Graphs In Data Structure
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A graph G is composed of vertices V connected by edges E. It can be represented using an adjacency matrix or adjacency lists. Graph search algorithms like depth-first search (DFS) and breadth-first search (BFS) are used to traverse the graph and find paths between vertices. DFS recursively explores edges until reaching the end of a branch before backtracking, while BFS explores all neighbors at each level before moving to the next.
![Adjacency Matrix
Let G=(V,E) be a graph with n vertices.
The adjacency matrix of G is a two-dimensional
n by n array, say adj_mat
If the edge (vi, vj) is in E(G), adj_mat[i][j]=1
If there is no such edge in E(G), adj_mat[i][j]=0
The adjacency matrix for an undirected graph is
symmetric; the adjacency matrix for a digraph
need not be symmetric](https://image.slidesharecdn.com/graphs1-130126021226-phpapp01/85/Graphs-In-Data-Structure-18-320.jpg)
![Merits of Adjacency Matrix
From the adjacency matrix, to determine the
connection of vertices is easy
n −1
The degree of a vertex is ∑ adj _ mat[i][ j ]
j=0
For a digraph (= directed graph), the row sum is
the out_degree, while the column sum is the
in_degree
n −1 n −1
ind (vi ) = ∑ A[ j , i ] outd (vi ) = ∑ A[i , j ]
j =0 j =0](https://image.slidesharecdn.com/graphs1-130126021226-phpapp01/85/Graphs-In-Data-Structure-20-320.jpg)
![Adjacency Lists (data structures)
Each row in adjacency matrix is represented as an adjacency list.
#define MAX_VERTICES 50
typedef struct node *node_pointer;
typedef struct node {
int vertex;
struct node *link;
};
node_pointer graph[MAX_VERTICES];
int n=0; /* vertices currently in use */](https://image.slidesharecdn.com/graphs1-130126021226-phpapp01/85/Graphs-In-Data-Structure-21-320.jpg)
Graphs In Data Structure
- 1. GRAPH
- 2. What is a Graph? • A graph G = (V,E) is composed of: V: set of vertices E: set of edges connecting the vertices in V • An edge e = (u,v) is a pair of vertices • Example: a b V= {a,b,c,d,e} E= {(a,b),(a,c), c (a,d), (b,e),(c,d),(c,e), (d,e)} d e
- 3. Applications CS16 • electronic circuits • networks (roads, flights, communications) JFK LAX STL HNL DFW FTL
- 4. Terminology: Adjacent and Incident • If (v0, v1) is an edge in an undirected graph, – v0 and v1 are adjacent – The edge (v0, v1) is incident on vertices v0 and v1 • If <v0, v1> is an edge in a directed graph – v0 is adjacent to v1, and v1 is adjacent from v0 – The edge <v0, v1> is incident on v0 and v1
- 5. Terminology: Degree of a Vertex The degree of a vertex is the number of edges incident to that vertex For directed graph, the in-degree of a vertex v is the number of edges that have v as the head the out-degree of a vertex v is the number of edges that have v as the tail if di is the degree of a vertex i in a graph G with n vertices and e edges, the number of edges is n −1 ∑d ) / 2 Why? Since adjacent vertices each e =( i count the adjoining edge, it will be 0 counted twice
- 6. Examples 0 3 2 0 1 2 3 3 3 1 2 3 3 4 5 6 31 G 1 1 G2 1 1 3 0 in:1, out: 1 directed graph in-degree out-degree 1 in: 1, out: 2 2 in: 1, out: 0 G3
- 7. Terminology: Path • path: sequence of 3 2 vertices v1,v2,. . .vk such that consecutive vertices vi 3 and vi+1 are adjacent. 3 3 a b a b c c d e d e abedc bedc 7
- 8. More Terminology • simple path: no repeated vertices a b bec c d e • cycle: simple path, except that the last vertex is the same as the first vertex a b acda c d e
- 9. Even More Terminology •connected graph: any two vertices are connected by some path connected not connected • subgraph: subset of vertices and edges forming a graph • connected component: maximal connected subgraph. E.g., the graph below has 3 connected components.
