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### An efficient upper bound of the rotation distance of binary trees

Jean Pallo
2000 Information Processing Letters
A polynomial time algorithm is developed for computing an upper bound for the rotation distance of binary trees and equivalently for the diagonal-flip distance of convex polygons triangulations.  ...  In Section 2, we have exhibited an upper bound δ(T , T ) of the rotation distance d(T , T ) using ordinal tools. According to the above theorem, an other upper bound is given by δ(f (T ), f (T )).  ...  If f (T ) is the flexed tree corresponding to T , then we have d(f (T ), f (T )) = d(T , T ). We deduce in Section 4 an efficient algorithm for computing an upper bound µ(T , T ) of d(T , T ).  ...

### A Linear-Time Approximation Algorithm for Rotation Distance

Sean Cleary, Katherine St. John
2010 Journal of Graph Algorithms and Applications
In this short note, we give an efficient, linear-time approximation algorithm, which estimates the rotation distance, within a provable factor of 2, between ordered rooted binary trees.  ...  Rotation distance between rooted binary trees measures the number of simple operations it takes to transform one tree into another.  ...  By rotating to right caterpillar trees, Culik and Wood  gave an immediate upper bound of 2n − 2 for the distance between two trees with n interior nodes.  ...

### A Linear-Time Approximation Algorithm for Rotation Distance [article]

Sean Cleary, Katherine St. John
2009 arXiv   pre-print
We give an efficient, linear-time approximation algorithm, which estimates the rotation distance, within a provable factor of 2, between ordered rooted binary trees. .  ...  Rotation distance between rooted binary trees measures the number of simple operations it takes to transform one tree into another.  ...  By rotating to right caterpillar trees, Culik and Wood  gave an immediate upper bound of 2n − 2 for the distance between two trees with n interior nodes.  ...

### Expected maximum vertex valence in pairs of polygonal triangulations

Timothy Chu, Sean Cleary
2015 Involve. A Journal of Mathematics
Edge-flip distance between triangulations of polygons is equivalent to rotation distance between rooted binary trees. Both distances measure the extent of similarity of configurations.  ...  The best known exact universal upper bounds on rotation distance arise from measuring the maximum total valence of a vertex in the corresponding triangulation pair obtained by a duality construction.  ...  Rotations in binary trees correspond to edge-flip moves in such triangulations of polygons, so the rotation distance between two rooted ordered binary trees corresponds exactly to the edge-flip distance  ...

### Bounding restricted rotation distance

Sean Cleary, Jennifer Taback
2003 Information Processing Letters
Cleary exhibited a linear upper bound of 12n for the restricted rotation distance between two trees with n interior nodes, and a lower bound of (n − 1)/3 if the two trees satisfy a reduction condition.  ...  We obtain a significantly improved sharp upper bound of 4n − 8 for restricted rotation distance between two rooted binary trees with n interior nodes, and a significantly improved sharp lower bound of  ...  Using metric estimates in this group, Cleary obtains an upper bound of 12n on the restricted rotation distance between two binary trees with n nodes.  ...

### The Fermat star of binary trees

Fabrizio Luccio, Linda Pagli
2009 Information Processing Letters
As an efficient procedure for determining the rotation distance of two trees is unknown, we will study upper bounds on the total number δ of rotations needed for transforming the given trees to a Fermat  ...  The technique of  was then adopted by Baril and Pallo for evaluating lower and upper bounds on this distance efficiently  .  ...

### Distributions of restricted rotation distances

2021 The Art of Discrete and Applied Mathematics
There are linear upper and lower bounds on restricted rotation distance with respect to the sizes of the reduced tree pairs.  ...  The associated restricted rotation graph has an efficient distance algorithm.  ...  We note that trees realizing the lower bound of restricted rotation distance from  would have an RRD ratio limiting to 1, and those realizing the upper bound would have an RRD ratio limiting to 4  ...

