John Perry. Fall 2009

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1 Lecture 11: Recursion University of Southern Mississippi Fall 2009

2 Outline You should be in worksheet mode to repeat the examples.

3 Outline 1 2 3

4 re + cursum: return, travel the path again (Latin) Two (equivalent) views: mathematical: a function defined using itself; computational: an algorithm that invokes itself.

5 When? At least one base case with no All recursive chains terminate at base case

6 Proof by induction Prove P (n) for all n : Inductive Base: Show P (1) Inductive Hypothesis: Assume P (i) for 1 i < n Inductive Step: Show P (n) using P (i) for 1 i < n

7 Fibonacci s Bunnies Leonardo da Pisa, called Fibonacci, describes in Liber Abaci a population of bunnies: first month: one pair of bunnies;

8 Fibonacci s Bunnies Leonardo da Pisa, called Fibonacci, describes in Liber Abaci a population of bunnies: first month: one pair of bunnies; second month: pair matures; third month: mature pair produces new pair;

9 Fibonacci s Bunnies Leonardo da Pisa, called Fibonacci, describes in Liber Abaci a population of bunnies: first month: one pair of bunnies; second month: pair matures; third month: mature pair produces new pair; fourth month: second pair matures, first pair produces new pair;

10 Fibonacci s Bunnies Leonardo da Pisa, called Fibonacci, describes in Liber Abaci a population of bunnies: first month: one pair of bunnies; second month: pair matures; third month: mature pair produces new pair; fourth month: second pair matures, first pair produces new pair; fifth month: third pair matures, two mature pairs produce new pairs;...

11 How many pairs? month mature pairs immature pairs new pairs 1 total pairs 1

12 How many pairs? month mature pairs immature pairs 1 new pairs 1 total pairs 1 1

13 How many pairs? month mature pairs 1 immature pairs 1 new pairs 1 1 total pairs 1 1 2

14 How many pairs? month mature pairs 1 1 immature pairs 1 1 new pairs total pairs

15 How many pairs? month mature pairs immature pairs new pairs total pairs

16 How many pairs? month mature pairs immature pairs new pairs total pairs

17 How many pairs? month mature pairs immature pairs new pairs total pairs

18 Describing it month mature pairs immature pairs new pairs total pairs total = (# mature + # immature) + # new

19 Describing it month mature pairs immature pairs new pairs total pairs total = (# mature + # immature) + # new total = # one month ago + # new

20 Describing it month mature pairs immature pairs new pairs total pairs total = (# mature + # immature) + # new total = # one month ago + # new total = # one month ago + # mature now

21 Describing it month mature pairs immature pairs new pairs total pairs total = (# mature + # immature) + # new total = # one month ago + # new total = # one month ago + # mature now total = # one month ago + # two months ago

22 Describing it month mature pairs immature pairs new pairs total pairs total = (# mature + # immature) + # new total = # one month ago + # new total = # one month ago + # mature now total = # one month ago + # two months ago F now = F one month ago + F two months ago, or

23 Describing it month mature pairs immature pairs new pairs total pairs total = (# mature + # immature) + # new total = # one month ago + # new total = # one month ago + # mature now total = # one month ago + # two months ago F now = F one month ago + F two months ago, or F i = F i 1 + F i 2

24 Fibonacci Sequence 1, i = 1,2; F i = F i 1 + F i 2, i 3.

25 Example F i = Fibonacci Sequence 1, i = 1,2; F i 1 + F i 2, i 3. F 5 = F 4 + F 3 = F 3 + F 2 + F2 + F 1 = F 2 + F 1 + F2 + F2 + F 1 = 3F 2 + 2F 1 = 5.

26 Example F i = Fibonacci Sequence 1, i = 1,2; F i 1 + F i 2, i 3. F 5 = F 4 + F 3 = F 3 + F 2 + F2 + F 1 = F 2 + F 1 + F2 + F2 + F 1 = 3F 2 + 2F 1 = 5. F 100 = F 99 + F 98 =... = F F 1 =

27 Pseudocode Easy to implement : algorithm Fibonacci inputs n outputs the nth Fibonacci number do if n > 2 return Fibonacci(n 2) + Fibonacci(n 1) else return 1

28 Implementation sage: def fibonacci(n): if (n>2): return fibonacci(n-2) + fibonacci(n-1) else: return 1

29 Implementation sage: def fibonacci(n): if (n>2): return fibonacci(n-2) + fibonacci(n-1) else: return 1 sage: fibonacci(5) 5 sage: fibonacci(20) 6765 sage: fibonacci(30)

30 Outline 1 2 3

31 Infinite loops must stop eventually

32 Infinite loops must stop eventually Wasted computation fibonacci(20) requires fibonacci(19) and fibonacci(18) fibonacci(19) also requires fibonacci(18) fibonacci(18) computed twice!

