Graphs. Ric Glassey

Transcription

1 Graphs Ric Glassey

2 Overview Graphs Aim: Introduce graphs as a fundamental data structure that can be applied in many real-world problems Applications of graphs Definition and types of graphs Underlying representation & performance

3 Applications of Graphs

4 Applications of Graphs Application Item Connection Computer Network Node Network Link Web Page Link Software Method Call Social Network Person Friendship T-bana Map Stop Rail

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7 Modelling with Graphs Many problems can be tackled with graphs Scale and structure of graph can range massively Operations on items and connections are the same irrespective of scale/structure Challenge is to find the best representation to fit the application requirements

8 Definitions

9 Graph A graph is a set of vertices and a collection of edges that each connect a pair of vertices V is the unique set of vertices E is the collection of edges v-w represents the connection between vertex v and vertex w

10 Graph Vertex 3 V = [0, 1, 2, 3, 4, 5, 6] Edge Value E = [ ] 0-0, 0-4, 1-2, 1-5, 2-1, 3-4, 4-0, 4-3, 4-6, 5-1, 5-4, 5-6,

11 Anomalies We can find self-loops and parallel edges Applications may need these, but care is required when processing Self-loop Parallel Edges 0 1

12 Terminology Adjacent: One V connected to another V Degree: The number of E connected to any V Subgraph: A subset of a larger graph V(4) has degree '3' V(4,5,6) is a subgraph 5 V(1) is adjacent to V(5) 1

13 Terminology Path: sequence of V connected by E Simple Path: no repeated visits to any V Cycle: A path that starts and ends at one V

14 Terminology A graph is Connected if there is a path from every V to any other V A graph is Acyclic if there are no cycles A graph is a Tree when it is both connected and acyclic

15 Terminology The Density of a graph is the proportion of all possible pairs of V connected by E A graph is Sparse if there are relatively few E present A graph is Dense graph has relatively few edges missing V=50, E=200 V=50, E=1000

16 Types of Graphs

17 Types Undirected Graphs (graphs) Simple connection between V and E Directed Graphs (digraphs) * Direction of connection is significant between V Edge-weighted Graphs * Each connection E has an associated weight Edge-weighted Digraphs * Direction and weight on connections

18 Types - Graph, Digraph

19 Types - Edge-weighted

20 Graph API

21 Public class Graph Graph(int V) Graph(InputStream in) int V( ) int E( ) void addedge(int v, int w) Iterable<Integer> adjacent(int v) String tostring( ) Create a V vertex graph with no edges Create a graph from input stream (e.g. from File) Number of V Number of E Add edge v-w to this graph Return collection of V adjacent to v Return a string representation of graph

22 public static int degree(graph G, int v) { int degree = 0; for (int w : G.adjacent(v)) { degree++; } return degree; } public static int maxdegree(graph G) { int max = 0; for (int v = 0; v < G.V(); v++){ if (degree(g, v) > max) { max = degree(g, v); } } return max; } public static int avgdegree(graph G){ return 2 * G.E() / G.V(); }

23 Underlying Representation

24 Implementation of Graph API Recall List data structure API: Add(), Next(), Remove() etc We could choose from an array-based implementation or a linked list of objects Each represents an underlying representation Each had differing performance implications

25 Data Structure Array Linked List Operation Search O(1) O(n) Insert O(n) O(1) Delete O(n) O(1)

26 Implementation of Graph API Graph API has two main underlying representations Adjacency Matrix Adjacency List Both focus on modelling the adjacency of vertices, but each has implications on space/time complexity

27 Adjacency Matrix

28 Adjacency Matrix V by V array A True (1) entry in table indicates there is an edge between v and w

29 Adjacency Matrix Advantages: Finding if vertices are adjacent is fast More appropriate for dense graphs Disadvantages Add/Remove potentially affects many cells Less appropriate for sparse graphs

30 Adjacency Matrix A lot of wasted space

31 Adjacency List Adj [ ]

32 Adjacency List Adj [ ] ?

33 Adjacency List Advantages Space efficient representation Most operations are faster than matrix Disadvantages Edge removal and adjacency testing are less efficient

34 Performance Comparison Data Structure Operation Adjacency Matrix Adjacency List Storage O(V 2 ) O(V+E) Add Vertex O(V 2 ) O(1) Add Edge O(1) O(1) Remove Vertex O(V 2 ) O(E) Remove Edge O(1) O(E) Are Adjacent O(1) O(V)

35 Adj [ ]

36 Readings Algorithms and Data Structures *required reading* Nilsson Graphs Introduction to Algorithms, 3rd Edition Corman et al Chapter 22, Elementary Graph Algorithms Algorithms 4th Edition Sedgwick & Wayne Chapter 4, Graphs