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Geography and CS

Geography and CS. Philip Chan. How do I get there?. Navigation Which web sites can give you turn-by-turn directions?. Navigation [Problem understanding]. Finding a route from the origin to the destination “Static” directions Mapquest, Google maps “Dynamic” on-board directions

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Geography and CS

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  1. Geography and CS Philip Chan

  2. How do I get there? • Navigation • Which web sites can give you turn-by-turn directions?

  3. Navigation[Problem understanding] • Finding a route from the origin to the destination • “Static” directions • Mapquest, Google maps • “Dynamic” on-board directions • GPS navigation • if the car deviates from the route, it finds a new route

  4. Consider a Simpler Problem • A national map with only • Cities and • Highways • That is, ignoring • smaller streets and intersections in a city • small roads between cities • …

  5. Navigation[Problem Formulation] • Given (input) • Map (cities and highways) • Origin city • Destination city • Find (output) • City-by-city route between origin and destination cities

  6. Graph Problem • A graph has vertices and edges • Cities -> vertices • Highways -> edges • City-by-city route -> shortest path

  7. Shortest Path Problem • How would you solve the shortest path problem?

  8. Algorithm 1 • Greedy algorithm • Pick the closest city • Go to the city • Repeat until the destination city is reached • Does this always find the shortest path? • If not, what could be a counter example?

  9. Problem with Algorithm 1 • What is the main problem?

  10. Problem with Algorithm 1 • What is the main problem? • Committing to the next city too soon • Any ideas for improvement?

  11. Algorithm 2 • Keep track of alternative paths • Commit to the next city when • we are sure it is shortest • no other paths are shorter

  12. Algorithm 2 • Let the current city be the origin • While the current city is not the destination • Explore the neighboring non-committed cities of the current city • Compare alternative path to current path • Update if alternative path is shorter • Find the non-committed city that has the shortest total distance from origin • Commit that city • Update the current city to that city

  13. Algorithm 2 • Why does it guarantee to find the shortest path? • The shortest path to city X is committed • When?

  14. Algorithm 2 • Why does it guarantee to find the shortest path? • The shortest path to city X is committed • when every path to the “non-committed” cities is longer

  15. Algorithm 2 • Why does it guarantee to find the shortest path? • The shortest path to city X is committed • when every path to the “non-committed” cities is longer • no way to get to city X with a shorter path via “non-committed” cities

  16. Dijkstra’s shortest path algorithm • Interesting applet to demonstrate the alg: • http://www.dgp.toronto.edu/people/JamesStewart/270/9798s/Laffra/DijkstraApplet.html

  17. Implementation 1 • Simplified problem • What is the shortest distance between origin and destination? • We will worry about the intermediate cities later. • Consider • what we need to keep track (data) • how to keep track (instructions)

  18. What to keep track (data)?

  19. What to keep track (data)? • Whether a city is committed

  20. What to keep track (data)? • Whether a city is committed • What is the shortest distance so far for a city

  21. What to keep track (data)? • Whether a city is committed • What is the shortest distance so far for a city • How to implement the data storage?

  22. What to keep track (data)? • Whether a city is committed • What is the shortest distance so far for a city • How to implement the data storage? • committed[city] • shortestDistance[city]

  23. How to keep track (instructions)? • Sketching on whiteboard • Java in HW6

  24. Implementation 2 • We would like to know the intermediate cities as well • the shortest path, not just the shortest distance

  25. What to keep track (data)? • Whether a city is committed • What is the shortest distance so far for a city • What else?

  26. What to keep track (data)? • Whether a city is committed • What is the shortest distance so far for a city • What else? • What do you notice for each of the intermediate city?

  27. What to keep track (data)? • Whether a city is committed • What is the shortest distance so far for a city • What else? • What do you notice for each of the intermediate city? • Each was committed • What do you notice when we commit a city and update the shortest distance?

  28. What to keep track (data)? • Whether a city is committed • What is the shortest distance so far for a city • What else? • What do you notice for each of the intermediate city? • Each was committed • What do you notice when we commit a city and update the shortest distance? • We know the previous city

  29. What to keep track (data)? • Whether a city is committed • What is the shortest distance so far for a city • What is the previous city • How to implement the data storage? • committed[city] • shortestDistance[city]

  30. What to keep track (data)? • Whether a city is committed • What is the shortest distance so far for a city • What is the previous city • How to implement the data storage? • committed[city] • shortestDistance[city] • previousCity[city]

  31. Storing the map • How to store the distance between two cities • so that, given two cities, we can find the distance quickly?

  32. How to keep track (instructions)? • Sketching on whiteboard • Java in HW6

  33. Storing the Map • How to store the distance between two cities • so that, given two cities, we can find the distance quickly?

  34. Storing the Map (graph) • How to store the distance between two cities • so that, given two cities, we can find the distance quickly? • Adjacency matrix • Table (2D array) • Rows and columns are cities • Cells have distance

  35. Number of comparisons (speed of algorithm) • Comparing: • Shortest distance so far and • Distance of an alternative path • For updating what?

  36. Number of comparisons (speed of algorithm) • Comparing: • Shortest distance so far and • Distance of an alternative path • For updating what? • Shortest distance so far

  37. Number of comparisons (speed of algorithm) • Worst –case scenario • When does it occur?

  38. Number of comparisons (speed of algorithm) • N is the number of cities • Worst –case scenario • When does it occur? • Every city is connected to every city • Maximum numbers of neighbors to explore

  39. Worst-case scenario (speed of algorithm) • How many comparisons? • How many non-committed neighbors from the origin (in the first round)?

  40. Worst-case scenario (speed of algorithm) • How many comparisons? • How many non-committed neighbors from the origin (in the first round)? • N – 1 comparisons • How many in the second round?

  41. Worst-case scenario (speed of algorithm) • How many comparisons? • How many non-committed neighbors from the origin (in the first round)? • N – 1 comparisons • How many in the second round? • N – 2 comparisons • ... • How many in total?

  42. Worst-case scenario (speed of algorithm) • How many comparisons? • How many non-committed neighbors from the origin (in the first round)? • N – 1 comparisons • How many in the second round? • N – 2 comparisons • ... • How many in total? • (N-1) + (N-2) + … + 1

  43. Worst-case scenario (speed of algorithm) • How many comparisons? • How many non-committed neighbors from the origin (in the first round)? • N – 1 comparisons • How many in the second round? • N – 2 comparisons • ... • How many in total • (N-1) + (N-2) + … + 1 • (N-1)N/2 = (N2 – N)/2

  44. Shortest Path Algorithm • Dijkstra’s Algorithm • In terms of vertices (cities) and edges (highways) in a graph • 1959 • more than 50 years ago • Navigation optimization • Cheapest (tolls) route? • Least traffic route? • Many applications

  45. Summary • Navigation problem • Turn-by-turn directions • Simplified: city-by-city directions • Algorithms: • Greedy: might not yield shortest path • Dijkstra’s: always yield shortest path • Reasons for guarantee • Data structures in implementation • Quadratic comparisons in # of cities

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