DOCSLIB.ORG

Elementary Graph Algorithms (Chapter 22) of Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Elementary Graph Algorithms (Chapter 22) of Graphs CSCE423/823 Computer Science & Engineering 423/823 Introduction Design and Analysis of Algorithms Types of Graphs Representations Lecture 03 | Elementary Graph Algorithms (Chapter 22) of Graphs Elementary Graph Algorithms Applications Stephen Scott (Adapted from Vinodchandran N. Variyam) 1 / 29 [email protected] Introduction CSCE423/823 Introduction Types of Graphs are abstract data types that are applicable to numerous Graphs problems Representations of Graphs Can capture entities, relationships between them, the degree of the Elementary relationship, etc. Graph Algorithms This chapter covers basics in graph theory, including representation, Applications and algorithms for basic graph-theoretic problems We'll build on these later this semester 2 / 29 Types of Graphs CSCE423/823 Introduction A (simple, or undirected) graph G = (V; E) consists of V , a Types of nonempty set of vertices and E a set of unordered pairs of distinct Graphs Representations vertices called edges of Graphs D E Elementary V={A,B,C,D,E} Graph Algorithms Applications E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)} A B C 3 / 29 Types of Graphs (2) CSCE423/823 A directed graph (digraph) G = (V; E) consists of V , a nonempty set of vertices and E a set of ordered pairs of distinct vertices called Introduction edges Types of Graphs Representations of Graphs Elementary Graph Algorithms Applications 4 / 29 Types of Graphs (3) CSCE423/823 A weighted graph is an undirected or directed graph with the Introduction additional property that each edge e has associated with it a real Types of number w(e) called its weight Graphs 3 Representations of Graphs Elementary 12 Graph Algorithms 0 Applications -6 7 4 3 Other variations: multigraphs, pseudographs, etc. 5 / 29 Representations of Graphs CSCE423/823 Introduction Types of Graphs Representations Two common ways of representing a graph: Adjacency list and of Graphs Adjacency List adjacency matrix Adjacency Matrix Let G = (V; E) be a graph with n vertices and m edges Elementary Graph Algorithms Applications 6 / 29 Adjacency List CSCE423/823 For each vertex v 2 V , store a list of vertices adjacent to v Introduction For weighted graphs, add information to each node Types of Graphs How much is space required for storage? Representations of Graphs Adjacency List Adjacency a b a b c d Matrix Elementary Graph b a e Algorithms Applications c c a d c d a c e dee b c d 7 / 29 Adjacency Matrix CSCE423/823 Use an n × n matrix M, where M(i; j) = 1 if (i; j) is an edge, 0 otherwise Introduction Types of If G weighted, store weights in the matrix, using 1 for non-edges Graphs How much is space required for storage? Representations of Graphs Adjacency List Adjacency Matrix a b Elementary a b c d e Graph a Algorithms 0 1 1 1 0 Applications c b 1 0 0 0 1 c 1 0 0 1 1 d 1 0 1 0 1 d e e 0 1 1 1 0 8 / 29 Breadth-First Search (BFS) CSCE423/823 Introduction Given a graph G = (V; E) (directed or undirected) and a source node Types of s 2 V , BFS systematically visits every vertex that is reachable from s Graphs Uses a queue data structure to search in a breadth-first manner Representations of Graphs Creates a structure called a BFS tree such that for each vertex Elementary Graph v 2 V , the distance (number of edges) from s to v in tree is the Algorithms Breadth-First shortest path in G Search Depth-First Search Initialize each node's color to white Applications As a node is visited, color it to gray () in queue), then black () finished) 9 / 29 BFS(G; s) CSCE423/823 for each vertex u 2 V n fsg do 1 color[u] = white 2 d[u] = 1 Introduction 3 π[u] = nil 4 end Types of Graphs 5 color[s] = gray 6 d[s] = 0 Representations 7 π[s] = nil of Graphs 8 Q = ; Elementary 9 Enqueue(Q; s) Graph 10 while Q 6= ; do Algorithms 11 u = Dequeue(Q) Breadth-First 12 for each v 2 Adj[u] do Search 13 if color[v] == white then Depth-First Search 14 color[v] = gray 15 d[v] = d[u] + 1 Applications 16 π[v] = u 17 Enqueue(Q; v) 18 19 end 20 color[u] = black 21 end 10 / 29 BFS Example CSCE423/823 Introduction Types of Graphs Representations of Graphs Elementary Graph Algorithms Breadth-First Search Depth-First Search Applications 11 / 29 BFS Example (2) CSCE423/823 Introduction Types of Graphs Representations of Graphs Elementary Graph Algorithms Breadth-First Search Depth-First Search Applications 12 / 29 BFS Properties CSCE423/823 Introduction What is the running time? Types of Graphs Hint: How many times will a node be enqueued? Representations After the end of the algorithm, d[v] = shortest distance from s to v of Graphs Elementary ) Solves unweighted shortest paths Graph Can print the path from s