Lecture Notes on “Analysis of Algorithms”: Directed Minimum Spanning Trees
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Directed Graph Algorithms
Directed Graph Algorithms CSE 373 Data Structures Readings • Reading Chapter 13 › Sections 13.3 and 13.4 Digraphs 2 Topological Sort 142 322 143 321 Problem: Find an order in which all these courses can 326 be taken. 370 341 Example: 142 Æ 143 Æ 378 Æ 370 Æ 321 Æ 341 Æ 322 Æ 326 Æ 421 Æ 401 378 421 In order to take a course, you must 401 take all of its prerequisites first Digraphs 3 Topological Sort Given a digraph G = (V, E), find a linear ordering of its vertices such that: for any edge (v, w) in E, v precedes w in the ordering B C A F D E Digraphs 4 Topo sort – valid solution B C Any linear ordering in which A all the arrows go to the right F is a valid solution D E A B F C D E Note that F can go anywhere in this list because it is not connected. Also the solution is not unique. Digraphs 5 Topo sort – invalid solution B C A Any linear ordering in which an arrow goes to the left F is not a valid solution D E A B F C E D NO! Digraphs 6 Paths and Cycles • Given a digraph G = (V,E), a path is a sequence of vertices v1,v2, …,vk such that: ›(vi,vi+1) in E for 1 < i < k › path length = number of edges in the path › path cost = sum of costs of each edge • A path is a cycle if : › k > 1; v1 = vk •G is acyclic if it has no cycles. -
1 Steiner Minimal Trees
Steiner Minimal Trees¤ Bang Ye Wu Kun-Mao Chao 1 Steiner Minimal Trees While a spanning tree spans all vertices of a given graph, a Steiner tree spans a given subset of vertices. In the Steiner minimal tree problem, the vertices are divided into two parts: terminals and nonterminal vertices. The terminals are the given vertices which must be included in the solution. The cost of a Steiner tree is de¯ned as the total edge weight. A Steiner tree may contain some nonterminal vertices to reduce the cost. Let V be a set of vertices. In general, we are given a set L ½ V of terminals and a metric de¯ning the distance between any two vertices in V . The objective is to ¯nd a connected subgraph spanning all the terminals of minimal total cost. Since the distances are all nonnegative in a metric, the solution is a tree structure. Depending on the given metric, two versions of the Steiner tree problem have been studied. ² (Graph) Steiner minimal trees (SMT): In this version, the vertex set and metric is given by a ¯nite graph. ² Euclidean Steiner minimal trees (Euclidean SMT): In this version, V is the entire Euclidean space and thus in¯nite. Usually the metric is given by the Euclidean distance 2 (L -norm). That is, for two points with coordinates (x1; y2) and (x2; y2), the distance is p 2 2 (x1 ¡ x2) + (y1 ¡ y2) : In some applications such as VLSI routing, L1-norm, also known as rectilinear distance, is used, in which the distance is de¯ned as jx1 ¡ x2j + jy1 ¡ y2j: Figure 1 illustrates a Euclidean Steiner minimal tree and a graph Steiner minimal tree. -
Graph Varieties Axiomatized by Semimedial, Medial, and Some Other Groupoid Identities
Discussiones Mathematicae General Algebra and Applications 40 (2020) 143–157 doi:10.7151/dmgaa.1344 GRAPH VARIETIES AXIOMATIZED BY SEMIMEDIAL, MEDIAL, AND SOME OTHER GROUPOID IDENTITIES Erkko Lehtonen Technische Universit¨at Dresden Institut f¨ur Algebra 01062 Dresden, Germany e-mail: [email protected] and Chaowat Manyuen Department of Mathematics, Faculty of Science Khon Kaen University Khon Kaen 40002, Thailand e-mail: [email protected] Abstract Directed graphs without multiple edges can be represented as algebras of type (2, 0), so-called graph algebras. A graph is said to satisfy an identity if the corresponding graph algebra does, and the set of all graphs satisfying a set of identities is called a graph variety. We describe the graph varieties axiomatized by certain groupoid identities (medial, semimedial, autodis- tributive, commutative, idempotent, unipotent, zeropotent, alternative). Keywords: graph algebra, groupoid, identities, semimediality, mediality. 2010 Mathematics Subject Classification: 05C25, 03C05. 1. Introduction Graph algebras were introduced by Shallon [10] in 1979 with the purpose of providing examples of nonfinitely based finite algebras. Let us briefly recall this concept. Given a directed graph G = (V, E) without multiple edges, the graph algebra associated with G is the algebra A(G) = (V ∪ {∞}, ◦, ∞) of type (2, 0), 144 E. Lehtonen and C. Manyuen where ∞ is an element not belonging to V and the binary operation ◦ is defined by the rule u, if (u, v) ∈ E, u ◦ v := (∞, otherwise, for all u, v ∈ V ∪ {∞}. We will denote the product u ◦ v simply by juxtaposition uv. Using this representation, we may view any algebraic property of a graph algebra as a property of the graph with which it is associated. -
