ITEC 2620M Introduction to Data Structures

1. ITEC 2620MIntroduction to Data Structures Instructor: Prof. Z. Yang Course Website: http://people.math.yorku.ca/~zyang/itec2620m.htm Office: TEL 3049

2. Graphs

3. Key Points of this Lecture • Graph Algorithms • Definitions, representations, analysis • Shortest paths • Minimum-cost spanning tree

4. Basic Definitions • A graph G = ( V, E ) consists of a set of vertices V and a set of edges E – each edge E connects a pair of vertices in V. • Graphs can be directed or undirected • redraw above with arrows – first vertex is source • Graphs may be weighted • redraw above with weights, combine definitions • A vertex vi is adjacent to another vertex vj if they are connected by an edge in E. These vertices are neighbours • A path is a sequence of vertices in which each vertex is adjacent to its predecessor and successor • The length of a path is the number of edges in it • The cost of a path is the sum of edge weights in the path

5. Basic Definitions (Cont’d) • A cycle is a path of length greater than one that begins and ends at the same vertex • A simple cycle is a cycle of length greater than three that does not visit any vertex (except the start/finish) more than once • Two vertices are connected if there as a path between them • A subset of vertices S is a connected component of G if there is a path from each vertex vi to every other distinct vertex vj in S • The degree of a vertex is the number of edges incident to it – the number of vertices that it is connected to • A graph is acyclic if it has no cycles (e.g. a tree) • A directed acyclic graph is called a DAG or digraph(useful for unidirectional relationships

6. Representations • The adjacency matrix of graph G = ( V, E ) for vertices numbered 0 to n-1 is an n x n matrix M where M[i][j] is 1 if there is an edge from vi to vi, and 0 otherwise. • The adjacency list of graph G = ( V, E ) for vertices numbered 0 to n-1 consists of an array of n linked lists. The ith linked list includes the node j if there is an edge from vi to vi. • Example

7. Comparisons and Analysis • Space • adjacency matrix uses O(|V|2) space (constant) • adjacency list uses O(|V| + |E|) space (note: pointer overhead) • better for sparse graphs (graphs with few edges) • Access Time • Is there an edge connecting vi to vj? • adjacency matrix – O(1) • adjacency list – O(d) • Visit all edges incident to vi • adjacency matrix – O(n) • adjacency list – O(d) • Primary operation of algorithm and density of graph determines more efficient data structure • complete graphs should use adjacency matrix • traversals of sparse graphs should use adjacency list

8. Spanning Tree and Shortest Paths • Minimum-Cost Spanning Tree • assume weighted (undirected) connected graph • use Prim’s algorithm (a greedy algorithm) • from visited vertices, pick least-cost edge to an unvisited vertex • Shortest Paths • assume weighted (undirected) connected graph • use Dijkstra’s algorithm (a greedy algorithm) • build paths from unvisited vertex with least current cost