China
Africa
Explore chapters and articles related to this topic
Trees and Cycles
Published in Kenneth H. Rosen, Graphs, Algorithms, and Optimization, 2005
The Cheriton-Tarjan algorithm is a modification of Kruskal’s algorithm designed to reduce the O(ε log ε) term. It also grows a spanning forest, beginning with a forest of n — |G| components each consisting of a single node. Now the term O(ε log ε) comes from selecting the minimum edge from a heap of ε edges. Since every component Tu must eventually be connected to another component, this algorithm keeps a separate heap PQu for each component Tu, so that initially n smaller heaps are used. Initially, PQu will contain only Deg(u) edges, since Tu consists only of vertex u. When Tu and Tv are merged, PQu and PQV must also be merged. This requires a modification of the data structures, since heaps cannot be merged efficiently. This is essentially because merging heaps reduces to building a new heap. Any data structure in which a minimum element can be found efficiently is called a priority queue. A heap is one form of priority queue, in which elements are stored as an array, but viewed as a binary tree. There are many other forms of priority queue. In this algorithm, PQu will stand for a priority queue which can be merged. The Cheriton-Tarjan algorithm can be described as follows.
Introduction
Published in Randall L. Eubank, Ana Kupresanin, Statistical Computing in C++ and R, 2011
Randall L. Eubank, Ana Kupresanin
The simplest efficient implementation of a priority queue uses the data structure called a heap. A heap is a binary tree that at each node stores a member of the set and satisfies the properties every node has two children, except for the nodes in the lowest two levels of the tree,the lowest level of the tree is filled left to right (see Figure 9.1) andthe key value of every node is no greater than that of either of its children (known as the heap property).
Applications of Graph Theory
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
The heap sort algorithm also uses a special kind of binary tree, called a ‘heap', whose vertices are again members of the list to be sorted. The ‘shape' of a heap is such that it has the smallest possible height for the number of its vertices. This is achieved by ensuring that, if we ignore its highest level, the remainder of the tree is both complete and full. In addition, the leaf vertices at the highest level are situated as far to the left of the diagram as possible. These two conditions determine the shape of the diagram of a heap (see figure 11.10 below). The last condition in the following definition refers to the relationships between the vertices.
Bayesian network-based vulnerability assessment of a large-scale bridge network using improved ORDER-II-Dijkstra algorithm
Published in Structure and Infrastructure Engineering, 2021
Jie Wang, Kun Fang, Shunlong Li, Shaoyang He
The minimum heap is a complete binary tree, where the weight of each node is no larger than that of its children nodes (if they exist). Therefore, the weight at the root must be minimum. The required priority queue is obtained step by step by extracting the root node of the continuously updated minimum heap. For more information and rules regarding adding or deleting ordered state combinations of EqBs to the minimum heap including reordering procedures, see Reference (Lam & Li, 1986). It stops when the sum of the probabilities of the obtained state combinations is greater than or equal to the accuracy a Therefore, it effectively controls the sample size and running time using the precision a. In this study, a = 0.999.
An efficient algorithm for solving the median problem on real road networks
Published in Engineering Optimization, 2020
Saeed Ghanbartehrani, J. David Porter
The maximum number of iterations performed by the MTD algorithm in the worst case scenario is k.|E|, since an independent search is performed from each demand point. At each iteration, a node is removed and expanded from the open list. These operations have a time complexity of O(log|V|), assuming the heap implementation for priority queues. Other operations, such as adding items to open lists, would also have the same time complexity of O(log|V|). Therefore, the resulting time complexity of the MTD algorithm is O(k.|E|.log|V|).