to v by recursively following π[v], π[π[v]], Algorithms Breadth-First etc. Search Depth-First Search If d[v] == 1, then v not reachable from s Applications ) Solves reachability 13 / 29 Depth-First Search (DFS) CSCE423/823 Introduction Types of Another graph traversal algorithm Graphs Representations Unlike BFS, this one follows a path as deep as possible before of Graphs backtracking Elementary Graph Where BFS is \queue-like," DFS is \stack-like" Algorithms Breadth-First Search Tracks both \discovery time" and “finishing time" of each node, Depth-First Search which will come in handy later Applications 14 / 29 DFS(G) CSCE423/823 for each vertex u 2 V do Introduction 1 color[u] = white Types of Graphs 2 π[u] = nil Representations of Graphs 3 end Elementary 4 time = 0 Graph Algorithms 5 for each vertex u 2 V do Breadth-First Search 6 if color[u] == white then Depth-First Search 7 DFS-Visit(u) Applications 8 9 end 15 / 29 DFS-Visit(u) CSCE423/823 color[u] = gray Introduction 1 time = time + 1 Types of Graphs 2 d[u] = time Representations 3 for each v 2 Adj[u] do of Graphs 4 if color[v] == white then Elementary Graph 5 π[v] = u Algorithms 6 (v) Breadth-First DFS-Visit Search Depth-First 7 Search Applications 8 end 9 color[u] = black 10 f[u] = time = time + 1 16 / 29 DFS Example CSCE423/823 Introduction Types of Graphs Representations of Graphs Elementary Graph Algorithms Breadth-First Search Depth-First Search Applications 17 / 29 DFS Example (2) CSCE423/823 Introduction Types of Graphs Representations of Graphs Elementary Graph Algorithms Breadth-First Search Depth-First Search Applications 18 / 29 DFS Properties CSCE423/823 Introduction Types of Graphs Time complexity same as BFS: Θ(jV j + jEj) Representations Vertex u is a proper descendant of vertex v in the DF tree iff of Graphs d[v] < d[u] < f[u] < f[v] Elementary Graph ) Parenthesis structure: If one prints \(u" when discovering u and Algorithms Breadth-First \u)" when finishing u, then printed text will be a well-formed Search Depth-First parenthesized sentence Search Applications 19 / 29 DFS Properties (2) CSCE423/823 Classification of edges into groups A tree edge is one in the depth-first forest Introduction A back edge (u; v) connects a vertex u to its ancestor v in the DF Types of tree (includes self-loops) Graphs A forward edge is a nontree edge connecting a node to one of its DF Representations tree descendants of Graphs A cross edge goes between non-ancestral edges within a DF tree or Elementary Graph between DF trees Algorithms See labels in DFS example Breadth-First Search Depth-First Example use of this property: A graph has a cycle iff DFS discovers a Search Applications back edge (application: deadlock detection) When DFS first explores an edge (u; v), look at v's color: color[v] == white implies tree edge color[v] == gray implies back edge color[v] == black implies forward or cross edge 20 / 29 Application: Topological Sort CSCE423/823 A directed acyclic graph (dag) can represent precedences: an edge (x; y) implies that event/activity x must occur before y Introduction Types of Graphs Representations of Graphs Elementary Graph Algorithms Applications Topological Sort Strongly Connected Components 21 / 29 Application: Topological Sort (2) CSCE423/823 Introduction Types of Graphs A topological sort of a dag G is an linear ordering of its vertices such Representations of Graphs that if G contains an edge (u; v), then u appears before v in the ordering Elementary Graph Algorithms Applications Topological Sort Strongly Connected Components 22 / 29 Topological Sort Algorithm CSCE423/823 1 Call DFS algorithm on dag G Introduction 2 Types of As each vertex is finished, insert it to the front of a linked list Graphs 3 Return the linked list of vertices Representations of Graphs Elementary Graph Thus topological sort is a descending sort of vertices based on DFS Algorithms finishing times Applications Topological Sort Why does it work? Strongly Connected Components When a node is finished, it has no unexplored outgoing edges; i.e. all its descendant nodes are already finished and inserted at later spot in final sort 23 / 29 Application: Strongly Connected Components CSCE423/823 Given a directed graph G = (V; E), a strongly connected component (SCC) of G is a maximal set of vertices C ⊆ V such that for every pair of Introduction vertices u; v 2 C u is reachable from v and v is reachable from u Types of Graphs Representations of Graphs Elementary Graph Algorithms Applications Topological Sort Strongly Connected Components What are the SCCs of the above graph? 24 / 29 Transpose Graph CSCE423/823 Our algorithm for finding SCCs of G depends on the transpose of G, denoted GT Introduction GT is simply G with edges reversed Types of Graphs Fact: GT and G have same SCCs.
