Restricted rotation distance between binary trees

Computing the rotation distance between two binary trees with n internal nodes efficiently (in poly( n) time) is a long standing open question in the study of height balancing in tree data structures. In this paper, we initiate the study of this problem bounding the rank of the trees given at the input (defined in [1] in the context of decision trees). We define the rank-bounded rotation distance between two given full binary trees T1 and T2 (with n internal nodes) of rank at most r = max{rank( T1), rank( T2)}, denoted by dR( T1, T2), as the length of the shortest sequence of rotations that transforms T1 to T2 with the restriction that the intermediate trees must be of rank at most r. We show that the rotation distance problem reduces in polynomial time to the rank bounded rotation distance problem. This motivates the study of the problem in the combinatorial and algorithmic frontiers. Observing that trees with rank 1 coincide exactly with skew trees (full binary trees where every internal node has at least one leaf as a child), we show the following results in this frontier: We present an O( n2) time algorithm for computing dR( T1, T2). That is, when the given full binary trees are skew trees (we call this variant the skew rotation distance problem) - where the intermediate trees are restricted to be skew as well. In particular, our techniques imply that for any two skew trees dR( T1, T2) ≤ n2. We show the following upper bound: for any two full binary trees T1 and T2 of rank r1 and r2 respectively, we have that: dR( T1, T2) ≤ n2(1 + (2 n + 1)( r1 + r2 – 2)) where r = max{ r1, r2}. This bound is asymptotically tight for r = 1. En route to our proof of the above theorems, we associate full binary trees to permutations and relate the rotation operation on trees to transpositions in the corresponding permutations. We give exact combinatorial characterizations of permutations that correspond to full binary trees and full skew binary trees under this association. We also precisely characterize the transpositions that correspond to the set of rotations in full binary trees. We also study bi-variate polynomials associated with binary trees (introduced by [2]), and show characterizations and algorithms for computing rotation distances for the case of full skew trees using them.

There are various ways to measure the shape difference between two n-node rooted binary trees (binary trees for short). A rotation on a binary tree is a local restructuring that changes the tree into another one preserving the in-order sequence. The rotation distance between two binary trees is the minimum number of rotations needed to transform one into another. Till now, no polynomial–time algorithm exists for computing the rotation distance between any two binary trees. Recently, Lucas (Comput. J., 47, 259–269, 2004) presented an O(n2)–time algorithm for finding the rotation distance between two binary trees, where the source tree is a degenerate tree and the destination tree is an angle tree. This paper improves the time-complexity to O(n) under this constraint.

We study restricted rotation distance between ternary and higher-valence trees using approaches based upon generalizations of Thompson's group $F$. We obtain bounds and a method for computing these distances exactly in linear time, as well as a linear-time algorithm for computing rotations needed to realize these distances. Unlike the binary case, the higher-valence notions of rotation distance do not give Tamari lattices, so there are fewer tools for analysis in the higher-valence settings. Higher-valence trees arise in a range of database and filesystem applications where balance is important for efficient performance.

Lower bounds on the rotation distance of binary trees

Encoding the shape of a binary tree is a basic step in a number of algorithms in integrated circuit design, automated theorem proving, and game playing. We propose cost-optimal parallel algorithms to solve the binary tree encoding/decoding problem. Specifically, we encode the relevant shape information of an n-node binary tree in a 2n bitstring. Conversely, given an arbitrary 2n bitstring we reconstruct the shape of the corresponding binary tree, if such a tree exists. All our algorithms run in O (log n) time using O (n/log n) processors in the EREW-PRAM model of computation.

Phylogenetic trees are a central tool in many areas of life science and medicine. They demonstrate evolutionary patterns among species, genes, and patterns of ancestry among sets of individuals. The tree shapes and branch lengths of phylogenetic trees encode evolutionary and epidemiological information. To extract information from tree shapes and branch lengths, representation and comparison methods for phylogenetic trees are needed. Representing and comparing tree shapes and branch lengths of phylogenetic trees are challenging, for a tree shape is unlabeled and can be displayed in numerous different forms, and branch lengths of a tree shape are specific to edges whose positions vary with respect to the displayed forms of the tree shape. In this article, we introduce representation and comparison methods for rooted unlabeled phylogenetic trees based on a tree lattice that serves as a coordinate system for rooted binary trees with branch lengths and a graph polynomial that fully characterizes tree shapes. We show that the introduced tree representations and metrics provide distance-based likelihood-free methods for tree clustering, parameter estimation, and model selection and apply the methods to analyze phylogenies reconstructed from virus sequences. [Graph polynomial; likelihood-free inference; phylogenetics; tree lattice; tree metrics.].

