1 / 11

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

waneta
Télécharger la présentation

Graph Theory - I

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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/

More Related