 1 Loop and Set operations

Creating a loop Loops are used to repeat a computation. Let’s see the example of factorial computation. Let’s compute 10! = 10*9*8*…*1 2

Creating a loop 1) Make a list of evaluation step by step 10 10*9 (10*9)*8 ((10*9)*8)*7 … 3

Creating a loop 2) Assign intermediate results into variables f10 = 10 f9 = 10*9 f8 = (10*9)*8 f7 = ((10*9)*8)*7 … f1 = ((10*9)*8)*7* … *1 4

Creating a loop 3) Replace an expression with an equivalent variable. f10 = 10 f9 = f10*9 f8 = f9*8 f7 = f8*7 … f1 = f2*1 5 f10 = 10 f9 = 10*9 f8 = (10*9)*8 f7 = ((10*9)*8)*7 … f1 = ((10*9)*8)*7* … *1

Creating a loop 4) Check if variables can be reduced into a single variable. f10 = 10 f10 = f10*9 f10 = f10*8 f10 = f10*7 … f10 = f10*1 6 f = 10 f = f*9 f = f*8 f = f*7 … f = f*1 If possible, use a meaningful variable name

Creating a loop 5) Identify numbers or values which changes. factorial = 10 factorial = factorial*9 factorial = factorial*8 factorial = factorial*7 … factorial = factorial*1 7

Creating a loop 6) Find a general formula or sequence to generate those values. factorial = 10 factorial = factorial*9 factorial = factorial*8 factorial = factorial*7 … factorial = factorial*1 8

Creating a loop 7) In this example, the changing sequence is [10,9,8,…,1] factorial = 10 factorial = factorial*9 factorial = factorial*8 factorial = factorial*7 … factorial = factorial*1 9

Creating a loop 8) Replace values with a variable; L = [10,9,8,…,1] factorial = L[0] factorial = factorial*L[1] factorial = factorial*L[2] factorial = factorial*L[3] … factorial = factorial*L[9] 10

Creating a loop 9) Initialize the output variable and make the recurring formula as general as possible factorial = 1 factorial = factorial*L[0] factorial = factorial*L[1] factorial = factorial*L[2] factorial = factorial*L[3] … factorial = factorial*L[9] 11

Creating a loop 10) Introduce for or while statement to make a loop for recurrences L = [10,9, …,1] factorial = 1 for e in L: factorial = factorial * e 12

Understanding a loop L = [10,9, …,1] factorial = 1 for e in L: factorial = factorial * e Identify counter variables 13

Understanding a loop L = [10,9, …,1] factorial = 1 for e in L: factorial = factorial * e Here, the counter variable is e; [10, 9, …, 1] 14

Understanding a loop L = [10,9, …,1] factorial = 1 for e in L: factorial = factorial * e Expand the repeated block with examples of counter variables 15

Understanding a loop L = [10,9, …,1] factorial = 1 factorial = factorial * 10 factorial = factorial * 9 factorial = factorial * 8 … Expand the repeated block with examples of counter variables 16

Understanding a loop L = [10,9, …,1] factorial = 1 factorial = factorial * 10 factorial = factorial * 9 factorial = factorial * 8 … Replace the overwritten variable with actual expressions 17

Understanding a loop L = [10,9, …,1] factorial = 1 factorial = 1 * 10 factorial = (1 * 10) * 9 factorial = ((1 * 10) * 9) * 8 … Find out the last expression! 18

Understanding a loop L = [10,9, …,1] factorial = 1 factorial = 1 * 10 … factorial = 1*10*9*8*7* … *2*1 = 10! If possible, think an equivalent mathematical expression 19

Example: conditional counting Count 2 from [2,3,1,0,2,1,0,4,2] Devise a function which works with one or two arguments. Sometimes, it is not obvious at first. Then, start from the end some_func(, 2) = 3; (for this case) Guess the signature of some_func: some_func(, ) = 20

Example: conditional counting The signature is … some_func(, ) = Guess/Imagine what will happen with specific examples some_func(,2) = + 1; why? some_func(,1) = ; why? Find out the missing type ; 21

Example: conditional counting Now, the signature is complete some_func(, ) = Let’s try to draw specific exemplar cases: some_func(,2) = + 1 some_func(,1) = Introduce a variable, namely, count, to replace the type 22

Example: conditional counting Concrete exemplar cases are: some_func(count,2) = count + 1 some_func(count,1) = count Wire exemplar cases with if statement some_func(count, n) = count + 1 # if n equals to 2 count # if n is not 2 23

Example: conditional counting Define a function based on the previous analysis def count_2(count, n): if n == 2: return count + 1 return count 24

Example: conditional counting Test if it can be applied with an example list as we did at the beginning of this class. L = [1,2,0] count_2(0,1) = 0 ; note the initial counter 0 count_2(0,2) = 1 ; this zero is different from the above count_2(1,0) = 1 ;  count_2(count_2(count_2(0,1),2),0) 25

Example: conditional counting L = [1,2,0] count_2(0,1) = 0 # note the initial counter 0 count_2(0,2) = 1 # this zero is different from the above count_2(1,0) = 1 # Identify changing variable; 1, 2, 0 Identify accumulated values (or production values): 0,1,1 26

Example: conditional counting L = [1,2,0] count = count_2(0,L[0]) = 0 count = count_2(0,L[1]) = 1 count = count_2(1,L[2]) = 1 Replace changing values with a loop variable. Replace the accumulating value with a variable. 27

Example: conditional counting L = [1,2,0] count = 0 for e in L: count = count_2(count, e) Introduce a for or while statement. However, sometimes, the function is not necessary. Try to expand the function within the loop 28

