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9/28/20 An Announcements Co Computational Structures in Data Science Midterm 10/7, 7-9pm PT Alternate: 8-10am 10/8 Recu Re cursion on See Ed: https://us.edstem.org/courses/2362/discussion/134919 If you have a time conflict,

  1. 9/28/20 An Announcements Co Computational Structures in Data Science • Midterm 10/7, 7-9pm PT – Alternate: 8-10am 10/8 Recu Re cursion on – See Ed: https://us.edstem.org/courses/2362/discussion/134919 – If you have a time conflict, or are in a timezone where you can’t take the exam, request an alternate by 10/5. UC Berkeley EECS Lecturer • Recursion is on the midterm, but not super advanced Michael Ball recursion. • No Live lecture on 10/7. Take a break or prep. :) UC Berkeley | Computer Science 88 | Michael Ball | https://cs88.org UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org 2/25/19 UCB CS88 Sp19 L5 2 1 2 Co Computing In The News Computational Structures in Data Science Co Stanford researchers combine CAT scans and advanced computing to fight wildfires Recu Re cursion on Andrew Winters, Stanford (Jr.) University, Sept 22, 2020 UC Berkeley EECS Lecturer As wildfires rage across much of the American Michael Ball West, researchers at Stanford have used CAT scanners, the same instruments used in medicine to peer inside the human body, to understand the process of smoldering – the state of burning without flame that often leads to fire. They then folded this deeper understanding of burning into computer models to predict where wildfires might strike next. These models could help firefighters allocate precious resources, reduce the loss of property and help save lives, the researchers say. UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UC Berkeley | Computer Science 88 | Michael Ball | https://cs88.org 3 3 4 Wh Why Recursion? To Today: Recursion • Recursive structures exist (sometimes hidden) in nature and therefore in data! • It’s mentally and sometimes computationally more efficient to process recursive structures using recursion. • Sometimes, the recursive definition is easier to understand or write, even if it is computationally slower. • Recursive function calls itself, directly or indirectly UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UCB CS88 Sp19 L5 2/25/19 5 5 6 1

  2. 9/28/20 The Recursive Process Th Computational Structures in Data Science Co Recu Re cursion on § Recursive solutions involve two major parts: ú Base case(s), the problem is simple enough to be solved UC Berkeley EECS Lecturer directly Michael Ball ú Recursive case(s). A recursive case has three components: Divide the problem into one or more simpler or smaller parts Invoke the function (recursively) on each part, and Combine the solutions of the parts into a solution for the problem. UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UC Berkeley | Computer Science 88 | Michael Ball | https://cs88.org 7 8 Le Learning Ob Objectives Ite Iterati tion vs Re Recursion: Sum Nu Numbers • Compare Recursion and Iteration to each other – Translate some simple functions from one method to another • Write a recursive function For loop: – Understand the base case and a recursive case def sum(n): s=0 for i in range(0,n+1): s=s+i return s UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org 9 9 10 Ite Iterati tion vs Re Recursion: Sum Numbers Ite Iterati tion vs Re Recursion: Sum Numbers Recursion: While loop: def sum(n): def sum(n): if n == 0: s=0 i=0 return 0 while i<n: return n+sum(n-1) i=i+1 s=s+i return s UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org 11 12 2

  3. 9/28/20 Ite Iterati tion vs Recursion: Cheati ting! The Recursive Process Th Sometimes it’s best to just use a formula! But that’s not always the point. J § Recursive solutions involve two major parts: ú Base case(s), the problem is simple enough to be solved def sum(n): directly return (n * (n + 1)) / 2 ú Recursive case(s). A recursive case has three components: Divide the problem into one or more simpler or smaller parts Invoke the function (recursively) on each part, and Combine the solutions of the parts into a solution for the problem. UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org 13 14 13 14 Re Recall: Iteration Recursion Key concepts – by Re by example 1. Initialize the “base” case of no iterations 1. Test for simple “base” case 2. Solution in simple “base” case 2. Starting value def sum_of_squares(n): def sum_of_squares(n): accum = 0 if n < 1: 3. Ending value for i in range(1,n+1): return 0 else: accum = accum + i*i return sum_of_squares(n-1) + n**2 return accum 4. New loop variable value 3. Assume recusive solution 4. ”Combine” the simpler part of to simpler problem the solution, with the recursive case UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org 15 16 In In words Why does it work Wh • The sum of no numbers is zero • The sum of 1 2 through n 2 is the – sum of 1 2 through (n-1) 2 sum_of_squares(3) – plus n 2 # sum_of_squares(3) => sum_of_squares(2) + 3**2 def sum_of_squares(n): # => sum_of_squares(1) + 2**2 + 3**2 # => sum_of_squares(0) + 1**2 + 2**2 + 3**2 if n < 1: # => 0 + 1**2 + 2**2 + 3**2 = 14 return 0 else: return sum_of_squares(n-1) + n**2 UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UCB CS88 Sp19 L5 2/25/19 17 UCB CS88 Sp19 L5 2/25/19 18 17 18 3

  4. 9/28/20 Re Review: Functions Ho How w does it wo work? • Each recursive call gets its own local variables def <function name> ( <argument list> ) : – Just like any other function call • Computes its result (possibly using additional calls) expression – Just like any other function call return • Returns its result and returns control to its caller – Just like any other function call def concat(str1, str2): return str1+str2; • The function that is called happens to be itself concat(“Hello”,”World”) – Called on a simpler problem – Eventually stops on the simple base case • Generalizes an expression or set of statements to apply to lots of instances of the problem • A function should do one thing well UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org 19 20 Qu Questions Tr Trust … • In what order do we sum the squares ? • The recursive “leap of faith” works as long as we • How does this compare to iterative approach ? hit the base case eventually def sum_of_squares(n): accum = 0 for i in range(1,n+1): accum = accum + i*i What happens if we don’t? return accum def sum_of_squares(n): def sum_of_squares(n): if n < 1: if n < 1: return 0 return 0 else: else: return sum_of_squares(n-1) + n**2 return n**2 + sum_of_squares(n-1) UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org 21 22 Wh Why Recursion? Recursion (unwanted) Re • “After Abstraction, Recursion is probably the 2 nd biggest idea in this course” • “It’s tremendously useful when the problem is self-similar” • “It’s no more powerful than iteration, but often leads to more concise & better code” • “It’s more ‘mathematical’” • “It embodies the beauty and joy of computing” • … UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org 23 24 4

  5. 9/28/20 Ex Example I An Another Example indexing an element of a sequence List all items on your hard disk def first(s): """Return the first element in a sequence.""" • Files return s[0] • Folders contain def rest(s): """Return all elements in a sequence after the first""" – Files return s[1:] – Folders Slicing a sequence of elements def min_r(s): “””Return minimum value in a sequence.””” if len(s) == 1: Base Case return first(s) Recursion! else: return min(first(s), min_r(rest(s))) Recursive Case • Recursion over sequence length, rather than number magnitude UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org 2/25/19 UCB CS88 Sp19 L5 29 25 29 Wh Why Recursion? More Reasons • Recursive structures exist (sometimes hidden) in nature and therefore in data! • It’s mentally and sometimes computationally more efficient to process recursive structures using recursion. UC Berkeley | Computer Science 88 | Michael Ball | http://cs88.org UCB CS88 Sp19 L5 2/25/19 32 32 5

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