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.