Chapter 05 Introduction to Graph And Search Algorithms

2  A Graph is a data structure which consists of a set of vertices, and a set of edges that connect (some of) them.  That is, G = ( V, E ), Where V - set of vertices, E - set of edges  The edges indicate how we can move through the graph Vertex (Node) Edge V = {1, 2, 3, 4, 5} E = { (1,2), (1,3), (1,4), (2,3), (3,5), (4,5) } What is Graph?

3  Computer Networks  Electrical Circuits  Road Map Computer Resistor/Inductor/… City Applications

4  A Directed Graph or Digraph is a graph where each edge has a direction  The edges in a digraph are called Arcs or Directed Edges  A directed graph edges’ allow travel in one direction  Example:  G = (V, E), where V = {1, 2, 3, 4, 5, 6} and E = {(1,4), (2,1), (2,3), (3,2), (4,3), (4,5), (4,6), (5,3), (6,1), (6,5)}  (1, 4) = 1→4 where 1 is the tail and 4 is the head Graph Categorization

5  An Undirected Graph is a graph where the edges have no directions  The edges in an undirected graph are called Undirected Edges  An undirected graph edges’ allow travel in either direction  Example:  G = (V, E), where V = {1, 2, 3, 4, 5} and E = {(1,2), (1,3), (1,4), (2,3), (3,5), (4,5)} Graph Categorization (Contd.)

 A Weighted Graph is a graph where all the edges are assigned weights.  If the same pair of vertices have more than one edge, that graph is called a Multigraph  Edges with the same end vertices are called parallel edges Graph Categorization (Contd.) 6

7  Adjacent vertices: :two vertices are adjacent if they are connected by an edge  An edge (i,j) is Incident to vertices i and j.  Vertices 2 and 5 are not adjacent  Loop or self edges: An edge ( i,i ) is called a self edge or a loop.  In graphs loops are not permitted  ( 1,1 ) and ( 4,4 ) are self edges Graph Terminology

8  Adjacent Edges: two non parallel edges are adjacent if they have vertex in common  An edge (i,j) is Incident to vertices i and j Graph Terminology

9  Path: A sequence of edges in the graph. A path is a subset of E that is a series of edges between two vertices  There can be more than one path between two vertices.  Vertex A is reachable from B if there is a path from A to B.  Paths from B to D  B, A, D  B, C, D  shortest path is the minimum number edges needed to connect two vertices  Simple Path: A path where all the vertices are distinct.( without loops and parallel edges)  1,4,5,3 is a simple path.  But 1,4,5,4 is not a simple path. A D C F E B G Graph Terminology (Contd.)

10  Length : Sum of the lengths of the edges on the path.  The length of a path in a graph is the number of edges in the path  Length of the path 1,4,5,3 is 3  Circuit: A path whose first and last vertices are the same.  The path 3,2,1,4,5,3 is a circuit.  Cycle: A circuit where all the vertices are distinct except for the first (and the last) vertex.  1,4,5,3,1 is a cycle, but 1,4,5,4,1 is not a cycle.  Hamiltonian Cycle: A Cycle that contains all the vertices of the graph.  1,4,5,3,2,1 is a Hamiltonian Cycle Graph Terminology (Contd.)

11  Degree of a Vertex : In an undirected graph, the no. of edges incident to the vertex  In-degree: The no. of edges entering the vertex in a digraph  Out-Degree: The no. of edges leaving the vertex in a digraph  In-degree of 1 is 3  Out-degree of 1 is 1  A Subgraph of graph G=(V,E) is a graph H=(U,F) such that U Є V and F Є E G=(V,E) H=(U,F) Graph Terminology (Contd.)

12  A graph is said to be Connected if there is at least one path from every vertex to every other vertex in the graph.  Tree: A connected undirected graph that contains no cycles  Forest: A graph that does not contain a cycle Connected Unconnected Tree Forest Graph Terminology (Contd.)

13  The Spanning Tree of a Graph G is a subgraph of G that is a tree and contains all the vertices of G Graph Spanning Tree Graph Terminology (Contd.)

14  Adjacency Matrix (A)  The Adjacency Matrix A=(a i,j ) of a graph G=(V,E) with n nodes is an nXn matrix  Each element of A is either 0 or 1, depending on the adjacency of the nodes  a ij = 1, if (i,j) Є E, = 0, otherwise  Example: Find the adjacency matrices of the following graphs Representation of Graphs

15  Adjacency Matrix of a Weighted Graph  The weight of the edge can be shown in the matrix when the vertices are adjacent  A nil value (0 or ∞) depending on the problem is used when they are not adjacent  Example: To find the minimum distance between nodes... ∞4∞ ∞∞5 95∞ Representation of Graphs (Contd.)

16  Adjacency List  An Adjacency list is an array of lists, each list showing the vertices a given vertex is adjacent to….  Adjacency List of a Weighted Graph  The weight is included in the list Representation of Graphs (Contd.)

Searching Graphs 17

 Why do we need to search graphs?  To find paths  To look for connectivity  Depth-First Search (DFS)  Start from an arbitrary node  Visit (Explore) an unvisited adjacent edge  If the node visited is a dead end, go back to the previous node (Backtrack)  Stop when no unvisited nodes are found and no backtracking can be done  Implemented using a Stack  Explore if possible, Backtrack otherwise… Depth-First Search 18

Algorithm DFS(G) { for each vertex u Є V[G] do { Color[u] := white; Parent[u] := nil; } for each vertex u Є V[G] do if (Color[u] = white) then DFS_Visit(u); } Algorithm DFS_Visit(u) { Color[u] := gray for each vertex v Є Adj[u] do if (Color[v] = white) then { Parent[v] := u; DFS_Visit(v); } Color[u] := black; } white - Unvisited gray - Discovered black - Finished DFS Algorithm 19

AC D B E H G F A, B, F, C, G, E Example 20

 Start from an arbitrary node  Visit all the adjacent nodes (distance=1)  Visit the nodes adjacent to the visited nodes (distance=2, 3 etc.)  Stop when a dead end is met  Implemented using a Queue  Explore all nodes at distance d… Breadth-First Search (BFS) 21

Algorithm BFS(G, s) { for each vertex u Є V[G] – {s} do { Color[u] := white; Distance[u] := ∞; Parent[u] := nil; } Color[s] := gray; Distance[s] := 0; Parent[s] := nil; Q := Ø; Enqueue (Q, s); while (Q ≠ Ø) do { u := Dequeue (Q); for each v Є Adj[u] do if (Color[v] = white) then { Color[v] := gray; Distance[v] := Distance[u] + 1; Parent[v] := u;Enqueue (Q,v); } Color[u] := black; } white - Unvisited gray - Discovered black - Finished s - Source Vertex Q - FIFO Queue BFS Algorithm 22

AC D B E H G F A, B, C, E, F, G Example and Analysis 23

Depth-first searching A depth-first search (DFS) explores a path all the way to a leaf before backtracking and exploring another path For example, after searching A, then B, then D, the search backtracks and tries another path from B Node are explored in the order A B D E H L M N I O P C F G J K Q N will be found before J 24 LM N OP G Q H J IK FED BC A

Breadth-first searching A breadth-first search (BFS) explores nodes nearest the root before exploring nodes further away For example, after searching A, then B, then C, the search proceeds with D, E, F, G Node are explored in the order A B C D E F G H I J K L M N O P Q J will be found before N 25 LM N OP G Q H J IK FED BC A