Jean Marcel Pallo

On the rotation distance in the lattice of binary trees

A natural idea to compare tree structures is to compute a distance based on the least number of admissible transformations needed to transform a tree into another [8]. Distance measures of leastmoves type are conceptually simple, yet computationally difficult [2]. In this paper, as in [4], the basis for the distance measure is the well-known rotation tree transformation [7]. In a previous paper [10], it is proved that this rotation induces a lattice structure on binary trees. Unfortunately, algebraic results cannot be used since this lattice is neither modular nor graded [5,8]. Using the weight sequences introduced in [10], we give an O(n 2) algorithm for computing efficient lower and upper bounds of the rotation distance. The rotation distance can exactly be computed using an admissible searching Algorithm D*, which uses the lower bound as an evaluation function. tively external) node. The weight q; of a tree T is the number of external nodes of T. Let B~ denote the set of binary trees with n internal nodes (i.e., n + 1 external nodes). The external nodes of a tree T are numbered by a pre-order traversal of T. Given T ~ B n, the weight sequence of T is the integer sequence (WT(1), WT(2),...,wT(n)) where WT(i ) is the weight of the largest subtree of T, whose last external node is the external node i [10]. An ordered pair of integers (i, j ) is said to be a segment of a tree T if there exists a subtree of T with i (respectively j) as first (respectively last) external node. Every T ~ B~ has n segments (i, j ) satisfying the following properties (where we set WT(n + I) = n + 1):

An efficient upper bound of the rotation distance of binary trees

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. Ordinal tools are used.

Right-arm rotation distance between binary trees

We consider a transformation on binary trees, named right-arm rotation, which is a special instance of the well-known rotation transformation. Only rotations at nodes of the right arm of the trees are allowed. Using ordinal tools, we give an efficient algorithm for computing the right-arm rotation distance between two binary trees, i.e., the minimum number of rightarm rotations necessary to transform one tree into the other.

A shortest path metric on unlabeled binary trees

Abstract We consider a transformation on rooted unlabeled binary trees which is a special instance of the well-known general rotation. Using lattice-theoretic results, we give an efficient algorithm for computing the shortest path distance between two binary trees.

Efficient lower and upper bounds of the diagonal-flip distance between 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. We present an efficient algorithm for computing lower and upper bounds of this distance between a pair of triangulations.

A distance metric on binary trees using lattice-theoretic measures

A so called height function which is a strictly antitone supervaluation is defined on binary trees. Via lattice-theoretic results and using the height function, we can define a distance metric on binary trees of size n which can be computed in expected time O(n 3/2 )

Generating binary trees by Glivenko classes on Tamari lattices

Using algebraic-theoretic results, we give an algorithm for generating binary trees within Glivenko classes in Tamari lattices. Tamari lattices are lattices of binary trees endowed by the well-known rotation transformation.

The number of coverings in four catalan lattices

We consider in this note a series of lattices that are enumerated by the well-known Catalan numbers. We give formulas for the number of coverings, i.e. the number of edges in the Hasse diagrams of these lattices.

A-transformation dans les arbres n-aires

We introduce into the set of binary trees a transformation which always moves the branch nodes in the same direction. To study this transformation amounts to solving the following question: given two trees @s and @t, can we tell whether @s -> @t? Here we propose a characterisation of @s -> @t through a comparison on sequences of positive integers and the result is generalized to n-ary trees.

Parallel Algorithms for Listing Well-Formed Parentheses Strings

We present two cost-optimal parallel algorithms generating the set of all well-formed parentheses strings of length 2n with constant delay for each generated string. In our first algorithm we generate in lexicographic order well-formed parentheses strings represented by bitstrings, and in the second one we use the representation by weight sequences. In both cases the computational model is based on an architecture CREW PRAM, where each processor performs the same algorithm simultaneously on a different set of data. Different processors can access the shared memory at the same time to read different data in the same or different memory locations, but no two processors are allowed to write into the same memory location simultaneously. These results complete a recent parallel generating algorithm for well-formed parentheses strings in a linear array of processors model, due to Akl and Stojmenovic.