GRAPHS

Definition The Graph Data Structure Drawing of a Graph set V of vertices collection E of edges (pairs of vertices in V) Drawing of a Graph vertex <-> circle/oval edge <-> line connecting the vertex pair

Sample Uses Airports and flights Persons and acquaintance relationships Intersections and streets Computers and connections; the Internet

Graph Example Graph G=(V,E): V={a,b,c,d}, E={(a,b),(b,c),(b,d),(a,d)}

Adjacency Vertices connected by an edge are adjacent An edge e is incident on a vertex v (and vice-versa) if v is an endpoint of e Degree of a vertex v, deg(v) number of edges incident on v if directed graph, indeg(v) and outdeg(v)

Graph Properties Graph G with n vertices and m edges v in G deg(v) = 2m v in G indeg(v) = v in G outdeg(v) = m m is O(n2) m <= n(n-1)/2 if G is undirected m <= n(n-1) if G is directed

Subgraphs Graph H a subgraph of graph G Spanning subgraph vertex set of H is a subset of vertex set of G edge set of H is a subset of edge set of G Spanning subgraph vertex set of H identical to vertex set of G

Paths and Cycles (simple) Path (simple) Cycle sequence of distinct vertices such that consecutive vertices are connected through edges (simple) Cycle same as path except that first and last vertices are identical Directed paths and cycles edge directions matter

Connectivity Connected graph there is a path between any two vertices

Graph Methods Container methods General or Global methods numVertices(),numEdges(),vertices(),edges() Directed Edge methods indegree(v), inadjacentVertices(v), inincidentEdges(v) Update methods adding/removing vertices or edges

General Methods numvertices() -returns the number of vertices in G numedges() - returns the number of edges in G vertices() - return an enumeration of the vertex positions in G edges() - returns an enumeration of the edge positions in G

General Methods Degree (v) - returns the degree of v adjacentvertices(v) - returns an enumeration of the vertices adjacent to v incidentedges(v) - returns an enumeration of the edges incident upon v endvertices(e) - return an array of size 2 storing the end vertices of e

Directed Edges Directededges() - return an enumeration of all directed edges indegree(v) - return the indegree of v outdegree(v) - return the outdegree of v origin (v) - return the origin of directed edge e other functions - inadjacentvertices(), outadjacentvertices(), destination()

Updating Graphs Insertedge(v,w,o) - insert and return an undirected edge between vertices v and w storing the object o at this position Insertvertex(o) - insert and return a new vertex storing the object o at this position removevertex (v) - remove the vertex v and all incident edges removeedge(e) - remove edge e

Implementation of a graph data structure

Graph Implementations Edge List vertices and edges are nodes edge nodes refer to incident vertex nodes Adjacency List same as edge list but vertex nodes also refer to incident edge nodes Adjacency Matrix 2D matrix of vertices; entries are edges

Implementation Examples Illustrate the following graph using the different implementations 4 a b 5 1 3 c d 2

Edge List

Adjacency List

Adjacency Matrix

Implementation Examples- Query Illustrate the following graph using the different implementations 4 a b 8 5 e 1 3 6 c 7 d 2

Comparing Implementations Edge List simplest but most inefficient Adjacency List many vertex operations are O(d) time where d is the degree of the vertex Adjacency Matrix adjacency test is O(1) addition/removal of a vertex: resize 2D array

Graph Traversals Traversal - A systematic procedure for exploring a graph by examining all of its vertices and edges Consistency - rules to determine the order of visits (top-down, alphabetical,etc.) - make the path consistent with the actual graph (don’t change conventions)

Graph Traversals Depth First Search (DFS) Breadth First Search (BFS) visit a starting vertex s recursively visit unvisited adjacent vertices Breadth First Search (BFS) visit starting vertex s visit unvisited vertices adjacent to s, then visit unvisited vertices adjacent to them, and so on … (visit by levels)

DFS Traversal Depth First Search - recursively visit (traverse) unvisited vertices Dead End - reaching a vertex where all the incident edges go to a previously visited vertex Back Track - If a dead-end vertex is reached, go to the previous vertex

Traversal Examples DFS: a,b,c,d,h,e,f,g,l,k,j,i c b d a e g f h i j l

DFS Traversal Path of the DFS traversal - path=a,b,c,d,h - h is a dead end vertex - back track to d,c,b,a and go to another adjacent (unvisited) vertex -> e - path=a,e,f,g,k,j,i - i is a dead end vertex - back track to all the way back to a (also a dead end vertex) finished

BFS Traversal Visit adjacent vertices by levels discovery edges - edges that lead to an unvisited vertex cross edge - edges that lead to a previously visited vertex

Traversal Examples BFS: a,b,e,i,c,f,j,d,g,k,h,l c b d a e g f h i j l

DFS Algorithm Algorithm DFS(v) Input: Vertex v Output: Depends on the application mark v as visited; perform processing for each vertex w adjacent to v do if w is unmarked then DFS(w)

BFS Algorithm Algorithm BFS(s) Input: Starting vertex s Output: Depends on the application // traversal algorithm uses a queue Q // loop terminates when Q is empty

BFS Algorithm continued Q  new Queue() mark s as visited and Q.enqueue(s) while not Q.isEmpty() v  Q.dequeue() perform processing on v for each vertex w adjacent to v do if w is unmarked then mark w as visited and Q.enqueue(w)

Query Let G be a graph whose vertices are the integers 1 through 8 and let the adjacent vertices of each vertex be given by the table below : Vertex Adjacent Vertices 1 (2,3,4) 2 (1,3,4) 3 (1,2,4) 4 (1,2,3,6) 5 (6,7,8) 6 (4,5,7) 7 (5,6,8) 8 (5,7)

Query Assume that in the traversal of G, the adjacent vertices of a given vertex are returned in the same order as they are listed in the above table: a) Draw G b) Give the sequence of vertices of G visited using a DFS traversal order starting at vertex 1 c) Give the sequence of vertices visited using a BFS traversal starting at vertex 1

Weighted Graphs Overview

Weighted Graphs Graph G = (V,E) such that there are weights/costs associated with each edge w((a,b)): cost of edge (a,b) representation: matrix entries <-> costs a 20 35 e b 12 32 c 65 d

Problems on Weighted Graphs Minimum-cost Spanning Tree Given a weighted graph G, determine a spanning tree with minimum total edge cost Single-source shortest paths Given a weighted graph G and a source vertex v in G, determine the shortest paths from v to all other vertices in G Path length: sum of all edges in path

Weighted Graphs-Concerns Minimum Cost Shortest Path

Weighted Graphs Shortest Path - Dijkstra’s Algorithm Minimum Spanning Trees - Kruskal’s Algorithm