ECE 242. Data Structures

Transcription

1 ECE 242 Data Structures Lecture 28 Introduction to Graphs Overview Problem: How do we represent irregular connections between locations? Graphs Definition Directed and Undirected graph Simple path and cycle Connected and Unconnected graph Weighted graph Graph Representation Graph Traversal

2 Abstract Data Type We have discussed: List Tree Today we will talk about Graphs Northwest Airline Flight Plan Seattle Anchorage SF Minneapolis Hartford Boston Austin Atlanta

3 Computer Network Or Internet AT&T MCI Regional Network Intel Campus Umass Lots of Interesting Problems Graph traversal Shortest path between two nodes Anything else?

4 Concepts of Graphs node or vertex edges (weight) Graph Definition G = (V, E) V is the vertex set Vertices are also called nodes or points E is the edge set Each edge connects two vertices Edges are also called arcs or lines

5 Graphs, Vertices and Edges Undirected vs. Directed Graph Undirected Graph no oriented edge Directed Graph every edge has oriented vertex

6 Subgraph Subgraph: subset of vertices and edges Simple Path A simple path traverse a node no more than once ABCD is a simple path B A C D path

7 Cycle A cycle is a path that starts and ends at the same point CBDC is a cycle B A C D Connected vs. Unconnected Graph Connected Graph Unconnected Graph

8 Directed Acyclic Graph Directed Acyclic Graph (DAG) : directed graph without cycle Examples Course Requirement Graph: DAG Directed Acyclic Graph This is not DAG ABCD is a cycle D A C B

9 Weighted Graph Weighted graph: a graph with numbers assigned to its edges Weight: cost, distance, travel time, hop, etc Representation Of Graph Two representations Adjacency Matrix Adjacency List

10 Adjacency Matrix N nodes in graph Use Matrix A[0 N-1][0 N-1] if vertex i and vertex j are adjacent in graph, A[i][j] = 1, otherwise A[i][j] = 0 if vertex i has a loop, A[i][i] = 1 if vertex i has no loop, A[i][i] = 0 Example of Adjacency Matrix A[i][j] So, Matrix A =

11 Undirected vs. Directed Undirected graph adjacency matrix is symmetric A[i][j] always equals A[j][i] Directed graph adjacency matrix may not be symmetric A[i][j] may not equal A[j][i] Directed Graph Matrix Representation A[i][j] So, Matrix A =

12 Weighted Graph A[i][j] So, Matrix A = Adjacency Matrices

13 Adjacency List An array of list the ith element of array is a list of vertices that connect to vertex i Example of Adjacency List vertex 0 connect to vertex 1, 2 and vertex 1 connects to vertex 2 connects to

14 Weighted Graph Weighted graph: extend each node with an addition field: weight Adjacency List

15 Compare Two Representations Given two vertices: u, v find out if u and v are adjacent Given a vertex: u enumerate all neighbors of u For all vertices enumerate all neighbors of each vertex Comparison Of Representations Cost Adjacency Matrix Adjacency List Given two vertices u and v: find out whether u and v are adjacent O(1) degree of node O(N) Given a vertex u: enumerate all neighbors of u For all vertices: enumerate all neighbors of each vertex O(N) O(N 2 ) degree of node O(N) Summations of all node degree O(E)

16 Complete Graph There is an edge between any two vertices Total number of edges in graph: E = N(N-1)/2 = O(N 2 ) Sparse Graph There are very small no. of edges in the graph For example: E = N-1= O(N)

17 Space Requirement of Representations Memory space: adjacency matrix O(N 2 ) adjacency list O(E) Sparse graph adjacency list is better Dense graph same running time Summary We will spend the rest of the semester discussing graphs Many important problems represented as graphs Mapquest, chip routing, Google, maze type games Next step: search for a node in a graph Other problem: shortest path to node, shortest path to all nodes