A lower bound on the path length of binary trees (original) (raw)

Some upper bounds for minimal trees

Information Processing Letters, 1991

This paper introduces two upper bounds for the length of two kinds of minimal trees. The first upper bound, a(n) < T([((n-I)/IT)*/~] + l/2) + l/2, is related to the Euclidean Steiner Minimal Tree obtained from any set of n points enclosed inside a circumference of radius 1. The second upper bound, G(n) 6 2(n-1)/k, is related to the Rectilinear Minimal Tree obtained from any set of n = !(k(k + 2) + 1)/21 points (k = 1, 2, 3,.. .), enclosed in a square with sides equal to 1. Moreover, we also present the only case where the well-known bound u(n) = 1 + fi is attained for the length of the Rectilinear Steiner Minimal Tree for any set of n = t2 points (t = 2, 3,.. _) enclosed in a square with sides equal to 1.

Treewidth computations II. Lower bounds

Information and Computation, 2011

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.

Lower bounds on the size of general branch-and-bound trees

Mathematical Programming, 2022

A general branch-and-bound tree is a branch-and-bound tree which is allowed to use general disjunctions of the form ⊤ ≤ 0 ∨ ⊤ ≥ 0 + 1, where is an integer vector and 0 is an integer scalar, to create child nodes. We construct a packing instance, a set covering instance, and a Traveling Salesman Problem instance, such that any general branch-and-bound tree that solves these instances must be of exponential size. We also verify that an exponential lower bound on the size of general branch-and-bound trees persists when we add Gaussian noise to the coefficients of the cross polytope, thus showing that polynomial-size "smoothed analysis" upper bound is not possible. e results in this paper can be viewed as the branch-and-bound analog of the seminal paper by Chvátal et al. [7], who proved lower bounds for the Chvátal-Gomory rank.

Bounds on trees

Discrete Mathematics, 2011

We prove a finitary version of the Halpern-Läuchli Theorem. We also prove partition results about strong subtrees. Both results give estimates on the height of trees.

On the Optimality of a Family of Binary Trees

2011

In this paper we present an analysis of the complexity of a class of algorithms. These algorithms recursively explore a binary tree and need to make two recursive calls for one of the subtrees and only one for the other. We derive the complexity of these algorithms in the worst and in the best case and show the tree structures for which these cases happen.

Note on the weighted internal path length of b-ary trees

Discrete Mathematics & Theoretical Computer Science

Analysis of Algorithms International audience In a recent paper Broutin and Devroye (2005) have studied the height of a class of edge-weighted random trees.This is a class of trees growing in continuous time which includes many wellknown trees as examples. In this paper we derive a limit theorem for the internal path length for this class of trees.For the proof we extend a limit theorem in Neininger and Rüschendorf (2004) to recursive sequences of random variables with continuous time parameter.

Lower bounds on the rotation distance of binary trees

Information Processing Letters, 2010

The rotation distance d(S, T) between two binary trees S, T of n vertices is the minimum number of rotations to transform S into T. While it is known that d(S, T) 2n − 6, a wellknown conjecture states that there are trees for which this bound is sharp for any value of n 11. We are unable to prove the conjecture, but we give here some simple criteria for lower bound evaluation, leading for example to individuate some "regular" tree structures for which d(S, T) = 3n/2 − O (1), or d(S, T) = 5n/3 − O (1).

A branch and bound algorithm for exact, upper, and lower bounds on treewidth

Lecture Notes in Computer Science, 2006

In this paper, a branch and bound algorithm for computing the treewidth of a graph is presented. The method incorporates extensions of existing results, and uses new pruning and reduction rules, based upon properties of the adopted branching strategy. We discuss how the algorithm can not only be used to obtain exact bounds for the treewidth, but also to obtain upper and/or lower bounds. Computational results of the algorithm are presented.

Treewidth computations I. Upper bounds

2010

For more and more 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.

An Inquistive Result on DFS Problem of Binary Trees

Let be a complete binary tree of height ,then is called a full binary tree, if all the leaves in T are at level h. Let denote the full binary tree of height .Let denote the time taken by DFS with respect topreorder traversing to find the vertex in the graph. In this paper we analytically compute the for being a full binary tree of height , .