“The coolest phones we’ve seen from the world’s largest mobile show so far include quad-core chips, larger screens, and even one with a projector built in.” ve_of_smartphones.html#tk.hp_pop The Beauty and Joy of Computing Lecture #11 Recursion II UC Berkeley iSchool Grad Student Dan Garcia

UC Berkeley “The Beauty and Joy of Computing” : Recursion II (2) Garcia  Factorial(n) = n! Inductive definition:  n! = 1, n = 0  n! = n * (n-1)!, n > 0  Let’s act it out…  “Little people”, or “subcontractor” model  5! How the Computer Works … n! nn!

UC Berkeley “The Beauty and Joy of Computing” : Recursion II (3) Garcia  Inductive definition:  fib(n) = n, n < 2  fib(n) = fib(n-1)+fib(n-2), n > 1  Let’s act it out…  “contractor” model  fib(5) How the Computer Works … fib(n) nfib(n) Leonardo de Pisa aka, Fibonacci en.wikipedia.org/wiki/Fibonacci_number Let’s now: trace… (gif from

UC Berkeley “The Beauty and Joy of Computing” : Recursion II (4) Garcia  Given coins {50, 25, 10, 5, 1} how many ways are there of making change?  5: 2 (N,5 P)  10  4 (D, 2N, N 5P, 10P)  15  6 (DN,D5P,3N,2N5P,1N10 P,15P)  100? Counting Change (thanks to BH)

UC Berkeley “The Beauty and Joy of Computing” : Recursion II (5) Garcia Call Tree for “Count Change 10 (10 5 1)” 10 (10 5 1) 10 (5 1) 10 (1)10 ()09 (1)9 ()08 (1)8 ()07 (1)7 ()06 (1)6 ()05 (1)5 ()04 (1)4 ()03 (1)3 ()02 (1)2 ()01 (1)1 ()00 (1)15 (5 1)5 (1) 5 ()04 (1)4 ()03 (1)3 ()02 (1)2 ()01 (1)1 ()00 (1)1 0 (5 1)1 0 (10 5 1) 1  Skip Coin Use Coin  D NN N 5P 10P D? N? N? P? P? P? P? P? P? P? P? P? P? P? P? P? P? P?

UC Berkeley “The Beauty and Joy of Computing” : Recursion II (6) Garcia  It’s important to understand the machine model  It’s often the cleanest, simplest way to solve many problems  Esp those recursive in nature!  Recursion is a very powerful idea, and one way to separate good from great Summary Menger Cube by Dan Garcia