Announcements

• Midterm is on Thursday, May 10. If you have an exam accommodation and haven't received an email from Tyler, please email us.
• Midterm review session on Tuesday - come with questions!
• Please give us feedback! https://vptleval.stanford.edu/auth/evaluation.php?id=14262

Midterm

• Reference sheet and cover page are available
• Covers everything in the course through today's lecture
• Questions are similar to the homeworks: mix of code and short answers
• In class, in room 420-41 (same as lecture)

Plan for Today

Today, we're discussing the theory of computer science

• Are there problems computers can't solve?
• What is an algorithm? What makes a "good" algorithm?
• Picture Contest!

Alan Turing

• Developed the Universal Turing Machine to prove that some problems are unsolvable
• Broke Enigma encryption
• Turing Test in Artificial Intelligence
• Turing Award ("Nobel Prize of CS") named after him

Turing Machines

• Abstraction (representation) of a computer
• Anything computable can be computed with a Turing Machine
• Idea: Infinite "tape" (memory)
• Move from one state (circle) to another by following the arrows and changing the memory as specified
• Accept or reject input - answers yes or no questions
• Anything we can do with computers we can do with Turing Machines

Turing Machine Fun

Google Doodle commemorating Alan Turing's 100th birthday

Idea: the "code" can change the input

Unsolvable Problems

• Turing machines can simulate running other Turing Machines (just like your computer can pretend to act as another computer)
• We use this fact to prove that we can never definitively say that a program will end
• Result: not all problems are solvable!
• Another undecidable problem: finding the cheapest sequence of flights for a trip

"Good" Algorithms

• Algorithm: way to solve a problem
• Two main resources: time and space (memory)
• Good algorithms run faster with respect to input size
  • Getting the red value of a pixel always takes the same amount of time (constant)
  • Green screen: double the width and height of the image, quadruple the number of pixels to look at (polynomial)
  • Listing all the two digit numbers takes one-tenth the time as listing all the three digit numbers (exponential)

Algorithms in practice: Search

• We want to see if a certain value is in a sorted list of elements
• Could look through all the values one by one (polynomial)
• Binary Search
  • Look at the middle value
  • Only need to look at the left values if it's too big
  • Only need to look at the right values if it's too large

Algorithms in practice: Sorting

• Bogosort: randomly pick all the elements, then check if the list is in order (really, really slow)
• Insertion Sort: Add the elements one-by-one
• Bucket sort: pick categories for the data, then put each item in one of those buckets
  • Commonly used with exams
  • Idea: faster if we relax our definition of "sorted"

P vs. NP

• Biggest unsolved question in Computer Science (million dollar prize)
• Basic idea: is it just as fast to solve a problem (NP) as it is to check that the solution is correct (P)?

P ≠ NP: Implications

• Encryption would break (we'll talk about this more in a couple weeks)
• Much better optimization (everything's faster!)
• Better transportation layouts

Recap

• Abstractions can help computer scientists think about problems differently.
• Some problems in computer science don't have any solutions (computers are not all-powerful).
• Some problems don't have any "good" solutions.
• There are lots of ways to solve problems! Computer scientists constantly try to find better ways.