Load more

Recommended publications
  • Bidirected Graph from Wikipedia, the Free Encyclopedia Contents
    Bidirected graph From Wikipedia, the free encyclopedia Contents 1 Bidirected graph 1 1.1 Other meanings ............................................ 1 1.2 See also ................................................ 2 1.3 References ............................................... 2 2 Bipartite double cover 3 2.1 Construction .............................................. 3 2.2 Examples ............................................... 3 2.3 Matrix interpretation ......................................... 4 2.4 Properties ............................................... 4 2.5 Other double covers .......................................... 4 2.6 See also ................................................ 5 2.7 Notes ................................................. 5 2.8 References ............................................... 5 2.9 External links ............................................. 6 3 Complex question 7 3.1 Implication by question ........................................ 7 3.2 Complex question fallacy ....................................... 7 3.2.1 Similar questions and fallacies ................................ 8 3.3 Notes ................................................. 8 4 Directed graph 10 4.1 Basic terminology ........................................... 11 4.2 Indegree and outdegree ........................................ 11 4.3 Degree sequence ............................................ 12 4.4 Digraph connectivity .......................................... 12 4.5 Classes of digraphs .........................................
    [Show full text]
  • Graph Algorithms
    Graph Algorithms PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Wed, 29 Aug 2012 18:41:05 UTC Contents Articles Introduction 1 Graph theory 1 Glossary of graph theory 8 Undirected graphs 19 Directed graphs 26 Directed acyclic graphs 28 Computer representations of graphs 32 Adjacency list 35 Adjacency matrix 37 Implicit graph 40 Graph exploration and vertex ordering 44 Depth-first search 44 Breadth-first search 49 Lexicographic breadth-first search 52 Iterative deepening depth-first search 54 Topological sorting 57 Application: Dependency graphs 60 Connectivity of undirected graphs 62 Connected components 62 Edge connectivity 64 Vertex connectivity 65 Menger's theorems on edge and vertex connectivity 66 Ear decomposition 67 Algorithms for 2-edge-connected components 70 Algorithms for 2-vertex-connected components 72 Algorithms for 3-vertex-connected components 73 Karger's algorithm for general vertex connectivity 76 Connectivity of directed graphs 82 Strongly connected components 82 Tarjan's strongly connected components algorithm 83 Path-based strong component algorithm 86 Kosaraju's strongly connected components algorithm 87 Application: 2-satisfiability 88 Shortest paths 101 Shortest path problem 101 Dijkstra's algorithm for single-source shortest paths with positive edge lengths 106 Bellman–Ford algorithm for single-source shortest paths allowing negative edge lengths 112 Johnson's algorithm for all-pairs shortest paths in sparse graphs 115 Floyd–Warshall algorithm
    [Show full text]
  • Graph Theory) 15 3.1 Oriented Graphs
    Strong orientation From Wikipedia, the free encyclopedia Contents 1 Directed acyclic graph 1 1.1 Mathematical properties ....................................... 2 1.1.1 Reachability, transitive closure, and transitive reduction ................... 2 1.1.2 Topological ordering ..................................... 3 1.1.3 Combinatorial enumeration ................................. 3 1.1.4 Related families of graphs .................................. 3 1.2 Computational problems ....................................... 3 1.2.1 Topological sorting and recognition ............................. 3 1.2.2 Construction from cyclic graphs ............................... 4 1.2.3 Transitive closure and transitive reduction .......................... 4 1.2.4 Closure problem ....................................... 4 1.3 Applications .............................................. 4 1.3.1 Path algorithms ........................................ 4 1.3.2 Scheduling .......................................... 4 1.3.3 Data processing networks .................................. 5 1.3.4 Causal structures ....................................... 5 1.3.5 Genealogy and version history ................................ 5 1.3.6 Data compression ...................................... 6 1.4 References .............................................. 6 1.5 External links ............................................. 8 2 Directed graph 9 2.1 Basic terminology ........................................... 10 2.2 Indegree and outdegree .......................................
