Number Of External Nodes Research Articles

Profile of Random Exponential Binary Trees

We introduce the random exponential binary tree (EBT) and study its profile. As customary, the tree is extended by padding each leaf node (considered internal), with the appropriate number of external nodes, so that the outdegree of every internal node is made equal to 2. In a random EBT, at every step, each external node is promoted to an internal node with probability p, stays unchanged with probability 1 - p, and the resulting tree is extended. We study the internal and external profiles of a random EBT and get exact expectations for the numbers of internal and external nodes at each level. Asymptotic analysis shows that the average external profile is richest at level \(\frac {2p}{p+1}n\), and it experiences phase transitions at levels a n, where the a’s are the solutions to an algebraic equation. The rates of convergence themselves go through an infinite number of phase changes in the sublinear range, and then again at the nearly linear levels.

Binary dynamics on star networks under external perturbations.

We study a binary dynamical process that is a representation of the voter model with two candidates and opinion makers. The voters are represented by nodes of a network of social contacts with internal states labeled 0 or 1 and nodes that are connected can influence each other. The network is also perturbed by opinion makers, a set of external nodes whose states are frozen in 0 or 1 and that can influence all nodes of the network. The quantity of interest is the probability of finding m nodes in state 1 at time t. Here we study this process on star networks, which are simple representations of hubs found in complex systems, and compare the results with those obtained for networks that are fully connected. In both cases a transition from disordered to ordered equilibrium states is observed as the number of external nodes becomes small. For fully connected networks the probability distribution becomes uniform at the critical point. For star networks, on the other hand, we show that the equilibrium distribution splits in two peaks, reflecting the two possible states of the central node. We obtain approximate analytical solutions for the equilibrium distribution that clarify the role of the central node in the process. We show that the network topology also affects the time scale of oscillations in single realizations of the dynamics, which are much faster for the star network. Finally, extending the analysis to two stars we compare our results with simulations in simple scale-free networks.

The Variance of the Profile in Digital Search Trees

Analysis of Algorithms What today we call digital search tree (DST) is Coffman and Eve's sequence tree introduced in 1970. A digital search tree is a binary tree whose ordering of nodes is based on the values of bits in the binary representation of a node's key. In fact, a digital search tree is a digital tree in which strings (keys, words) are stored directly in internal nodes. The profile of a digital search tree is a parameter that counts the number of nodes at the same distance from the root. In this paper we concentrate on external profile, i.e., the number of external nodes at level k when n strings are sorted. By assuming that the n input strings are independent and follow a (binary) memoryless source the asymptotic behaviour of the average profile was determined by Drmota and Szpankowski (2011). The purpose of the present paper is to extend their analysis and to provide a precise analysis of variance of the profile. The main (technical) difference is that we have to deal with an inhomogeneous part in a proper functional-differential equations satisfied by the second moment and Poisson variance. However, we show that the variance is asymptotically of the same order as the expected value which implies concentration. These results are derived by methods of analytic combinatorics such as generating functions, Mellin transform, Poissonization, the saddle point method and singularity analysis.

Formula omitted]: The internal extent formula for compacted tries

It is well known (Knuth, 1997 [5, pp. 399–400]) that in a binary tree the external path length minus the internal path length is exactly 2 n − 2 , where n is the number of external nodes. We show that a generalization of the formula holds for compacted tries, replacing the role of paths with the notion of extent, and the value 2 n − 2 with the trie measure, an estimation of the number of bits that are necessary to describe the trie.

A Node Scheduling Scheme Based on Virtual Coordinate in Sensor Networks

Firstly a maximum similarity distribution model is proposed.Secondly,a new definition,named virtual coordinate of a node which composes of the minimum hop counts to several special nodes,is introduced in instead of absolute physical coordinate of a node.Based on the theories,a distributed location-independent node scheduling scheme is proposed.The scheme consists of a coverage algorithm and a connection algorithm.The coverage algorithm takes advantage of the nodes' virtual coordinate to divide all nodes into several subsets.Without using location information,this scheduling scheme not only has the ability that sensor nodes in subsets are more uniformly distributed in the target region than other schemes,but also guarantees that all subsets are connective. The simulation results show that the scheme outperforms the randomized node scheduling scheme,on the coverage rate and network lifetime,as well as the number of external nodes when maintaining subsets to be connective.

Embedding DSP control algorithms in PSpice

A new method is developed to embed DSP control algorithms into PSpice providing accurate time stamped simulation of real-time applications. The proposed method allows to build new PSpice models with an arbitrary number of external nodes using PSpice's Device Equation option. Sampled systems can be simulated including the effects of discrete sampling times as well as propagation delays caused by processors. Control code developed in C-language can be embedded into this PSpice control model prior to downloading into the real-time DSP environment. To verify the simulation approach, simulation results and experimental results of a shunt active filter for harmonic current compensation in a three-phase system are presented.

Martingales and large deviations for binary search trees

We establish an almost sure large deviations theorem for the depth of the external nodes of binary search trees (BSTs). To achieve this, a parametric family of martingales is introduced. This family also allows us to get asymptotic results on the number of external nodes at deepest level. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 112–127, 2001

Percolation of coalescing random walks

We study the shape of the binary tree containing 0 that is created from percolation of coalescing random walks. The key result is a duality lemma describing the shape of the tree. Furthermore, we show that and where A(R 0), M(R 0), L(R 0) are respectively the area, the number of external nodes and the length of the longest path of the tree R 0.

Percolation of coalescing random walks

We study the shape of the binary tree containing 0 that is created from percolation of coalescing random walks. The key result is a duality lemma describing the shape of the tree. Furthermore, we show thatandwhereA(R0),M(R0),L(R0) are respectively the area, the number of external nodes and the length of the longest path of the treeR0.