1. Problems and Algorithms
Introduction to:
Problem
Algorithms
Complexity

2. Problem: Get to NYC
Need to travel from the Jersey Shore to New York City
How would you do this?

3. Algorithm (Step-by-Step Solution)
Drive in a car from Jersey shore to Miami
Drive from Miami to Los Angeles
Drive from Los Angeles to San Francisco
Drive From San Francisco to New York
Analysis
Will this algorithm solve the problem?
Is this a good algorithm to use?
Best way? Gas? Toll? Mileage? Time?

4. Problem: Go from Jersey to San Fran
Some choices for algorithms
Walk the entire way
Drive by car the entire way
Drive to the nearest airport, fly to the nearest airport to San Fran, the rent a car and drive again

5. Problem: Go from class to the Gym
Go from this classroom to the Gym

6. Algorithm: To get to the Gym
Drive a car?
Take a plane?
Walk?
Analysis:
If planes are faster than cars, should you solve all travel problems by using a plane?
Know how to use planes, cars, bicycles, and walk and use your head.

7. Problem: Find Max Instance: A collection of numbers Problem:
Print out the maximum number of all numbers

8. Algorithm for Find Max Algorithm: Sort all the numbers
Print out A[n-1]
Analysis of the algorithm:
How do we sort the numbers? How long does this take.
Printing out the answer is simply the time it takes to print.

9. Algorithm for Find Max (cont)
Suppose we sort the number by repeatedly
Taking all permutations of the numbers
Check if this is correctly sorted
Stop if it is sorted
Analysis of the algorithm
How many permutations are there? N!
20! = 2.43 X 10^18
On a 1 gig processor would take approx. 76 years.

10. Better algorithm for Find Max?
Suggestions?
Best possible algorithm (a.k.a. Completeness)
How do you know someone else will not come along later and come up with a better algorithm

11. Problem: Skyline Silhouette
Instance:
A collection of rectangular building (heights and widths)
Problem:
Print out the silhouette of the skyline from one point of view.

12. Skyline input and output
building one     left 3 right 7 height 4 building two     left 11 right 20 height 8 building three   left 15 right 29 height 3          **********          *        *          *        *          *        *  *****   *        *  *   *   *   ***************  *   *   *   *    *         *  *   *   *   *    *        * ===============================  3           29      <---- left/right edge positions
The output should be the points (3, 0) (3, 4) (7, 4) (7, 0) (11, 0) (11, 8) (20, 8) (20, 3) (29, 3)  (29, 0)

13. Algorithm for Skyline problem
Hints:
Sort the building in some order
Move buildings from one data structure to another
Have a data structure that can always find the max and anytime