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Graph Theory - I

Graph Theory - I. Today’s Plan. Introduction Special Graphs Various Representations Depth First Search Solve a problem from GCJ Breadth First Search Solve a problem from SPOJ Dijkstra’s Algorithm Solve a problem from SPOJ. Special Graphs. Undirected Graphs Edge Weighted Graphs

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Graph Theory - I

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  1. Graph Theory - I

  2. Today’s Plan • Introduction • Special Graphs • Various Representations • Depth First Search • Solve a problem from GCJ • Breadth First Search • Solve a problem from SPOJ • Dijkstra’s Algorithm • Solve a problem from SPOJ

  3. Special Graphs • Undirected Graphs • Edge Weighted Graphs • Directed Graphs • Trees • Directed Acyclic Graphs • Bi-Partite Graphs

  4. Representation - I • Adjacency matrix • 2 D Array M of size |V|x|V| • M[i][j] – 1 if Vi and Vj are connected by and edge and 0 otherwise. • Adjacency List • Each vertex maintains a list of vertices that are adjacent to it. • We can use: vector< vector<int> >

  5. Representation - II • Edge List • Checking if edge (Vi,Vj) is present in G. • Adjacency Matrix – O(1) • Adjacency List – O(min(deg(Vi),deg(Vj))) • Iterating through the list of neighbours of Vi • Adjacency Matrix – O(|V|) • Adjacency List – O(deg(Vi))

  6. Representation - III • Implicit graphs • Two squares on an 8x8 chessboard. Determine the shortest sequence of knight moves from one square to the other. • Tricks: • Use Dx[] , Dy[] for generating the neighbors of a position in grid problems.

  7. Depth First Search • Finding Connected Components • Implemented using • Stack • Recursion (Most Frequently used) • Complexity • Time: O(|V|+|E|) • Space: O(|V|) [to maintain the vertices visited till now] • Google Code Jam Problem • http://code.google.com/codejam/contest/dashboard?c=90101#s=p1

  8. Breadth First Search • Finding a Path with Minimum # of edges from starting vertex to any other vertex. • Used to Solve Shortest Path problem in un weighted graphs • Implemented using queue • Same Time and Space Complexity as DFS. • SPOJ Problem • http://www.spoj.pl/problems/PPATH/

  9. Dijkstra’s Algorithm • Used to solve Shortest Path problem in Weighted Graphs • Only for Graphs with positive edge weights • Greedy strategy • Use priority_queue<node> for implementing Dijkstra’s • SPOJ Problem • http://www.spoj.pl/problems/CHICAGO

  10. Practice problems • http://www.spoj.pl/problems/PARADOX/ • http://www.spoj.pl/problems/HERDING/ • http://www.spoj.pl/problems/COMCB/ • http://www.spoj.pl/problems/PT07Y/ • http://www.spoj.pl/problems/PT07Z/

  11. More Practice Problems • SRM 453.5 Division 2 500 Pt • http://www.codechef.com/problems/N4 • http://www.spoj.pl/problems/ONEZERO • http://www.spoj.pl/problems/CERC07K/

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