    [Show full text]
  • Strongly Connected Components Example: Strongly Connected Components
    Data Structures – LECTURE 14 Connected components • Find the largest components (sub-graphs) such that Strongly connected components there is a path from any vertex to any other vertex. • Applications : networking, communications. • Definition and motivation • Undirected graphs : apply BFS/DFS (inner function) • Algorithm from a vertex, and mark vertices as visited . Upon termination, repeat for every unvisited vertex. • Directed graphs : strongly connected components, not Chapter 22.5 in the textbook (pp 552—557). just connected: a path from u to v AND from v to u, which are not necessarily the same! Data Structures, Spring 2006 © L. Joskowicz 1 Data Structures, Spring 2006 © L. Joskowicz 2 Example: strongly connected components Example: strongly connected components b e b e a a d g d g c c h f h f Data Structures, Spring 2006 © L. Joskowicz 3 Data Structures, Spring 2006 © L. Joskowicz 4 Strongly connected components Strongly connected components graph • Definition : the strongly connected components • Definition : the SCC graph G~ = ( V~,E~) of the (SCC) C1, …, Ck of a directed graph G = ( V,E) are graph G = ( V,E) is as follows: ~ the largest disjoint sub-graphs (no common vertices – V = { C1, …, Ck}. Each SCC is a vertex. or edges) such that for any two vertices u and v in ~ ∈ ∈ ∈ – E = {( Ci,C j)| i≠j and ( x,y) E, where x Ci and y Cj}. Ci, there is a path from u to v and from v to u. A directed edge between components corresponds to a • Equivalence classes of the binary relation path (u,v) directed edge between them from any of their vertices.
    [Show full text]
  • Graph Factorization from Wikipedia, the Free Encyclopedia Contents
    Graph factorization From Wikipedia, the free encyclopedia Contents 1 2 × 2 real matrices 1 1.1 Profile ................................................. 1 1.2 Equi-areal mapping .......................................... 2 1.3 Functions of 2 × 2 real matrices .................................... 2 1.4 2 × 2 real matrices as complex numbers ............................... 3 1.5 References ............................................... 4 2 Abelian group 5 2.1 Definition ............................................... 5 2.2 Facts ................................................. 5 2.2.1 Notation ........................................... 5 2.2.2 Multiplication table ...................................... 6 2.3 Examples ............................................... 6 2.4 Historical remarks .......................................... 6 2.5 Properties ............................................... 6 2.6 Finite abelian groups ......................................... 7 2.6.1 Classification ......................................... 7 2.6.2 Automorphisms ....................................... 7 2.7 Infinite abelian groups ........................................ 8 2.7.1 Torsion groups ........................................ 9 2.7.2 Torsion-free and mixed groups ................................ 9 2.7.3 Invariants and classification .................................. 9 2.7.4 Additive groups of rings ................................... 9 2.8 Relation to other mathematical topics ................................. 10 2.9 A note on the typography ......................................
    [Show full text]
  • Graph Algorithms
    Graph Algorithms Andreas Klappenecker [based on slides by Prof. Welch] 1 Directed Graphs Let V be a finite set and E a binary relation on V, that is, E⊆VxV. Then the pair G=(V,E) is called a directed graph. • The elements in V are called vertices. • The elements in E are called edges. • Self-loops are allowed, i.e., E may contain (v,v). 2 Undirected Graphs Let V be a finite set and E a subset of { e | e ⊆ V, |e|=2 }. Then the pair G=(V,E) is called an undirected graph. • The elements in V are called vertices. • The elements in E are called edges, e={u,v}. • Self-loops are not allowed, e≠{u,u}={u}. 3 Adjacency By abuse of notation, we will write (u,v) for an edge {u,v} in an undirected graph. If (u,v) in E, then we say that the vertex v is adjacent to the vertex u. For undirected graphs, adjacency is a symmetric relation. 4 Graph Representations • Adjacency lists • Adjacency matrix 5 Adjacency List Representation a b a b c e b a d e c d c d e + Space-efficient: just O(|V|) space for sparse graphs - Testing adjacency is O(|V|) in the worst case 6 Adjacency Matrix a b c d e a b a 0 1 1 0 0 e b 1 0 0 1 1 c 1 0 0 1 0 c d d 0 1 1 0 1 e 0 1 0 1 0 + Can check adjacency in constant time - Needs Ω(|V|2) space 7 Graph Traversals Ways to traverse or search a graph such that every node is visited exactly once 8 Breadth-First Search 9 Breadth First Search (BFS) Input: A graph G = (V,E) and source node s in V for each node v do mark v as unvisited od mark s as visited enq(Q,s) // first-in first-out queue Q while Q is not empty do u := deq(Q) for each unvisited neighbor v of u do mark v as visited; enq(Q,v); od od 10 BFS Example Visit the nodes in the a b order: s s c a, d d b, c Workout the evolution of the state of queue.
