Geography and CS Philip Chan. How do I get there? Navigation Which web sites can give you turn-by-turn directions?

Geography and CS

Philip Chan

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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 GPS navigation

if the car deviates from the route, it finds a new route

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 …

Navigation[Problem Formulation] Given (input)

Map (cities and highways) Origin city Destination city

Find (output) City-by-city route between origin and

destination cities

Graph Problem

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

Shortest Path Problem

How would you solve the shortest path problem?

Algorithm 1

Greedy algorithm1. Pick the closest city

2. Go to the city

3. Repeat until the destination city is reached

Algorithm 1

Greedy algorithm1. Pick the closest city

2. Go to the city

3. Repeat until the destination city is reached

Does this always find the shortest path?

If not, what could be a counter example?

Problem with Algorithm 1

What is the main problem?

Problem with Algorithm 1

What is the main problem? Committing to the next city too soon

Any ideas for improvement?

Algorithm 2

“Exhaustive” algorithm Explore/generate all possible paths

Not just the ones that look short Compare all possible paths

Greedy vs Exhaustive

Greedy doesn’t guarantee the shortest path


Greedy doesn’t guarantee the shortest path Greedy is faster


Greedy doesn’t guarantee the shortest path Greedy is faster Greedy requires less memory

Algorithm 3

“smart” algorithm Guarantees the shortest path Faster than Exhaustive algorithm

Any ideas on what we can ignore?

Algorithm 3

“Smart” algorithm Similar to Exhaustive

explore all alternatives from each city Ignore alternative paths that are not shorter

Algorithm 4

“Smarter” algorithm Dijkstra’s algorithm

Any ideas on additional alternatives that can be ignored?

Algorithm 4

“Smarter” algorithm Dijkstra’s algorithm

Any ideas on additional alternatives that can be ignored? Hint: we can commit certain cities earlier

Can’t have a shorter path to those cities Ignore the committed cities later

Algorithm 4

Keep track of alternative paths Commit to the next city when

we are sure it is shortest no other paths are shorter

Algorithm 4

Let the current city be the origin (I) While the current city is not the destination(C)

1. Explore neighboring non-committed cities X’s 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)

2. Find the non-committed city that has the shortest path length

3. Commit that city

4. Update the current city to the committed city (U)

Algorithm 4

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

When?

Algorithm 4


when every path to the “non-committed” cities is longer

Algorithm 4


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

Dijkstra’s shortest path algorithm

Interesting applet to demonstrate the alg:

http://www.dgp.toronto.edu/people/JamesStewart/270/9798s/Laffra/DijkstraApplet.html

Comparing the Four Algorithms

1. “Greedy” Commit to closest neighboring city

2. “Exhaustive” Consider all possible paths, compare all paths

3. “Smart” Consider all neighboring cities, compare old and

new paths, ignore the longer one

4. “Smarter” (Dijkstra’s) Consider only non-committed neighboring cities,

compare old and new paths, ignore the worse one

Implementation of Dijkstra’s Algorithm Keeping track of information needed by the

algorithm

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)

What to keep track (data)?


Whether a city is committed


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



How to implement the data storage?



How to implement the data storage? committed[city] pathLength[city]

aka shortestDistance[city]

How to keep track (instructions)?

Sketching on whiteboard

Implementation 2

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

distance


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?




Each was committed What do you notice when we commit a city

and update the shortest distance? We know the previous city


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


aka shortestDistance[city]


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


aka shortestDistance[city] parent[city]

aka previousCity[city]

Algorithm 4 with data tracking

Let the current city be the origin (I) While the current city is not the destination(C)

1. Explore neighboring non-committed cities X’s of the current city

if new path to X is shorter than current path to X Update current path to X (pathLength[X], parent[X])

2. Find the non-committed city that has the shortest path length

3. Commit that city (committed[city])

4. Update the current city to the committed city (U)

Storing the map

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

distance quickly?

How to keep track (instructions)?

Sketching on whiteboard

Storing the Map


distance quickly?

Storing the Map (graph)


distance quickly?

Adjacency matrix Table (2D array) Rows and columns are cities Cells have distance

Number of comparisons (speed of algorithm) Comparing:

Shortest distance so far and Distance of an alternative path

For updating what?

Number of comparisons (speed of algorithm) Comparing:

Shortest distance so far and Distance of an alternative path

For updating what? Shortest distance so far

Number of comparisons (speed of algorithm) Worst –case scenario

When does it occur?

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

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?





How many in total? (N-1) + (N-2) + … + 1





How many in total (N-1) + (N-2) + … + 1 (N-1)N/2 = (N2 – N)/2

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

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

Geography and CS Philip Chan. How do I get there? Navigation Which web sites can give you turn-by-turn directions?

Documents

city x

shortest path lengthcommit

shorter path

noncommitted city

xupdate current path

committed city ualgorithm

shortest pathfaster

current cityif new path