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Graph Introduction , Searching

Graph Introduction , Searching. Graph Theory Basics. - Anil Kishore. Graph. A collections of objects and pair wise relations between them A mathematical structure Objects are called Vertices ( or Nodes ) Relationship is shown using Edges. Graph. 2. 5. 1. 4. 6. 3.

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Graph Introduction , Searching

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  1. Graph Introduction , Searching Graph Theory Basics - Anil Kishore

  2. Graph • A collections of objects and pair wise relations between them • A mathematical structure • Objects are called Vertices ( or Nodes ) • Relationship is shown using Edges

  3. Graph 2 5 1 4 6 3 Vertex Set V = { 1, 2, 3, 4, 5, 6} Edge Set E = { 12, 14, 34, 45, 46, 25, 56 } We use N and M for the corresponding sizes |V| = N , |E| = M

  4. Storing a Graph • How to store a Graph ? The two popular representations are • Adjacency Matrix • N x N matrix A of 0s and 1s • Adjacency List • A list of neighbors for each of the vertices

  5. Adjacency Matrix 2 5 1 4 6 3 A[u][v] = 1, if vertex u and vertex v are adjacent = 0, otherwise A Symmetric Matrix Space : O(N2)

  6. Adjacency List 2 5 1 4 6 3 1 : { 2, 4 } 2 : { 1, 5 } 3 : { 4 } 4 : { 1, 3, 5, 6 } 5 : { 2, 4, 6 } 6 : { 4, 5 } Vertex u : list of all neighbors of u Space : O(N+2M)

  7. Directions and Weights • Edges can have direction • ( like one-way roads ) 2 5 1 4 6 7 2 5 3 3 2 5 1 1 6 4 6 • Edges and Vertices can have weight • e.g.: length of the road, • toll gate charge in a city. 4 3

  8. Path, Cycle 2 5 • A Path of length n-1 is a sequence of vertices u1, u2, … , unsuch that vertices ui and ui+1 are adjacent • e.g. 1 – 2 – 5 – 4 – 3 is a path 1 4 6 3 • A Cycleof length n is a sequence of vertices u1, u2, … , unsuch that vertices ui and ui+1 are adjacent and also u1 and un adjacent • e.g. 1 – 2 – 5 – 4 is a cycle

  9. Power of Adjacency Matrix • Number of paths ( possibly cyclic ) of length K from u to v F(u, v, K) = Sum of ( F(u, w, k-1) * F(w, v, 1) ) ( for all possible intermediate vertices w ) This is similar to the only computation step in Matrix Multiplication • Note that F(u, v, 1) = A[u][v] • AK[u][v] = Number of paths of length exactly K • from u to v

  10. Connected, Tree, Complete • A graph is said to be connected if there exists a path between all pairs of vertices • How many minimum edges we need to make a n vertices graph connected ? • Cycles introduce redundant edges • A Tree is a connected graph with out cycles ( acyclic ) • A tree on n vertices has exactly (n-1) edges • A Complete Graph has all possible edges present in it • A complete on n vertices (Kn) has nC2 edges

  11. Traversing a graph • Visit all the vertices in the graph in a particular order • Depth-first Search (DFS) • visit child nodes before visiting sibling nodes • Breadth-first Search (BFS) • visit sibling nodes before visiting child nodes

  12. Depth-first Search algorithm DFS( u ) // start time of u Mark u as ‘visited’ FOR each node v ∈ Adj.List(u) IF NOT visited(v) THEN par[v] := u DFS(v) ENDIF ENDFOR //end time of u END-DFS • Visit an unvisited neighbor, thus recursively traverse along depth of the graph • par[v] denotes the first preceding vertex from which vertex v was visited, and defines a DFS tree • Applications : • Checking connectivity • Finding Connected Components • Topological ordering • many more…

  13. DFS 5 2 1 4 6 3

  14. DFS 5 2 1 4 6 3

  15. DFS 5 2 1 4 6 3

  16. DFS 5 2 1 4 6 3

  17. DFS 5 2 1 4 6 3

  18. DFS 5 2 1 4 6 3

  19. DFS 5 2 1 4 6 3

  20. DFS 5 2 1 4 6 3

  21. DFS 5 2 1 4 6 3

  22. DFS 5 2 1 4 6 3

  23. DFS 5 2 1 4 6 3

  24. DFS 5 2 1 4 6 3

  25. DFS 5 2 1 4 6 3 DFS TREE

  26. DFS • Recursive algorithm • The active nodes in the recursion are Pushed and Popped, similar to a Stack • Instead of recursion, can implement using a Stack data structure • Complexity : O( N + M )

  27. Breadth-first Search algorithm BFS( s ) Mark all vertices u ‘unvisited’ Create an empty queue Q EnQueue(s, Q) mask s as ‘visited’ WHILE NOT Empty(Q) DO u:= DeQueue(Q) FOR each v ∈ Adj.List(u) DO IF NOT visited(v) EnQueue(v, Q) mask v as ‘visited’ ENDFOR ENDWHILE End-BFS • Visit the vertices in the order encountered • Vertices nearer to s are processed before farther ones • Applications : • Checking connectivity • Finding Connected Components • Shortest path in unweighted graphs • many more…

  28. BFS 5 2 1 4 6 3 Q :

  29. BFS 5 2 1 4 6 3 Q : 1

  30. BFS 5 2 1 4 6 3 Q : u = 1

  31. BFS 5 2 1 4 6 3 Q : 2, 4 u = 1

  32. BFS 5 2 1 4 6 3 Q : 4 u = 2

  33. BFS 5 2 1 4 6 3 Q : 4, 5 u = 2

  34. BFS 5 2 1 4 6 3 Q : 5 u = 4

  35. BFS 5 2 1 4 6 3 Q : 5, 3, 6 u = 4

  36. BFS 5 2 1 4 6 3 Q : 3, 6 u = 5

  37. BFS 5 2 1 4 6 3 Q : 6 u = 3

  38. BFS 5 2 1 4 6 3 Q : u = 6

  39. BFS 2 1 5 2 0 s 1 4 6 3 • Visit the nodes level by level ( level order traversal ) • All nodes at level k are the ones with • shortest path to s equals to k • Complexity : O( N + M )

  40. References • Introduction to Algorithms • Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein • www.derekroconnor.net/home/MMS406/ - End -

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