Analytic criterion and algorithm for the lowest common ancestor of two neighboring nodes in a complete binary tree

2013 ◽  
Author(s):  
Xingbo Wang ◽  
Jun Zhou
Keyword(s):  
Binary Tree ◽  
Common Ancestor ◽  
Complete Binary Tree ◽  
Lowest Common Ancestor ◽  
Analytic Criterion

scholarly journals On the Distribution of Depths in Increasing Trees

10.37236/409 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Markus Kuba ◽  
Stephan Wagner
Keyword(s):  
Common Ancestor ◽  
Short Note ◽  
Bijective Proof ◽  
Lowest Common Ancestor ◽  
Recursive Trees

By a theorem of Dobrow and Smythe, the depth of the $k$th node in very simple families of increasing trees (which includes, among others, binary increasing trees, recursive trees and plane ordered recursive trees) follows the same distribution as the number of edges of the form $j-(j+1)$ with $j < k$. In this short note, we present a simple bijective proof of this fact, which also shows that the result actually holds within a wider class of increasing trees. We also discuss some related results that follow from the bijection as well as a possible generalization. Finally, we use another similar bijection to determine the distribution of the depth of the lowest common ancestor of two nodes.

Binary Trees and the n-Cutset Property

1991 ◽  
Vol 34 (1) ◽  
pp. 23-30 ◽  
Author(s):  
Peter Arpin ◽  
John Ginsburg
Keyword(s):  
Positive Integer ◽  
Partially Ordered Set ◽  
Binary Tree ◽  
Ordered Set ◽  
Maximal Chain ◽  
Binary Trees ◽  
Complete Binary Tree ◽  
Maximal Elements ◽  
Extremal Case ◽  
Result Theorem

AbstractA partially ordered set P is said to have the n-cutset property if for every element x of P, there is a subset S of P all of whose elements are noncomparable to x, with |S| ≤ n, and such that every maximal chain in P meets {x} ∪ S. It is known that if P has the n-cutset property then P has at most 2n maximal elements. Here we are concerned with the extremal case. We let Max P denote the set of maximal elements of P. We establish the following result. THEOREM: Let n be a positive integer. Suppose P has the n-cutset property and that |Max P| = 2n. Then P contains a complete binary tree T of height n with Max T = Max P and such that C ∩ T is a maximal chain in T for every maximal chain C of P. Two examples are given to show that this result does not extend to the case when n is infinite. However the following is shown. THEOREM: Suppose that P has the ω-cutset property and that |Max P| = 2ω. If P — Max P is countable then P contains a complete binary tree of height ω

Two Models of Additional Adjacencies between the Root and Descendants in a Complete Binary Tree Minimizing Total Path Length

2009 ◽  
Author(s):  
Kiyoshi Sawada ◽  
Sio-Iong Ao ◽  
Alan Hoi-Shou Chan ◽  
Hideki Katagiri ◽  
Osca Castillo ◽  
...  
Keyword(s):  
Binary Tree ◽  
Path Length ◽  
Complete Binary Tree ◽  
Total Path Length

PARALLEL RANGE MINIMA ON COARSE GRAINED MULTICOMPUTERS

1999 ◽  
Vol 10 (04) ◽  
pp. 375-389
Author(s):  
H. MONGELLI ◽  
S. W. SONG
Keyword(s):  
Common Ancestor ◽  
Basic Problem ◽  
Coarse Grained ◽  
Time Range ◽  
Communication Overhead ◽  
Constant Number ◽  
Real Numbers ◽  
Graph Problems ◽  
Large N ◽  
Lowest Common Ancestor

Given an array of n real numbers A=(a0, a1, …, an-1), define MIN(i,j)= min {ai,…,aj}. The range minima problem consists of preprocessing array A such that queries MIN(i,j), for any 0≤i≤n-1 can be answered in constant time. Range minima is a basic problem that appears in many other important graph problems such as lowest common ancestor, Euler tour, etc. In this work we present a parallel algorithm under the CGM model (coarse grained multicomputer), that solves the range minima problem in O(n/p) time and constant number of communication rounds. The communication overhead involves the transmission of p numbers (independent of n). We show promising experimental results with speedup curves approximating the optimal for large n.

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