- 10. Subgraphs Examples 0 0 0 1 2 0 1 2 1 2 3 1 2 3 G1 (i) (ii) (iii) 3(iv) (a) Some of the subgraph of G1 0 0 0 0 0 1 1 1 1 2 2 2 (i) (ii) (iii) (iv) (b) Some of the subgraph of G3 G3
- 11. More… • tree - connected graph without cycles • forest - collection of trees tree tree forest tree tree
- 12. Connectivity • Let n = #vertices, and m = #edges • A complete graph: one in which all pairs of vertices are adjacent • How many total edges in a complete graph? – Each of the n vertices is incident to n-1 edges, however, we would have counted each edge twice! Therefore, intuitively, m = n(n -1)/2. • Therefore, if a graph is not complete, m < n(n -1)/2 n=5 m = (5 ∗ 4)/2 = 10
- 13. More Connectivity n = #vertices n=5 m = #edges m=4 • For a tree m = n - 1 If m < n - 1, G is not connected n=5 m=3
- 14. Oriented (Directed) Graph • A graph where edges are directed
- 15. Directed vs. Undirected Graph • An undirected graph is one in which the pair of vertices in a edge is unordered, (v0, v1) = (v1,v0) • A directed graph is one in which each edge is a directed pair of vertices, <v0, v1> != <v1,v0> tail head
- 16. ADT for Graph objects: a nonempty set of vertices and a set of undirected edges, where each edge is a pair of vertices functions: for all graph ∈ Graph, v, v1 and v2 ∈ Vertices Graph Create()::=return an empty graph Graph InsertVertex(graph, v)::= return a graph with v inserted. v has no incident edge. Graph InsertEdge(graph, v1,v2)::= return a graph with new edge between v1 and v2 Graph DeleteVertex(graph, v)::= return a graph in which v and all edges incident to it are removed Graph DeleteEdge(graph, v1, v2)::=return a graph in which the edge (v1, v2) is removed Boolean IsEmpty(graph)::= if (graph==empty graph) return TRUE else return FALSE List Adjacent(graph,v)::= return a list of all vertices that are adjacent to v
- 17. Graph Representations Adjacency Matrix Adjacency Lists
- 18. Adjacency Matrix Let G=(V,E) be a graph with n vertices. The adjacency matrix of G is a two-dimensional n by n array, say adj_mat If the edge (vi, vj) is in E(G), adj_mat[i][j]=1 If there is no such edge in E(G), adj_mat[i][j]=0 The adjacency matrix for an undirected graph is symmetric; the adjacency matrix for a digraph need not be symmetric
- 19. Examples for Adjacency Matrix 0 0 4 0 2 1 5 1 2 3 6 3 1 0 1 1 1 0 1 0 1 0 1 1 7 1 0 1 2 0 1 1 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 0 0 1 0 0 0 0 G2 G1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 symmetric 0 0 0 0 0 1 0 1 undirected: n2/2 0 0 0 0 0 0 1 0 directed: n2 G4
- 20. Merits of Adjacency Matrix From the adjacency matrix, to determine the connection of vertices is easy n −1 The degree of a vertex is ∑ adj _ mat[i][ j ] j=0 For a digraph (= directed graph), the row sum is the out_degree, while the column sum is the in_degree n −1 n −1 ind (vi ) = ∑ A[ j , i ] outd (vi ) = ∑ A[i , j ] j =0 j =0
- 21. Adjacency Lists (data structures) Each row in adjacency matrix is represented as an adjacency list. #define MAX_VERTICES 50 typedef struct node *node_pointer; typedef struct node { int vertex; struct node *link; }; node_pointer graph[MAX_VERTICES]; int n=0; /* vertices currently in use */
- 22. 0 0 4 2 1 5 1 2 3 6 3 7 0 1 2 3 0 1 2 1 0 2 3 1 0 3 2 0 1 3 2 0 3 3 0 1 2 3 1 2 G1 0 4 5 5 4 6 0 1 6 5 7 1 0 2 1 7 6 2 G3 G4 2 An undirected graph with n vertices and e edges ==> n head nodes and 2e list nodes
- 23. Some Operations degree of a vertex in an undirected graph –# of nodes in adjacency list # of edges in a graph –determined in O(n+e) out-degree of a vertex in a directed graph –# of nodes in its adjacency list in-degree of a vertex in a directed graph –traverse the whole data structure
- 24. Graph Traversal • Problem: Search for a certain node or traverse all nodes in the graph • Depth First Search – Once a possible path is found, continue the search until the end of the path • Breadth First Search – Start several paths at a time, and advance in each one step at a time
- 25. Depth-First Search A B C D E F G H I J K L M N O P
- 26. Exploring a Labyrinth Without Getting Lost • A depth-first search (DFS) in an undirected graph G is like wandering in a labyrinth with a string and a can of red paint without getting lost. • We start at vertex s, tying the end of our string to the point and painting s “visited”. Next we label s as our current vertex called u. • Now we travel along an arbitrary edge (u, v). • If edge (u, v) leads us to an already visited vertex v we return to u. • If vertex v is unvisited, we unroll our string and move to v, paint v “visited”, set v as our current vertex, and repeat the previous steps.
- 27. Breadth-First Search • Like DFS, a Breadth-First Search (BFS) traverses a connected component of a graph, and in doing so defines a spanning tree with several useful properties. • The starting vertex s has level 0, and, as in DFS, defines that point as an “anchor.” • In the first round, the string is unrolled the length of one edge, and all of the edges that are only one edge away from the anchor are visited. • These edges are placed into level 1 • In the second round, all the new edges that can be reached by unrolling the string 2 edges are visited and placed in level 2. • This continues until every vertex has been assigned a level. • The label of any vertex v corresponds to the length of the shortest path from s to v. 27
- 28. BFS - A Graphical Representation 0 0 1 A B C D A B C D a) E F G H E F G H b) I J K L I J K L M N O P M N O P 0 1 2 0 1 2 3 A B C D B C D A c) d) E F G H E F G H I J K L I J K L M N O P M N 28 O P
- 29. More BFS 0 1 2 3 0 1 2 3 A B C D A B C D E F G H E F G H 4 4 I J K L I J K L M N O P M N O P 5
- 30. Applications: Finding a Path • Find path from source vertex s to destination vertex d • Use graph search starting at s and terminating as soon as we reach d – Need to remember edges traversed • Use depth – first search ? • Use breath – first search?
- 31. DFS vs. BFS F B A start DFS Process E G D C destination C DFS on C D Call DFS on D B DFS on B B B Return to call on B A DFS on A A A A G Call DFS on G found destination - done! D Path is implicitly stored in DFS recursion B Path is: A, B, D, G A
- 32. DFS vs. BFS F B A start E BFS Process G D C destination rear front rear front rear front rear front A B D C D Initial call to BFS on A Dequeue A Dequeue B Dequeue C Add A to queue Add B Add C, D Nothing to add rear front G found destination - done! Dequeue D Path must be stored separately Add G