Networkx: Network Analysis with Python
NetworkX: Network Analysis with Python Salvatore Scellato Full tutorial presented at the XXX SunBelt Conference “NetworkX introduction: Hacking social networks using the Python programming language” by Aric Hagberg & Drew Conway Outline 1. Introduction to NetworkX 2. Getting started with Python and NetworkX 3. Basic network analysis 4. Writing your own code 5. You are ready for your project! 1. Introduction to NetworkX. Introduction to NetworkX - network analysis Vast amounts of network data are being generated and collected • Sociology: web pages, mobile phones, social networks • Technology: Internet routers, vehicular flows, power grids How can we analyze this networks? Introduction to NetworkX - Python awesomeness Introduction to NetworkX “Python package for the creation, manipulation and study of the structure, dynamics and functions of complex networks.” • Data structures for representing many types of networks, or graphs • Nodes can be any (hashable) Python object, edges can contain arbitrary data • Flexibility ideal for representing networks found in many different fields • Easy to install on multiple platforms • Online up-to-date documentation • First public release in April 2005 Introduction to NetworkX - design requirements • Tool to study the structure and dynamics of social, biological, and infrastructure networks • Ease-of-use and rapid development in a collaborative, multidisciplinary environment • Easy to learn, easy to teach • Open-source tool base that can easily grow in a multidisciplinary environment with non-expert users -
Structural Graph Theory Meets Algorithms: Covering And
Structural Graph Theory Meets Algorithms: Covering and Connectivity Problems in Graphs Saeed Akhoondian Amiri Fakult¨atIV { Elektrotechnik und Informatik der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Rolf Niedermeier Gutachter: Prof. Dr. Stephan Kreutzer Gutachter: Prof. Dr. Marcin Pilipczuk Gutachter: Prof. Dr. Dimitrios Thilikos Tag der wissenschaftlichen Aussprache: 13. October 2017 Berlin 2017 2 This thesis is dedicated to my family, especially to my beautiful wife Atefe and my lovely son Shervin. 3 Contents Abstract iii Acknowledgementsv I. Introduction and Preliminaries1 1. Introduction2 1.0.1. General Techniques and Models......................3 1.1. Covering Problems.................................6 1.1.1. Covering Problems in Distributed Models: Case of Dominating Sets.6 1.1.2. Covering Problems in Directed Graphs: Finding Similar Patterns, the Case of Erd}os-P´osaproperty.......................9 1.2. Routing Problems in Directed Graphs...................... 11 1.2.1. Routing Problems............................. 11 1.2.2. Rerouting Problems............................ 12 1.3. Structure of the Thesis and Declaration of Authorship............. 14 2. Preliminaries and Notations 16 2.1. Basic Notations and Defnitions.......................... 16 2.1.1. Sets..................................... 16 2.1.2. Graphs................................... 16 2.2. Complexity Classes................................ -
Katz Centrality for Directed Graphs • Understand How Katz Centrality Is an Extension of Eigenvector Centrality to Learning Directed Graphs
Prof. Ralucca Gera, [email protected] Applied Mathematics Department, ExcellenceNaval Postgraduate Through Knowledge School MA4404 Complex Networks Katz Centrality for directed graphs • Understand how Katz centrality is an extension of Eigenvector Centrality to Learning directed graphs. Outcomes • Compute Katz centrality per node. • Interpret the meaning of the values of Katz centrality. Recall: Centralities Quality: what makes a node Mathematical Description Appropriate Usage Identification important (central) Lots of one-hop connections The number of vertices that Local influence Degree from influences directly matters deg Small diameter Lots of one-hop connections The proportion of the vertices Local influence Degree centrality from relative to the size of that influences directly matters deg C the graph Small diameter |V(G)| Lots of one-hop connections A weighted degree centrality For example when the Eigenvector centrality to high centrality vertices based on the weight of the people you are (recursive formula): neighbors (instead of a weight connected to matter. ∝ of 1 as in degree centrality) Recall: Strongly connected Definition: A directed graph D = (V, E) is strongly connected if and only if, for each pair of nodes u, v ∈ V, there is a path from u to v. • The Web graph is not strongly connected since • there are pairs of nodes u and v, there is no path from u to v and from v to u. • This presents a challenge for nodes that have an in‐degree of zero Add a link from each page to v every page and give each link a small transition probability controlled by a parameter β. u Source: http://en.wikipedia.org/wiki/Directed_acyclic_graph 4 Katz Centrality • Recall that the eigenvector centrality is a weighted degree obtained from the leading eigenvector of A: A x =λx , so its entries are 1 λ Thoughts on how to adapt the above formula for directed graphs? • Katz centrality: ∑ + β, Where β is a constant initial weight given to each vertex so that vertices with zero in degree (or out degree) are included in calculations. -