    [Show full text]
  • (Gabow's Vs Kosaraju's) Based on Adjacency List by Saleh Alshomrani & Gulraiz Iqbal King Abdulaziz University, Saudi Arabia
    Global Journal of Computer Science and Technology Network, Web & Security Volume 13 Issue 11 Version 1.0 Year 2013 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 0975-4172 & Print ISSN: 0975-4350 An Extended Experimental Evaluation of SCC (Gabow's vs Kosaraju's) based on Adjacency List By Saleh Alshomrani & Gulraiz Iqbal King Abdulaziz University, Saudi Arabia Abstract - We present the results of a study comparing three strongly connected components algorithms. The goal of this work is to extend the understandings and to help practitioners choose appropriate options. During experiment, we compared and analysed strongly connected components algorithm by using dynamic graph representation (adjacency list). Mainly we focused on i. Experimental Comparison of strongly connected components algorithms. ii. Experimental Analysis of a particular algorithm. Our experiments consist large set of random directed graph with N number of vertices V and edges E to compute graph performance using dynamic graph representation. We implemented strongly connected graph algorithms, tested and optimized using efficient data structure. The article presents detailed results based on significant performance, preferences between SCC algorithms and provides practical recommenddations on their use. During experimentation, we found some interesting results particularly efficiency of Cheriyan-Mehlhorn- Gabow's as it is more efficient in computing strongly connected components then Kosaraju's algorithm. Keywords : graph algorithms, directed graph, SCC (strongly connected components), transitive closure. GJCST-E Classification : C.2.6 An Extended Experimental Evaluation of SCC Gabows vs Kosarajusbased on Adjacency List Strictly as per the compliance and regulations of: © 2013. Saleh Alshomrani & Gulraiz Iqbal.
    [Show full text]
  • Graph Theory, an Antiprism Graph Is a Graph That Has One of the Antiprisms As Its Skeleton
    Graph families From Wikipedia, the free encyclopedia Chapter 1 Antiprism graph In the mathematical field of graph theory, an antiprism graph is a graph that has one of the antiprisms as its skeleton. An n-sided antiprism has 2n vertices and 4n edges. They are regular, polyhedral (and therefore by necessity also 3- vertex-connected, vertex-transitive, and planar graphs), and also Hamiltonian graphs.[1] 1.1 Examples The first graph in the sequence, the octahedral graph, has 6 vertices and 12 edges. Later graphs in the sequence may be named after the type of antiprism they correspond to: • Octahedral graph – 6 vertices, 12 edges • square antiprismatic graph – 8 vertices, 16 edges • Pentagonal antiprismatic graph – 10 vertices, 20 edges • Hexagonal antiprismatic graph – 12 vertices, 24 edges • Heptagonal antiprismatic graph – 14 vertices, 28 edges • Octagonal antiprismatic graph– 16 vertices, 32 edges • ... Although geometrically the star polygons also form the faces of a different sequence of (self-intersecting) antiprisms, the star antiprisms, they do not form a different sequence of graphs. 1.2 Related graphs An antiprism graph is a special case of a circulant graph, Ci₂n(2,1). Other infinite sequences of polyhedral graph formed in a similar way from polyhedra with regular-polygon bases include the prism graphs (graphs of prisms) and wheel graphs (graphs of pyramids). Other vertex-transitive polyhedral graphs include the Archimedean graphs. 1.3 References [1] Read, R. C. and Wilson, R. J. An Atlas of Graphs, Oxford, England: Oxford University Press, 2004 reprint, Chapter 6 special graphs pp. 261, 270. 2 1.4. EXTERNAL LINKS 3 1.4 External links • Weisstein, Eric W., “Antiprism graph”, MathWorld.
    [Show full text]