Geography and CS

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Geography and CS

• Philip Chan

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How do I get there?

• Navigation
• Which web sites can give you turn-by-turn
  directions?

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NavigationProblem 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

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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

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NavigationProblem Formulation

• Given (input)
• Map (cities and highways)
• Origin city
• Destination city
• Find (output)
• City-by-city route between origin and destination
  cities

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Graph Problem

• A graph has vertices and edges
• Cities -gt vertices
• Highways -gt edges
• City-by-city route -gt shortest path

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Shortest Path Problem

• How would you solve the shortest path problem?

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Algorithm 1

• Greedy algorithm
• Pick the closest city
• Go to the city
• Repeat until the destination city is reached

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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?

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Problem with Algorithm 1

• What is the main problem?

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Problem with Algorithm 1

• What is the main problem?
• Committing to the next city too soon
• Any ideas for improvement?

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Algorithm 2

• Exhaustive algorithm
• Explore/generate all possible paths
• Not just the ones that look short
• Compare all possible paths

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Greedy vs Exhaustive

• Greedy doesnt guarantee the shortest path

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Greedy vs Exhaustive

• Greedy doesnt guarantee the shortest path
• Greedy is faster

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Greedy vs Exhaustive

• Greedy doesnt guarantee the shortest path
• Greedy is faster
• Greedy requires less memory

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Algorithm 3

• smart algorithm
• Guarantees the shortest path
• Faster than Exhaustive algorithm
• Any ideas on what we can ignore?

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Algorithm 3

• Smart algorithm
• Similar to Exhaustive
• explore all alternatives from each city
• Ignore alternative paths that are not shorter

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Algorithm 4

• Smarter algorithm
• Dijkstras algorithm
• Any ideas on additional alternatives that can be
  ignored?

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Algorithm 4

• Smarter algorithm
• Dijkstras algorithm
• Any ideas on additional alternatives that can be
  ignored?
• Hint we can commit certain cities earlier
• Cant have a shorter path to those cities
• Ignore the committed cities later

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Algorithm 4

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

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Algorithm 4

• Let the current city be the origin (I)
• While the current city is not the destination(C)
• Explore neighboring non-committed cities Xs of
  the current city
• if new path to X is shorter than current path to
  X
• Update current path to X (ie, ignore the longer
  path)
• Find the non-committed city that has the shortest
  path length
• Commit that city
• Update the current city to the committed city (U)

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Algorithm 4

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

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Algorithm 4

• 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

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Algorithm 4

• 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

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Dijkstras shortest path algorithm

• Interesting applet to demonstrate the alg
• http//www.dgp.toronto.edu/people/JamesStewart/270
  /9798s/Laffra/DijkstraApplet.html

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Comparing the Four Algorithms

• Greedy
• Commit to closest neighboring city
• Exhaustive
• Consider all possible paths, compare all paths
• Smart
• Consider all neighboring cities, compare old and
  new paths, ignore the longer one
• Smarter (Dijkstras)
• Consider only non-committed neighboring cities,
  compare old and new paths, ignore the worse one

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Implementation of Dijkstras Algorithm

• Keeping track of information needed by the
  algorithm

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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)

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What to keep track (data)?
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What to keep track (data)?

• Whether a city is committed

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What to keep track (data)?

• Whether a city is committed
• What is the shortest distance so far for a city

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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?

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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?
• committedcity
• pathLengthcity
• aka shortestDistancecity

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How to keep track (instructions)?

• Sketching on whiteboard

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Implementation 2

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

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What to keep track (data)?

• Whether a city is committed
• What is the shortest distance so far for a city
• What else?

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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?

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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?

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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

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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?
• committedcity
• pathLengthcity
• aka shortestDistancecity

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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?
• committedcity
• pathLengthcity
• aka shortestDistancecity
• parentcity
• aka previousCitycity

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Algorithm 4 with data tracking

• Let the current city be the origin (I)
• While the current city is not the destination(C)
• Explore neighboring non-committed cities Xs of
  the current city
• if new path to X is shorter than current path to
  X
• Update current path to X (pathLengthX,
  parentX)
• Find the non-committed city that has the shortest
  path length
• Commit that city (committedcity)
• Update the current city to the committed city (U)

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Storing the map

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

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How to keep track (instructions)?

• Sketching on whiteboard

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Storing the Map

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

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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

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Number of comparisons (speed of algorithm)

• Comparing
• Shortest distance so far and
• Distance of an alternative path
• For updating what?

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Number of comparisons (speed of algorithm)

• Comparing
• Shortest distance so far and
• Distance of an alternative path
• For updating what?
• Shortest distance so far

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Number of comparisons (speed of algorithm)

• Worst case scenario
• When does it occur?

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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

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Worst-case scenario (speed of algorithm)

• How many comparisons?
• How many non-committed neighbors from the origin
  (in the first round)?

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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?

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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?

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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

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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

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Shortest Path Algorithm

• Dijkstras 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

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Summary

• Navigation problem
• Turn-by-turn directions
• Simplified city-by-city directions
• Algorithms
• Greedy might not yield shortest path
• Dijkstras always yield shortest path
• Reasons for guarantee
• Data structures in implementation
• Quadratic comparisons in of cities