Algebraic graph algorithms : a practical guide using Python

• Main Author: Erciyes, K.
• Format: eBook Book
• Language: English
• Published: Cham Amazon 2021; Springer; Springer International Publishing AG; Springer International Publishing
• Edition: 1
• Series: Undergraduate Topics in Computer Science
• Subjects: Algorithm Analysis and Problem Complexity; Computer Science; Discrete Mathematics in Computer Science; Graph algorithms; Mathematical Applications in Computer Science; Python (Computer program language)
• ISBN: 3030878856; 9783030878856
• ISSN: 1863-7310; 2197-1781
• DOI: 10.1007/978-3-030-87886-3

This textbook discusses the design and implementation of basic algebraic graph algorithms, and algebraic graph algorithms for complex networks, employing matroids whenever possible. The text describes the design of a simple parallel matrix algorithm kernel that can be used for parallel processing of...

Table of Contents:

• 5.1 Parallel Processing -- 5.1.1 Architectures -- 5.1.2 Basic Communications -- 5.1.3 Performance -- 5.1.4 Parallel Matrix Computations -- 5.2 Parallel Computations with Python -- 5.2.1 Process-Based Model with Multiprocessing -- 5.2.2 Message Passing Interface mpi4py -- 5.3 Sparse Matrix Algorithms -- 5.3.1 Sparsity Calculation -- 5.3.2 Sparse Matrix Representations -- 5.3.3 Sparse Matrix Multiplication -- 5.4 Chapter Notes -- blackPart II Graph Algorithms-1pt -- 6 Trees -- 6.1 Introduction -- 6.2 Spanning Tree Construction -- 6.3 Minimum Spanning Trees -- 6.3.1 Jarnik-Prim Algorithm -- 6.3.2 Kruskal's Algorithm as a Matroid -- 6.3.3 Boruvka's Algorithm -- 6.4 Chapter Notes -- 7 Shortest Paths -- 7.1 Introduction -- 7.2 Breadth-First-Search -- 7.2.1 The Classical Algorithm -- 7.2.2 The Algebraic Algorithm -- 7.3 Dijkstra's Algorithm -- 7.4 Bellman-Ford Algorithm -- 7.5 Floyd-Warshall Algorithm -- 7.6 Transitive Closure -- 7.7 Chapter Notes -- 8 Connectivity and Matching -- 8.1 Connectivity -- 8.1.1 Theory -- 8.1.2 Undirected Graph Connectivity -- 8.1.3 Directed Graph Connectivity -- 8.2 Matching -- 8.2.1 Unweighted Matching -- 8.2.2 Weighted Matching -- 8.2.3 Rabin-Vazirani Algorithm -- 8.3 Chapter Notes -- 9 Subgraph Search -- 9.1 Independent Sets -- 9.1.1 A Greedy Algorithm -- 9.1.2 Lowest Degree First Algorithm -- 9.1.3 Luby's Algorithm -- 9.2 Dominating Sets -- 9.3 Vertex Cover -- 9.3.1 A Greedy Algorithm -- 9.3.2 A 2-Approximation Algorithm -- 9.4 Coloring -- 9.4.1 Vertex Coloring -- 9.4.2 Edge Coloring -- 9.5 Chapter Notes -- 10 Large Graph Analysis -- 10.1 Introduction -- 10.2 Degree Distribution -- 10.3 Density -- 10.4 Clustering Coefficient -- 10.5 Matching Index -- 10.6 Centrality -- 10.6.1 Degree Centrality -- 10.6.2 Closeness Centrality -- 10.6.3 Vertex Betweenness Centrality -- 10.6.4 Edge Betweenness Centrality
• Intro -- Preface -- Contents -- 1 Introduction -- 1.1 Graphs -- 1.2 Matrices -- 1.3 Algebraic Graph Algorithms -- 1.4 Python Language -- 1.5 Challenges -- 1.6 Outline of the Book -- 1.6.1 Background -- 1.6.2 Basic Graph Algorithms -- blackPart I Background-1pt -- 2 A Short Review of Python -- 2.1 Introduction -- 2.2 Data Structures -- 2.2.1 Strings -- 2.2.2 Lists -- 2.2.3 Tuples -- 2.2.4 Arrays -- 2.2.5 Dictionaries -- 2.2.6 Sets -- 2.3 Flow Control -- 2.3.1 The if Statement -- 2.3.2 Loops -- 2.4 Functions -- 2.5 Modules -- 2.6 Chapter Notes -- 3 Basic Matrix Computations -- 3.1 Introduction -- 3.2 Matrix Operations -- 3.3 Transpose of a Matrix -- 3.4 Matrix Properties -- 3.5 Types of Matrices -- 3.5.1 Diagonal Matrix -- 3.5.2 Identity Matrix -- 3.5.3 Triangular Matrix -- 3.6 Matrix Multiplication -- 3.6.1 Vector-Vector Multiplication -- 3.6.2 Matrix-Vector Multiplication -- 3.6.3 Matrix-Matrix Multiplication -- 3.6.4 Boolean Matrix Multiplication -- 3.7 Determinant of a Matrix -- 3.8 Matrix Inverse -- 3.9 Rank Computation -- 3.9.1 Powers of a Matrix -- 3.10 Eigenvalues and Eigenvectors -- 3.11 Block Matrices -- 3.12 Chapter Notes -- 4 Graphs, Matrices and Matroids -- 4.1 Introduction to Graphs -- 4.1.1 Degree of a Vertex -- 4.1.2 Subgraphs -- 4.2 Graph Operations -- 4.3 Types of Graphs -- 4.3.1 Digraph -- 4.3.2 Complete Graphs -- 4.3.3 Weighted Graphs -- 4.3.4 Bipartite Graphs -- 4.3.5 Regular Graphs -- 4.3.6 Graph Isomorphism -- 4.4 Walks, Paths, Cycles -- 4.5 Graphs and Matrices -- 4.5.1 Adjacency Matrix -- 4.5.2 Adjacency List -- 4.5.3 Incidence Matrix -- 4.5.4 Cycle Matrix -- 4.5.5 Path Matrix -- 4.5.6 The Laplacian -- 4.6 Python Graphs -- 4.7 Matroids -- 4.7.1 The Basic Matroid -- 4.7.2 The Graphic Matroid -- 4.7.3 Weighted Matroid -- 4.7.4 Graphs, Matroids and Matrices -- 4.8 Chapter Notes -- 5 Parallel and Sparse Matrix Computations
• 10.6.5 Eigenvalue Centrality -- 10.7 Network Models -- 10.7.1 Random Networks -- 10.7.2 Small-World Networks -- 10.7.3 Scale-Free Networks -- 10.8 Kronecker Graphs -- 10.9 Chapter Notes -- 11 Graph Partitioning and Clustering -- 11.1 Graph Partitioning -- 11.1.1 BFS-Based Partitioning -- 11.1.2 Spectral Bisection -- 11.1.3 Multilevel Graph Partitioning -- 11.2 Graph Clustering -- 11.2.1 MST-Based Clustering -- 11.2.2 Shared Nearest Neighbor Clustering -- 11.2.3 Edge Betweenness Clustering -- 11.2.4 k-Core Clustering -- 11.3 Chapter Notes -- Index