Example: conditional counting Do you remember? def count_2(count, n): if n == 2: return count + 1 return count 29

Example: conditional counting L = [1,2,0] count = 0 for e in L: if e == 2: count = count + 1 Maybe this is more comprehensive than before. Loops and function calls are essentially identical. Loops are successive function calls! 30

Example: conditional counting How to generalize this example? What is the base operation? ‘x == 2’ Replace the base operation with a function def test_2(x): return x==2 31

Example: conditional counting Make a big function; count_if(L, func) The signature of the function is … count_if(, )  Counting 2 is calling count_if(L, test_2) def test_2(x): return x==2 32

Example: conditional counting The complete version of count_if(L,cond) def count_if(L, cond): count = 0 for e in L: if cond(e): count = count + 1 return count count_if(L, test_2) # will count 2s in L 33

Example: conditional counting See some demo! 34

Example: conditional collection The variation of count_if(L,cond) def collect_if(L, cond): collection = [] for e in L: if cond(e): collection = collection + [e] return collection 35

Lambda Function Anonymous function lambda x: x + 2 Equivalent to def plus2(x): return x+2 Just a short form of a function without name and return 36

Example: conditional collection Collecting 2 or 3’s multiples: collect_if(range(1,100), \ lambda x: x%2==0 or x%3==0) Intersection of two lists L1 and L2: collect_if(L1, \ lambda x: x in L2) 37

Example: conditional collection Sometimes, we need [ f(x0), f(x1), …, f(xn) ] from [x0,x1,…,xn] Let’s tweak collect_if(L, cond) def collect_mapping_if(L,mapping,cond=None): out = [] for e in L: if cond and cond(e): out.append(mapping(e)) return out 38

Example: conditional collection See some demo 39

List Comprehension List comprehension is nothing but a collect_mapping_if. [ x for x in L ] is equivalent to collect_mapping_if(L,lambda x: x,None) [ x*x for x in L ] is equivalent to collect_mapping_if(L,lambda x:x*x) [ x for x in range(0,10) if x==2 ] is equivalent to collect_mapping_if(L,lambda x:x, lambda x:x==2) 40

List Comprehension See some demo List comprehensions could be nested 41

Can you? Can you reduce an expression into a variable? Can you expand a variable or a function into an embedded form? Then, you are free! 42

Revisiting Set operations What’s the definition of the intersection? As seen before, (but watch out for duplicates!) collect_if(L1, lambda x: x in L2)  for e in L1: if e in L2: o.append(e) 43

Revisiting Set operations What’s the definition of the subtraction? As seen before, subtract(L1,L2)  collect_if(L1, lambda x: x not in L2)  for e in L1: if (e not in L1) : o.append(e) 44

Revisiting Set operations What’s the definition of the union? The union is somewhat different because of duplicates. union(L1,L2) = L1 + subtract(L2,L1) or union(L1,L2) = L1+L2 – intersect(L1,L2) 45

Revisiting Set operations Using the list comprehension, intersection(L1,L2)  collect_if(L1, lambda x: x in L2)  [ e for e in L1 if x in L2 ] subtract(L1,L2)  collect_if(L1, lambda x: x not in L2)  [ e for e in L1 if x not in L2 ] 46

Revisiting the unique operation Finding the unique elements in a list L. Let’s define a function to determine an element’s uniqueness. For example, L = [ 1, 2, 1 ]; unique(L)  [1, 2] is_unique(, 1)  [ 1 ] is_unique(, 2)  [ 1, 2 ] is_unique(, 1)  [ 1, 2 ] 47

Revisiting the unique operation is_unique(, 1)  [ 1 ] is_unique(, 2)  [ 1, 2 ] is_unique(, 1)  [ 1, 2 ] Replace with previous results; is_unique(, 1)  [1] is_unique([1], 2)  [1, 2] is_unique([1,2], 2)  [1, 2] There is a pattern. Can you wire the above with if- statement? 48

Revisiting the unique operation is_unique(, 1)  [1] # don’t know is_unique([1], 2)  [1, 2] # 2 was added because 2 is not in [1] is_unique([1, 2], 2)  [1, 2] # 2 was not added because 2 is in [1,2] It’s somewhat complicated though is_unique(, 1)  [1] is_unique(, 2)  add 2 if has no 2 is_unique(, 2)  don’t add 2 if has 2 Do you see what I am seeing? 49

Revisiting the unique operation is_unique(, 1)  [1] is_unique(, 2)  add 2 if has no 2 is_unique(, 2)  don’t add 2 if has 2 We can code the above analysis as … def is_unique(L, e): if e not in L: return L + [e] # or L.append(e) Let’s expand with some examples: is_unique(is_unique(is_unique(?,1),2),2) 50

Revisiting the unique operation is_unique(is_unique(is_unique(?,1),2),2)  [1,2] I believe that you can make a loop from the above uniques = [] for e in L: uniques = is_unique(uniques, e) Let’s expand with the is_unique function uniques = [] for e in L: if e not in uniques: uniques.append(e) 51

Loops and Set operations Are these make sense to you? 52

One more thing! 53 List indexing An element of a list or a part of a list can be easily accessed by indexing [] operator. For example L = [1,2,3] print L[0], L[1], L[-1], L[-3] # will print 1, 2, 3, 1 print L[0:2], L[1:], L[:2], L[0:3:2], L[::-1] >>> [1,2], [2,3], [1,2], [1,3], [3,2,1]

54 Be prepared! Indexing is an evil… Sometimes there is no way to escape or avoid, unfortunately…

55 Next week, We will learn about sorting, searching, and files