A Gentle Introduction to Deep Learning for Graphs
A Gentle Introduction to Deep Learning for Graphs Davide Bacciua, Federico Erricaa, Alessio Michelia, Marco Poddaa aDepartment of Computer Science, University of Pisa, Italy. Abstract The adaptive processing of graph data is a long-standing research topic that has been lately consolidated as a theme of major interest in the deep learning community. The snap increase in the amount and breadth of related research has come at the price of little systematization of knowledge and attention to earlier literature. This work is a tutorial introduction to the field of deep learn- ing for graphs. It favors a consistent and progressive presentation of the main concepts and architectural aspects over an exposition of the most recent liter- ature, for which the reader is referred to available surveys. The paper takes a top-down view of the problem, introducing a generalized formulation of graph representation learning based on a local and iterative approach to structured in- formation processing. Moreover, it introduces the basic building blocks that can be combined to design novel and effective neural models for graphs. We com- plement the methodological exposition with a discussion of interesting research challenges and applications in the field. Keywords: deep learning for graphs, graph neural networks, learning for structured data arXiv:1912.12693v2 [cs.LG] 15 Jun 2020 1. Introduction Graphs are a powerful tool to represent data that is produced by a variety of artificial and natural processes. A graph has a compositional nature, being a compound of atomic information pieces, and a relational nature, as the links defining its structure denote relationships between the linked entities. -
Graph Mining: Social Network Analysis and Information Diffusion
Graph Mining: Social network analysis and Information Diffusion Davide Mottin, Konstantina Lazaridou Hasso Plattner Institute Graph Mining course Winter Semester 2016 Lecture road Introduction to social networks Real word networks’ characteristics Information diffusion GRAPH MINING WS 2016 2 Paper presentations ▪ Link to form: https://goo.gl/ivRhKO ▪ Choose the date(s) you are available ▪ Choose three papers according to preference (there are three lists) ▪ Choose at least one paper for each course part ▪ Register to the mailing list: https://lists.hpi.uni-potsdam.de/listinfo/graphmining-ws1617 GRAPH MINING WS 2016 3 What is a social network? ▪Oxford dictionary A social network is a network of social interactions and personal relationships GRAPH MINING WS 2016 4 What is an online social network? ▪ Oxford dictionary An online social network (OSN) is a dedicated website or other application, which enables users to communicate with each other by posting information, comments, messages, images, etc. ▪ Computer science : A network that consists of a finite number of users (nodes) that interact with each other (edges) by sharing information (labels, signs, edge directions etc.) GRAPH MINING WS 2016 5 Definitions ▪Vertex/Node : a user of the social network (Twitter user, an author in a co-authorship network, an actor etc.) ▪Edge/Link/Tie : the connection between two vertices referring to their relationship or interaction (friend-of relationship, re-tweeting activity, etc.) Graph G=(V,E) symbols meaning V users E connections deg(u) node degree: -
‣ Dijkstra's Algorithm ‣ Minimum Spanning Trees ‣ Prim, Kruskal, Boruvka ‣ Single-Link Clustering ‣ Min-Cost Arborescences
