k-Restricted rotation distance between binary trees

The restricted rotation distancedR(S,T) between two binary trees S, T of n vertices is the minimum number of rotations to transform S into T, where rotations take place at the root of S, or at the right child of the root. A sharp upper bound dR(S,T)?4n−8 is known, based on group theory [S. Cleary, J. Taback, Bounding restricted rotation distance, Information Processing Letters 88 (5) (2003) 251-256]. We refine this bound to a sharp dR(S,T)?4n−8−ρS−ρT, where ρS and ρT are the numbers of vertices in the rightmost vertex chains of the two trees, using an elementary transformation algorithm. We then generalize the concept to k-restricted rotation, by allowing rotations to take place at all the vertices of the highest k levels of the tree, and study the new distance for k=2. The case k?3 is essentially open.     

Restricted rotation distance between binary trees

Restricted rotation distance between pairs of rooted binary trees measures differences in tree shape and is related to rotation distance. In restricted rotation distance, the rotations used to transform the trees are allowed to be only of two types. Restricted rotation distance is larger than rotation distance, since there are only two permissible locations to rotate, but is much easier to compute and estimate. We obtain linear upper and lower bounds for restricted rotation distance in terms of the number of interior nodes in the trees. Further, we describe a linear-time algorithm for estimating the restricted rotation distance between two trees and for finding a sequence of rotations which realizes that estimate. The methods use the metric properties of the abstract group known as Thompson's group F.     

The Fermat star of binary trees

A Fermat point P is one that minimizes the sum δ of the distances between P and the points of a given set. The resulting arrangement, called here a Fermat star, is a particular Steiner tree with only one intermediate point. We extend these concepts to rooted binary trees under the known rotation distance that measures the difference in shape of such trees. Minimizing δ is hard, due to the intrinsic difficulty of computing the rotation distance. Then we limit our study to establishing significant upper bounds for δ. In particular, for m binary trees of n vertices, we show how to construct efficiently a Fermat star with δ?mn−3m, with a technique inherited from the studies on rotation distance.     

An efficient algorithm for estimating rotation distance between two binary trees

There are a number of ways of measuring the difference in shape between two rooted binary trees with the same number of leaves. Pallo (Computer Journal, 9, 171–175, 1986) introduced a left weight sequence, which is a sequence of positive integers, to characterize the structure of a binary tree. By applying the AVL tree transformation on binary trees, we develop an algorithm for the efficient transformation of the left weight sequences between two binary trees.     

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 right-arm rotations necessary to transform one tree into the other.     

Bounding restricted rotation distance

Restricted rotation distance between pairs of rooted binary trees quantifies differences in tree shape. 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 n−2, again with the requirement that the trees satisfy a reduction condition. These improvements use work of Fordham to compute the word metric in Thompson's group F.     

Note on the edge rotation distance between trees

In this paper, it is shown that the problem of computing the edge rotation distance between trees is NP-hard.     

Lower bounds for maximal and convex layers problems

In this paper we derive tight lower bounds for the maximal and convex layers problems in the plane. Our lower bound proofs for the maxima problem and convex hull problem are simpler than those previously known. We also obtain an (nlog n) lower bound for the maximal depth problem, and the convex depth problem, when the points are given in sorted order of their x-coordinates.     

The geometry and dynamics of binary trees

The modeling of a fully populated 3D tree able to regulate dynamically remains a relatively unexplored field. A non-dimensional representation of “autoregulation” coupled with an asymmetric binary tree algorithm has been developed. The tree has a defined topology as well as a spatial representation in 3D. An analysis using a simple linearization shows the systems dynamics when perturbed away from equilibrium. Results, based on previously published work by Karch and Schreiner are presented for a variety of parameters which provide different shapes of the tree and indicate a possible mechanism for “growing” the tree in specified directions. In addition the tree, through the use of local tagging has the ability to vary its size locally via a coupled set of conservation and reverting differential equations.     

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.     

An analytic approach to the height of binary search trees

By using analytic tools it is shown that the expected value of the heightH _n of binary search trees of sizen is asymptotically given by EH _n =c logn+ (log logn) and its variance by VH _n = ((log logn)²), wherec=4.31107 …. The same bounds have been obtained by Devroye and Reed [3] by completely different methods. Dedicated to Philippe Flajolet on the occasion of his 50th birthday This research was supported by the Austrian Science Foundation FWF, Grant P10187-MAT. Online publication September 22, 2000.     

Generalized non-recursive traversal of binary trees

A non-recursive algorithm for the traversal of a binary tree is presented in which the order of traversal is defined by an external data array, allowing any of the six possible orders to be selected without modification to the algorithm itself. The extra storage requirements are three pointer variables and two bits per node (or two bits per level if an auxiliary stack is used.) The algorithm is a generalization of the pointer reversal method of Schorr and Waite and is derived by transformation of a generalized recursive version. The algorithm is described using the notation and conventions of Pascal.     

Treewidth computations II. Lower bounds

For several applications, it is important to be able to compute the treewidth of a given graph and to find tree decompositions of small width reasonably fast. Good lower bounds on the treewidth of a graph can, amongst others, help to speed up branch and bound algorithms that compute the treewidth of a graph exactly. A high lower bound for a specific graph instance can tell that a dynamic programming approach for solving a problem is infeasible for this instance. This paper gives an overview of several recent methods that give lower bounds on the treewidth of graphs.     

The Variance of the height of binary search trees

By using analytic tools it is shown that the variance E(H_n−EH_n)² of the height H_n of binary search trees of size n is bounded.     

The semantic and syntactic manipulation of vacancies in associative binary trees

Execution of sub-processes within a program segment are subject to a partial ordering. In certain cases (such as expressions and assignment statements) this ordering reduces to a tree which, according to the characteristics of the operators present, may be manipulated to influence the extent to which parallel processing capabilities of multiple-processor configurations can be utilized in its evaluation. A strategy is presented which uses associativity of certain operators to adjust the shape of the trees to allow a degree of overlap between adjacent subtrees. Although only optimal in the local sense, the transformation yields significant improvements in the “parallel dimensions” of the tree and, more importantly, can be couched in syntactic terms. Consequently, it is possible in principle to perform these manipulations within the syntax analysis phase of compilation, regardless of other operational characteristics of the operators, or of the parallel capabilities of the target run-time system.     

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.