1. CSE 20 DISCRETE MATH Prof. Shachar Lovett http://cseweb.ucsd.edu/classes/wi15/cse20-a/ Clicker frequency: CA

2. Todays topics Proofs for algorithms Casting out nines Fast exponentiation Section 3.7 in Jenkyns, Stephenson

3. Casting out nines Goal: check if a number n is divisible by 9 Algorithm: Keep summing the digits (in base 10) Until we get a 1-digit number Which is easy to check Questions: Does the algorithm terminate? Does it return the correct answer? How fast?

4. Casting out nines

6. Casting out nines: termination

8. Casting out nines: correctness

12. Casting out nines: speed

15. Fast exponentiation

16. Questions: Does the algorithm terminate? Does it return the correct answer? How fast? Termination is simple: dividing b by 2 at each step (and rounding down), guarantees that eventually b=0 We will analyze correctness and speed

17. Fast exponentiation: correctness Let a*,b* denote original inputs

18. Fast exponentiation: correctness

19. Fast exponentiation: speed Question 3: how fast is the algorithm? Each iteration takes constant time How many iterations? A. a B. b C. log(b) D. log*(b) E. Other

20. Fast exponentiation: speed Question 3: how fast is the algorithm? Each iteration takes constant time How many iterations? A. a B. b C. log(b) D. log*(b) E. Other

21. Fast exponentiation: speed Algorithm to compute a b requires ~log(b) operations Compare to naïve algorithm, which multiplies a with itself b times, and requires b operations Example: compute 2 1000 Naïve: 1000 multiplications Fast: ~10 multiplications Fast exponentiation is indeed much faster, and hence is used widely in practice

22. Fast exponentiation: extensions

23. Next class More proofs for algorithms: Euclid’s algorithm Read section 3.7 in Jenkyns, Stephenson