### An Efficient Sampling Algorithm for Difficult Tree Pairs

Sean Cleary, Roland Maio
2022 Acta Cybernetica
between an arbitrary pair of trees, (S, T), can be efficiently reduced to the problem of computing the rotation distance between a difficult pair of trees (S', T'), where there is no known first step  ...  It is an open question whether there exists a polynomial-time algorithm for computing the rotation distances between pairs of extended ordered binary trees.The problem of computing the rotation distance  ...  In many of these very specific cases, analysis to that family of instances can give coincident upper and lower bounds on rotation distance, giving an exact calculation.  ...

### An efficient sampling algorithm for difficult tree pairs [article]

Sean Cleary, Roland Maio
2020 arXiv   pre-print
The problem of computing the rotation distance between an arbitrary pair of trees, (S, T), can be efficiently reduced to the problem of computing the rotation distance between a difficult pair of trees  ...  It is an open question whether there exists a polynomial-time algorithm for computing the rotation distances between pairs of extended ordered binary trees.  ...  One widely-considered tree distance metric on trees with a natural left-to-right order on leaves is that of the rotation distance between a pair of extended ordered binary trees.  ...

### Distributions of restricted rotation distances [article]

2020 arXiv   pre-print
There are linear upper and lower bounds on restricted rotation distance with respect to the sizes of the reduced tree pairs.  ...  The associated restricted rotation graph has an efficient distance algorithm.  ...  We note that trees realizing the the lower bound of restricted rotation distance from  would have an RRD ratio limiting to 1, and those realizing the the upper bound would have an RRD ratio limiting  ...

### Efficient lower and upper bounds of the diagonal-flip distance between triangulations

Jean-Luc Baril, Jean-Marcel Pallo
2006 Information Processing Letters
We present an efficient algorithm for computing lower and upper bounds of this distance between a pair of triangulations.  ...  There remains today an open problem whether the rotation distance between binary trees or equivalently the diagonal-flip distance between triangulations can be computed in polynomial time.  ...  Acknowledgements We are grateful to an anonymous referee for the improvement of the proofs of Theorems 4 and 5.  ...

### Restricted rotation distance between binary trees

Sean Cleary
2002 Information Processing Letters
We obtain linear upper and lower bounds for restricted rotation distance in terms of the number of interior nodes in the trees.  ...  Restricted rotation distance between pairs of rooted binary trees measures differences in tree shape and is related to rotation distance.  ...  With this minimal sufficient set of allowed rotations, we find an upper bound of 12n for the restricted rotation distance between any two rooted trees with n nodes, and a lower bound of (n − 1)/3 for the  ...

### BOUNDING RIGHT-ARM ROTATION DISTANCES

SEAN CLEARY, JENNIFER TABACK
2007 International journal of algebra and computation
We describe several types of rotation distance, and provide upper bounds on distances between trees with a fixed number of nodes with respect to each type.  ...  Rotation distance quantifies the difference in shape between two rooted binary trees of the same size by counting the minimum number of elementary changes needed to transform one tree to the other.  ...  This produces an upper bound of 4n − 8 < 4n − 8 on the restricted right-arm rotation distance between the two trees.  ...

### Page 6641 of Mathematical Reviews Vol. , Issue 90K [page]

1990 Mathematical Reviews
(I-SLRN) On the upper bound on the rotation distance of binary trees. Inform. Process. Lett. 31 (1989), no. 2, 57-60.  ...  Summary: “The rotation distance d(7\,7>) between two binary search trees 7; and 7> on nm nodes is the minimum number of rotations needed to transform 7; into 7>.  ...

### Rotation Distance is Fixed-Parameter Tractable [article]

Sean Cleary, Katherine St. John
2009 arXiv   pre-print
The proof relies on the kernalization of the initial trees to trees with size bounded by 7k.  ...  In the case of ordered rooted trees, we show that the rotation distance between two ordered trees is fixed-parameter tractable, in the parameter, k, the rotation distance.  ...  Conclusion We have shown that rotation distance of ordered trees is fixed-parameter tractable in parameter, k, the distance.  ...
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