4. GREEDY ALGORITHMS II ‣ Dijkstra's algorithm ‣ minimum spanning trees ‣ Prim, Kruskal, Boruvka ‣ single-link clustering ‣ min-cost arborescences Lecture slides by Kevin Wayne Copyright © 2005 Pearson-Addison Wesley http://www.cs.princeton.edu/~wayne/kleinberg-tardos Last updated on Feb 18, 2013 6:08 AM 4. GREEDY ALGORITHMS II ‣ Dijkstra's algorithm ‣ minimum spanning trees ‣ Prim, Kruskal, Boruvka ‣ single-link clustering ‣ min-cost arborescences SECTION 4.4 Shortest-paths problem Problem. Given a digraph G = (V, E), edge weights ℓe ≥ 0, source s ∈ V, and destination t ∈ V, find the shortest directed path from s to t. 1 15 3 5 4 12 source s 0 3 8 7 7 2 9 9 6 1 11 5 5 4 13 4 20 6 destination t length of path = 9 + 4 + 1 + 11 = 25 3 Car navigation 4 Shortest path applications ・PERT/CPM. ・Map routing. ・Seam carving. ・Robot navigation. ・Texture mapping. ・Typesetting in LaTeX. ・Urban traffic planning. ・Telemarketer operator scheduling. ・Routing of telecommunications messages. ・Network routing protocols (OSPF, BGP, RIP). ・Optimal truck routing through given traffic congestion pattern. Reference: Network Flows: Theory, Algorithms, and Applications, R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Prentice Hall, 1993. 5 Dijkstra's algorithm Greedy approach. Maintain a set of explored nodes S for which algorithm has determined the shortest path distance d(u) from s to u. ・Initialize S = { s }, d(s) = 0. ・Repeatedly choose unexplored node v which minimizes shortest path to some node u in explored part, followed by a single edge (u, v) ℓe v d(u) u S s 6 Dijkstra's algorithm Greedy approach. -
Converting MST to TSP Path by Branch Elimination
applied sciences Article Converting MST to TSP Path by Branch Elimination Pasi Fränti 1,2,* , Teemu Nenonen 1 and Mingchuan Yuan 2 1 School of Computing, University of Eastern Finland, 80101 Joensuu, Finland; [email protected] 2 School of Big Data & Internet, Shenzhen Technology University, Shenzhen 518118, China; [email protected] * Correspondence: [email protected].fi Abstract: Travelling salesman problem (TSP) has been widely studied for the classical closed loop variant but less attention has been paid to the open loop variant. Open loop solution has property of being also a spanning tree, although not necessarily the minimum spanning tree (MST). In this paper, we present a simple branch elimination algorithm that removes the branches from MST by cutting one link and then reconnecting the resulting subtrees via selected leaf nodes. The number of iterations equals to the number of branches (b) in the MST. Typically, b << n where n is the number of nodes. With O-Mopsi and Dots datasets, the algorithm reaches gap of 1.69% and 0.61 %, respectively. The algorithm is suitable especially for educational purposes by showing the connection between MST and TSP, but it can also serve as a quick approximation for more complex metaheuristics whose efficiency relies on quality of the initial solution. Keywords: traveling salesman problem; minimum spanning tree; open-loop TSP; Christofides 1. Introduction The classical closed loop variant of travelling salesman problem (TSP) visits all the nodes and then returns to the start node by minimizing the length of the tour. Open loop TSP is slightly different variant, which skips the return to the start node. -
A Faster Parameterized Algorithm for PSEUDOFOREST DELETION
A faster parameterized algorithm for PSEUDOFOREST DELETION Citation for published version (APA): Bodlaender, H. L., Ono, H., & Otachi, Y. (2018). A faster parameterized algorithm for PSEUDOFOREST DELETION. Discrete Applied Mathematics, 236, 42-56. https://doi.org/10.1016/j.dam.2017.10.018 Document license: Unspecified DOI: 10.1016/j.dam.2017.10.018 Document status and date: Published: 19/02/2018 Document Version: Accepted manuscript including changes made at the peer-review stage Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. -
CSE 421 Algorithms Warmup Dijkstra's Algorithm
Single Source Shortest Path Problem • Given a graph and a start vertex s – Determine distance of every vertex from s CSE 421 – Identify shortest paths to each vertex Algorithms • Express concisely as a “shortest paths tree” • Each vertex has a pointer to a predecessor on Richard Anderson shortest path 1 u u Dijkstra’s algorithm 1 2 3 5 s x s x 3 4 3 v v Construct Shortest Path Tree Warmup from s • If P is a shortest path from s to v, and if t is 2 d d on the path P, the segment from s to t is a a 1 5 a 4 4 shortest path between s and t 4 e e -3 c s c v -2 s t 3 3 2 s 6 g g b b •WHY? 7 3 f f Assume all edges have non-negative cost Simulate Dijkstra’s algorithm Dijkstra’s Algorithm (strarting from s) on the graph S = {}; d[s] = 0; d[v] = infinity for v != s Round Vertex sabcd While S != V Added Choose v in V-S with minimum d[v] 1 a c 1 Add v to S 1 3 2 For each w in the neighborhood of v 2 s 4 1 d[w] = min(d[w], d[v] + c(v, w)) 4 6 3 4 1 3 y b d 4 1 3 1 u 0 1 1 4 5 s x 2 2 2 2 v 2 3 z 5 Dijkstra’s Algorithm as a greedy Correctness Proof algorithm • Elements committed to the solution by • Elements in S have the correct label order of minimum distance • Key to proof: when v is added to S, it has the correct distance label.