree is common with the other tree. Second, ev ery in ternal edge of S that is not common with an internal edge of T needs a rotation (p ossibly more than one) to transform it to an edge in common in T . The n umber of internal edges o ccurring only in S is n − e − 1 and th us, is also a simple low er b ound. F or the upp er bound, w e use t wo facts from past w ork on rotation dis- tance. W e first let ( S 1 , T 1 ), ( S 2 , T 2 ), . . . , ( S e +1 , T e +1 ) b e the resulting tree pairs from removing the e edges S and T ha ve in common, where w e insert placeholder leav es to preserv e the extended binary tree prop ert y . Let n i b e the size of tree S i for i = 1 , 2 , . . . , e + 1. The first is the observ ation of Sleator et al. [12] used b efore: the rotation distance of the original tree pair ( S, T ) with a common edge is the sum of the rotation distances of the tw o tree pairs “abov e” and “below” the common edge. Extending this to e edges in common betw een S and T , w e hav e d R ( S, T ) = e +1 X i =1 d ( S i , T i ) ≤ e +1 X i =1 2 n i − 2 = 2 n − 2( e + 1) = 2( n − e − 1) The inequality follows b y the initial bound of 2 n − 2 on rotation distance b et w een trees with n in ternal no des of Culik and W o o d [5]. Th us, n − e − 1 ≤ d R ( S, T ) ≤ 2( n − e − 1).  W e note that using the sharp er b ound of 2 n − 6 for n > 12 from Sleator, T arjan and Thurston [12] together with the table of distances for n ≤ 12 can improv e this sligh tly still further. These reduction rules and counting the num b er of common edges can b e carried out in linear-time [2, 6], yielding the corollary: Corollary 2 L et S and T b e or der e d r o ote d tr e es with n internal no des. A 2 -appr oximation of their r otation distanc e c an b e c alculate d in line ar time. Pr o of: Let S and T b e tw o distinct ordered rooted n -leaf trees. Let n b e the num b er of internal no des and e the n um b er of edges in common for S and T . Then, b y Theorem 1, n − e − 1 ≤ d R ( S, T ) ≤ 2( n − e − 1). Since this is within a linear factor 2 from b oth b ounds, w e hav e the desired appro ximation.  W e note that this algorithm not only appro ximates rotation distance, it gives a sequence of rotations which realize the upp er b ound of the ap- pro ximation, again in linear time. The appro ximation algorithm uses the Culik-W o o d b ound on p oten tially sev eral pieces. On each piece, the 2 n − 2 4 b ound comes from rotating eac h internal no de whic h is not on the righ t side of the tree to obtain a righ t caterpillar, and then rotating the caterpillar to obtain the desired tree. This can b e accomplish simply in linear time. References [1] Jean-Luc Baril and Jean-Marcel P allo. Efficient lo w er and upper b ounds of the diagonal-flip distance b et ween triangulations. 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