Binary search trees built from the postorder traversal sequence of other binary search trees are characterized in terms of their binary tree structure. A connection is established between this structure and the Eulerian numbers. This yields considerable information concerning the “average” binary search tree with a given number of nodes. Periodicity is established for the process of repeatedly taking postorder sequences and building binary search trees. Finally, the average depth of a node in a postorder tree withn nodes is determined.

We investigate the search cost in terms of number of messages generated for routing queries in tree-based P2P structured systems including binary and k-ary tree structures with different arities and different degrees of imbalance in the tree shape. This work is motivated by the fact that k-ary balanced tree access structures can greatly reduce the number of hops for searching compared to the binary trees. We study to what extent the same fact is true when the tree-like structures for access in P2P environments are unbalanced. Another important issue related to P2P environments is how to build these structures in a self-organizing way. We propose a mechanism for constructing k-ary tree based decentralized access structure in a self-organizing way and based on local interactions only. The ability to search efficiently also on unbalanced k-ary trees opens interesting opportunities for load balancing as has been shown in earlier work on P-Grid, our approach to structured P2P systems.KeywordsCommunication CostAccess StructureSearch CostDistribute Hash TableRout Table EntryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper describes two types of problems related to tree shapes, as well as algorithms that can be used to solve these problems. The first problem is that of comparing the similarity of the unlabelled shapes instead of merely their degree of balance, in a manner analogous to that routinely used to compare topologies for labelled trees. There are possible practical applications for this comparison, such as determining, based on tree shape similarity alone, whether the taxa in two phylogenies are likely to have a correspondence (e.g. hosts and parasites with high specificity). It is shown that tree balance is insufficient for this task and that standard measures of topological difference (Robinson–Foulds distances, SPR distances or retention indices of the matrices representing the trees, MRPs) can be easily adapted to the problem. The second type of problem is to determine whether taxa of uncertain matching unique to two different phylogenies could correspond to each other (e.g. the same species in larvae and adults of metamorphic animals, fossils known from different body parts). This second problem can be solved by either relabelling taxa in such a way that the number of consensus nodes is maximized, or relabelling taxa in such a way that the sum of the number of steps in the MRP of each tree mapped onto the other is minimum.

On the rotation distance in the lattice of binary trees

This work is a case study of the formal verification and complexity analysis of some famous probabilistic algorithms and data structures in the proof assistant Isabelle/HOL. In particular, we consider the expected number of comparisons in randomised quicksort, the relationship between randomised quicksort and average-case deterministic quicksort, the expected shape of an unbalanced random Binary Search Tree, the randomised binary search trees described by Martínez and Roura, and the expected shape of a randomised treap. The last three have, to our knowledge, not been analysed using a theorem prover before and the last one is of particular interest because it involves continuous distributions.

The incrementally extensible hypercube (IEH) graph is a generalization of binary hypercube networks in that the number of nodes can be arbitrary in contrast to a strict power of 2. In this paper, the authors present an efficient model to fulfill the embedding of a full binary tree into a full IEH graph. As the model the authors proposed, an algorithm is developed for the embedding with expansion 1, load 1, congestion 2, and dilation 2. Moreover, the embedding algorithm can be also applied into hypercubes with one faulty node.

Kakkot Tree is a Binary Searching Tree with the caching concept added. It is used to speed up the searching process that is carried on a Binary Search Tree. The time complexity of BST varies between O(n) and O(log n) depends upon the number of nodes that are present in left and right children. Kakkot Tree is a new data structure which is a variant of BST with some initial nodes which act like caching.

On subtrees of trees

We design an efficient sublinear time parallel construction of optimal binary search trees. The efficiency of the parallel algorithm corresponds to its total work (the product time × processors). Our algorithm works in O(n1−ɛ log n) time with the total work O(n2−2ɛ), for an arbitrarily small constant 0 < e ≤ 1/2. This is optimal within a factor n2ɛ with respect to the best known sequential algorithm given by Knuth, which needs only O(n2) time due to a monotonicity property of optimal binary search trees, see [6]). It is unknown how to explore this property in an efficient NC construction of binary search trees. Here we show that it can be effectively used in sublinear time parallel computation. Our improvement also relies on the use (in independently processed small subcomputations) of the parallelism present in Knuth's algorithm. The best known sublinear time algorithms for the construction of binary search trees (as an instance of a more general problem) have O(n3) work for time larger than n3/4, see [3] and [7]. For time √n these algorithms need n4 work, while our algorithm needs for this time only n3 work, thus improving the known algorithms by a linear factor. Also if time is O(n1−ɛ) and e is very small our improvement is close to O(n). Such improvement is similar to the one implied by the monotonicity property in sequential computations (from n3 sequential time for a more general dynamic programming problem to n2 time for the special case of optimal binary search trees).