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Last Tutorial 11

Last Tutorial 11. Graph. Graph – What is it?. G(V,E) Collection of vertices V and edges E Not a totally new data structure, this is an extension from Linked List  Tree Very useful to model many (real-life) problems Internet Road Traffic Many others… Basic Terminologies (see q1 !)

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Last Tutorial 11

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  1. Last Tutorial 11 Graph

  2. Graph – What is it? • G(V,E) • Collection of vertices V and edges E • Not a totally new data structure, this is an extension from Linked List  Tree • Very useful to model many (real-life) problems • Internet • Road Traffic • Many others… • Basic Terminologies (see q1!) • Vertices, Edges • Path, Cycle • Directed/Un-directed • Weighted/Un-weighted • Connected/Un-connected • In-degree/Out-degree

  3. Graph – Basic Representations • Adjacency Matrix • 2-D array of vertices • M[i][j] = 1 if there is an edge between vertex i and j, 0 otherwise • For weighted graph, M[i][j] is the weight of the edges, not just 0 or 1! • Adjacency List • An array of V linked lists • List i represent the vertices connected to vertex i • For weighted graph, the list store both neighboring vertices and their edge costs • Discussed in Q1 • There are more representations • But we do not have time to discuss all…

  4. Student Presentation • Gr3 (average 2 times) • Gr4 (average 3 times) Overview of the questions: • Graph representations! (>1 students) • Graph traversals! (>1 students) • Topological Sort (1 student) • Mix and Match (floor) • Gr5 (average 4 times) • Gr6 (average 3 times)

  5. Q1 – Graph Representations • Explain: • Characteristics of the Graph • directed, weighted, connected*, has cyclenot dense, not too sparse either • Adjacency List • Adjacency Matrix • For small graph like this, both representations are ~ equally good. • SGUK (10 days)SGTokyo is cheaper than SGTokyo (10 days) HUB (3 in, 4 out) Dead end (1 in, 0 out) Weight can also be associated with vertex!

  6. Graph – Basic Algorithms (1) • Graph traversal algorithms: • Breadth First Search (BFS) O(V+E) • O(V + E) is to say “take the max between V or E”.For sparse graph, E < V.For dense graph, E can be V2! • Visit the graph breadth first • Usually implemented using queue • Depth First Search (DFS) O(V+E) • Visit the graph depth first • Usually implemented using recursion/stack • Discussed in Q2

  7. Q2 – Graph Traversals • Start from Singapore: • DFS • Answer: Singapore  Bali (cheapest),backtrack  Tokyo  Shanghai,backtrack, backtrack  London • BFS • Answer: It happens to be the same sequence:Singapore  Bali (cheapest)  Tokyo  Shanghai  London • If there is an edge (direct flight) of cost 9K from London to Shanghai • And we start from London • DFS • London  Shanghai  Tokyo  Singapore  Bali • BFS • London  (starting)Shanghai  Singapore  (1st layer)Tokyo  Bali (2nd layer) 9K

  8. Graph – Basic Algorithms (2) • Topological Sort • Topological Order: • Linear order that does not violate precedence constraint,e.g. underwear  outerwear * • The algorithm: O(V+E) • Insert vertex (or vertices) v with in-degree 0 in a queue • While queue is not empty() • Take/process vertex v located in front of the queue • Decrease the in-degree of vertices adjacent to v by 1 • If in-degree becomes 0, add it into the queue • Discussed in Q3

  9. Q3 – Topological Sort • Valid topological order (starting from Australian GP): • Australian GP, Malaysian GP, Monaco GP, French GP, British GP, Italian GP, Belgian GP, Singapore GP, Japanese GP, Chinese GP, Abu Dhabi GP • If Chinese GP  Japanese GP (a “funny” constraint) • We cannot use topological sort! (Graph must be Directed Acyclic Graph/DAG) Note: This constraint graph happens to be a DAG. However, not all constraint graphs are DAGs!

