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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
Applications of Graphs
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
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
Definitions
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
Graph
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Vertex
Value
Edge
V = [0, 1, 2, 3, 4, 5, 6]
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, 6-4 6-5]
Anomalies• We can find self-loops and parallel edges
• Applications may need these, but care is required when processing
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Parallel EdgesSelf-loop
Terminology• Adjacent: One V connected to another V
• Degree: The number of E connected to any V
• Subgraph: A subset of a larger graph
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V(4) has degree '3'
V(1) is adjacent to V(5)
V(4,5,6) is a subgraph
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
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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
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
Types of Graphs
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
Types - Graph, Digraph
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Types - Edge-weighted
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Graph API
Public class Graph
Graph(int V) Create a V vertex graph with no edges
Graph(InputStream in) Create a graph from input stream (e.g. from File)
int V( ) Number of V
int E( ) Number of E
void addEdge(int v, int w) Add edge v-w to this graph
Iterable<Integer> adjacent(int v) Return collection of V adjacent to v
String toString( ) Return a string representation of graph
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();
}
Underlying Representation
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
Data Structure
Operation
Array Linked List
Search O(1) O(n)
Insert O(n) O(1)
Delete O(n) O(1)
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
Adjacency Matrix
01234560|10001001|00100102|01000003|00001004|10010115|01001016|0000110
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Adjacency Matrix• V by V array
• A True (1) entry in table indicates there is an edge between v and w
01234560|10001001|00100102|01000003|00001004|10010115|01001016|0000110
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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
Adjacency Matrix
Adjacency Matrix
A lot of wasted space
01234560|10001001|00100102|01000003|00001004|10010115|01001016|0000110
Adjacency List
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Adj [ ]
0
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0 3 5 6
1 4 6
4 5
Adjacency List
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Adj [ ]
0
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4 5
?
Adjacency List
• Advantages
• Space efficient representation
• Most operations are faster than matrix
• Disadvantages
• Edge removal and adjacency testing are less efficient
Performance Comparison
Data Structure
Operation
Adjacency Matrix
Adjacency List
Storage O(V2) O(V+E)Add Vertex O(V2) O(1)Add Edge O(1) O(1)Remove Vertex O(V2) O(E)Remove Edge O(1) O(E)Are Adjacent O(1) O(V)
Adj [ ]
0
1
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0 4
2 5
1
4
0 3 5 6
1 4 6
4 5
01234560|10001001|00100102|01000003|00001004|10010115|01001016|0000110
Readings• Algorithms and Data Structures *required reading*
• Nilsson
• Graphs
• http://www.nada.kth.se/~snilsson/algoritmer/grafer/
• Introduction to Algorithms, 3rd Edition
• Corman et al
• Chapter 22, Elementary Graph Algorithms
• http://kth-primo.hosted.exlibrisgroup.com/KTH:KTH_SFX2560000000068328
• Algorithms 4th Edition
• Sedgwick & Wayne
• Chapter 4, Graphs
• http://kth-primo.hosted.exlibrisgroup.com/KTH:KTH_LMSMILL12447056
