The FKT algorithm refers to a specialized algorithm used primarily for computing flow in networks, specifically for solving the maximum flow problem. "FKT" stands for the authors of the algorithm: Fulkerson, Katz, and Tardos. The FKT algorithm is based on the "preflow" concept and uses a push-relabel method for determining maximum flow in a flow network.
The Floyd-Warshall algorithm is a classic algorithm used in computer science and graph theory to find the shortest paths between all pairs of vertices in a weighted directed or undirected graph. It is particularly useful for graphs with positive or negative edge weights, although it cannot handle graphs with negative cycles. ### Key Features: 1. **Dynamic Programming Approach**: The algorithm uses a dynamic programming technique to iteratively improve the shortest path estimates between every pair of vertices.
Force-directed graph drawing is a technique used to visualize graphs in a way that aims to position the vertices (nodes) of the graph in two-dimensional or three-dimensional space. The goal of this method is to create a visually appealing and easy-to-understand representation of the graph, where the edges (connections between nodes) are depicted as springs and the nodes themselves are treated as physical objects that repel or attract each other.
The Ford–Fulkerson algorithm is a method used to compute the maximum flow in a flow network. Developed by L.R. Ford, Jr. and D.R. Fulkerson in the 1950s, this algorithm is based on the concept of augmenting paths and works by iteratively increasing the flow in the network until no more augmenting paths can be found.
Fringe search is a graph search technique used in artificial intelligence and computer science, particularly in the context of search algorithms for problem-solving. It is closely related to other search methods like breadth-first search and depth-first search, but it has its own distinctive approach to exploring the search space.
The Gallai–Edmonds decomposition is a fundamental concept in graph theory, particularly in the study of matchings within bipartite graphs. It provides a structured way to analyze matchings and their properties, and it is named after mathematicians Claude Berge, who contributed to matching theory, and Laszlo Lovasz and others who contributed to its broader understanding.
The Girvan-Newman algorithm is a method used in network theory for detecting communities within a graph. It was developed by Michelle Girvan and Mark Newman in 2002. The algorithm identifies and extracts the community structure of a network by progressively removing edges based on the concept of edge betweenness, which measures the number of shortest paths that pass through an edge.
In computer science, particularly in the context of artificial intelligence and search algorithms, a **goal node** refers to a specific state or condition in a graph or search space that signifies the completion of a problem or a successful solution to a task. It is part of a broader framework often used in algorithms for pathfinding, problem solving, and decision-making processes.
A Gomory–Hu tree is a data structure that represents the minimum cuts of a weighted undirected graph. It is named after mathematicians Ralph Gomory and Thomas Hu, who introduced the concept in the early 1960s. The Gomory–Hu tree provides a compact representation of all maximum flow and minimum cut pairs in the graph. ### Key Features: 1. **Structure**: The Gomory–Hu tree is a binary tree.
Graph bandwidth is a measure of how "spread out" the vertices of a graph are when it is represented in a certain way, particularly in terms of the adjacency matrix of the graph. Specifically, for a given graph \( G \) with vertex set \( V \), if we arrange the vertices in a specific order or permutation, the bandwidth is defined as the maximum distance between any two vertices that are connected by an edge in that arrangement.
Graph Edit Distance (GED) is a measure used to quantify the difference or similarity between two graphs. It is defined as the minimum cost required to transform one graph into another through a series of allowable edit operations. These operations typically include: 1. **Node Insertion**: Adding a new node to one graph. 2. **Node Deletion**: Removing a node from one graph. 3. **Edge Insertion**: Adding a new edge between two nodes in one graph.
Graph embedding is a technique used to represent the nodes, edges, or entire graphs in a continuous vector space. The main idea behind graph embedding is to map discrete graph structures into a lower-dimensional space such that the semantic information and relationships within the graph are preserved as much as possible. This representation can then be used for various machine learning tasks, such as classification, clustering, or visualization. ### Key Concepts: 1. **Nodes and Edges**: In a graph, nodes represent entities (e.
A graph kernel is a method used in machine learning and pattern recognition that measures the similarity between two graphs. Graphs are data structures composed of nodes (or vertices) and edges connecting these nodes. They can represent various types of data, such as social networks, molecular structures, and more. Graph kernels are particularly useful for tasks involving graph-structured data, where traditional vector-based methods are not applicable.
A Graph Neural Network (GNN) is a type of neural network specifically designed to work with data represented as graphs. Graphs are mathematical structures consisting of nodes (or vertices) connected by edges, which can represent various types of relationships between entities. Common applications for GNNs include social networks, molecular chemistry, recommendation systems, and knowledge graphs. ### Key Features of Graph Neural Networks: 1. **Graph Structure**: Unlike traditional neural networks that operate on grid-like data (e.g.
Graph reduction is a concept that originates from computer science and mathematics, particularly in the fields of graph theory and functional programming. It involves simplifying or transforming a graph into a simpler or reduced form while preserving certain properties or relationships among its components. Here are some key aspects of graph reduction: 1. **Graph Theory Context**: In graph theory, graph reduction may involve removing certain nodes or edges from a graph to simplify its structure, often with the goal of making algorithms that operate on the graph more efficient.
Graph traversal is the process of visiting all the vertices (or nodes) in a graph in a systematic manner. This can be done for various purposes, such as searching for specific elements, exploring the structure of the graph, or performing computations based on the graph's topology. There are two primary methods for graph traversal: 1. **Depth-First Search (DFS)**: - DFS explores as far down a branch of the graph as possible before backtracking.
HCS stands for Hierarchical Clustering using Single-linkage. It is a type of hierarchical clustering algorithm that builds a hierarchy of clusters by progressively merging or splitting existing clusters based on some distance metric. Here’s a brief overview of how HCS operates: 1. **Distance Matrix**: The algorithm starts by calculating the pairwise distances between all data points, usually using a metric like Euclidean distance or Manhattan distance. This forms a distance matrix.
Hall-type theorems for hypergraphs are generalizations of Hall's Marriage Theorem, which originally deals with bipartite graphs. Hall's theorem states that a perfect matching exists in a bipartite graph if and only if for every subset of vertices in one part, the number of neighbors in the other part is at least as large as the size of the subset.
The Havel–Hakimi algorithm is a recursive algorithm used to determine whether a given degree sequence can represent the degree sequence of a simple, undirected graph. A degree sequence is a list of non-negative integers that represent the degrees (the number of edges incident to a vertex) of the vertices in a graph. ### Steps of the Havel–Hakimi Algorithm: 1. **Input**: A non-increasing sequence of non-negative integers, also known as the degree sequence.
Hierarchical clustering of networks is a method used to group nodes in a network into clusters based on their similarities and relationships. It is particularly useful in the analysis of complex networks, such as social networks, biological networks, and communication networks, where the goal is to uncover underlying structures or patterns within the data.