  10. Graph – Basic Algorithms (3) Shortest Path algorithm (not covered in CS1102 syllabus this sem) If the graph is un-weighted We can run BFS(), the level is the cost of the path O(V+E) If the graph is weighted Use Dijkstra Shortest Path Algorithm O((V+E) log V)

  11. Graph – Basic Algorithms (4) • There are many others… • We cannot study all in this CS1102 module… • Those who are interested to study more about Graph and other advanced algorithms like Dynamic Programming, Greedy Algorithm, etc should take CS3230 – Design and Analysis of Algorithms

  12. Q4 – Mix and Match (1)

  13. Q4 – Mix and Match (2)

  14. Q4 – Mix and Match (3)

  15. Our Class Photo  • You can see our photos here: • http://www.comp.nus.edu.sg/~stevenha/myteaching/record.html

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  17. Food For Thought • This is ‘the end’ of CS1102 tutorial • Remember that you still have final exam! • I hope you enjoy studying this module • Regardless your final grade :$ • All the best for your final exam and future endeavor • We may/may not meet again in the future • But if we do, please greet each other  • PS: Hopefully you are not in CS1102 class again next semester… • Take CS3230 if you are interested to study advanced algorithms

  18. Questions from Last Semester • Use the remaining slides as extra exercises!

  19. Q1 – Graph Representation Show this graph’s  Adjacency List: Adjacency Matrix:

  20. Q2 – Adjacency Matrix or List?? Would you use the adjacency list structure or the adjacency matrix structure in each of the following cases? Justify your choice. • The graph has 10,000 vertices and 20,000 edges.It is important to use as little space as possible. Answer: Use adjacency list as there are on average 2 edges on each vertex.We will waste a lot of memory space if adjacency matrix is used. • The graph has 10,000 vertices and 20,000,000 edges.It is important to use as little space as possible. Answer: Use adjacency matrix as there are on average 2000 edges on each vertex.It would be much faster to find the neighbors of each vertex. • You need to answer the query areAdjacent as fast as possible.No matter how much space you use. Answer: Use adjacency Matrix since space is not a problem.More importantly, the query areAdjacent can be answered in O(1) time. 

  21. Q3 – `Slow’ DFS?? Explain why the DFS traversal runs in O(n2) time On an n-vertex simple graph that is represented with the adjacency matrix structure. Answer: This is because for each vertex, We need to look at (n – 1) entries to see if they are the adjacent vertices. Using Adjacency List, this DFS runs in O(V+E) PS: A little note about V+E The + sign means: whoever greater takes over this asymptotic value. In sparse graph: E can be < V, O(V+E) = O(V) In dense graph: E can be as many as V2, thus E >> V, O(V+E) = O(V+V2) = O(V2)

  22. Q4 – DFS, BFS, Topological Sort • Do DFS and BFS starting from vertex `a’. • DFS/BFS traversal depends onthe ordering of neighbors… • Any valid DFS/BFS is accepted. • But usually we order neighbor alphabetically. • DFS: • a, b, c, d, f, e, g, i, h (my default answer) OR • a, b, c, d, h, f, g, e, i (if you adopt other vertex ordering) OR • a, b, c, e, i, g, d, h, f OR other possible sequences • BFS: • a, b, c, d, e, f, h, g, i (my default answer) OR • a, d, c, b, h, f, e, g, i (if you adopt other vertex ordering)OR other possible sequences • Note that a, b, c, d, f, e, h, g, i is not an acceptable sequence, why? • We are using a queue, e will be inside first after visiting c, so e is executed first before f… • Homework: • Try doing DFS/BFS from other nodes • Write the topological order of the vertices for the graph given above. • a, b, c, d, f, h, e, g, i (my default answer) OR • a, b, c, d, h, f, e, g, i (if you adopt other vertex ordering)

  23. Q5 – Dijkstra • Do Shortest Path from `a’ using Dijkstra’s algorithm! • Homework: • Try doing Shortest Path from ‘b’, ‘c’, ‘d’, etc…

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