33 Modify program: Example sage: def fibonacci(n): print 'computing fibonacci #', n, if (n>2): return fibonacci(n-2) + fibonacci(n-1) else: return 1

34 Modify program: Example sage: def fibonacci(n): print 'computing fibonacci #', n, if (n>2): return fibonacci(n-2) + fibonacci(n-1) else: return 1 sage: fibonacci(5) computing fibonacci # 5 computing fibonacci # 3 computing fibonacci # 1 computing fibonacci # 2 computing fibonacci # 4 computing fibonacci # 2 computing fibonacci # 3 computing fibonacci # 1 computing fibonacci # 2 5

35 Modify program: Example sage: def fibonacci(n): print 'computing fibonacci #', n, if (n>2): return fibonacci(n-2) + fibonacci(n-1) else: return 1 sage: fibonacci(5) computing fibonacci # 5 computing fibonacci # 3 computing fibonacci # 1 computing fibonacci # 2 computing fibonacci # 4 computing fibonacci # 2 computing fibonacci # 3 computing fibonacci # 1 computing fibonacci # F 3 computed 2 times; F 2, 3 times; F 1, 2 times

36 Maintain list of pre-computed values: algorithm Fibonacci_with_table globals F, a list of integers, initially [1,1] inputs n outputs the nth Fibonacci number do if n > #F Let a = Fibonacci_with_table(n 1) Let b = Fibonacci_with_table(n 2) Let F n = a + b return F n Workaround

37 Implementation sage: F = [1,1] sage: def fibonacci_with_table(n): if (n>len(f)): print 'computing fibonacci #', n, a = fibonacci_with_table(n-1) b = fibonacci_with_table(n-2) F.append(a + b) return F[n-1]

38 Implementation sage: F = [1,1] sage: def fibonacci_with_table(n): if (n>len(f)): print 'computing fibonacci #', n, a = fibonacci_with_table(n-1) b = fibonacci_with_table(n-2) F.append(a + b) return F[n-1] Example sage: fibonacci_with_table(5) computing fibonacci # 5 computing fibonacci # 4 computing fibonacci # 3 5

39 Avoid when possible can often rewrite as a loop can sometimes rewrite in closed form However...

40 Avoid when possible can often rewrite as a loop can sometimes rewrite in closed form Example Closed form for Fibonacci sequence: However... F n = n n 5.

41 Avoid when possible can often rewrite as a loop can sometimes rewrite in closed form Example Closed form for Fibonacci sequence: However... F n = n n 5. Coincidence? I think not = golden ratio 2

42 Looped Fibonacci: How? Recursive: backwards, then forwards again F n F n 1, F n 2 F 2, F 1 F n Looped: direct F 1, F 2 F 3 F n remember two previous computations remember? = variables

43 Looped Fibonacci: Pseudocode algorithm Looped Fibonacci inputs n outputs the nth Fibonacci number do Define the base case Let F prev = 1, F curr = 1 Use the formula to move forward to F n Let i = 2 while i < n do Compute next element, then move forward Let F next = F prev + F curr Let F prev = F curr, F curr = F next Increment i return F curr

44 sage: def looped_fibonacci(n): Fprev = 1 Fcurr = 1 i = 2 while (i < n): Fnext = Fprev + Fcurr Fprev = Fcurr Fcurr = Fnext i = i + 1 return Fcurr Looped Fibonacci: Implementation

45 sage: sage: def looped_fibonacci(n): Fprev = 1 Fcurr = 1 i = 2 while (i < n): Fnext = Fprev + Fcurr Fprev = Fcurr Fcurr = Fnext i = i + 1 return Fcurr looped_fibonacci(100) Looped Fibonacci: Implementation

46 sage: sage: def looped_fibonacci(n): Fprev = 1 Fcurr = 1 i = 2 while (i < n): Fnext = Fprev + Fcurr Fprev = Fcurr Fcurr = Fnext i = i + 1 return Fcurr looped_fibonacci(100) Looped Fibonacci: Implementation (Much faster than recursive version)

47 Recursive pros: simpler to write cons Looped Recursive vs. Looped vs. Closed-form slower memory intensive indefinite loop w/out loop structure pros: not too slow, not too complicated, loop can be definite cons: not (usually) as simple as recursive, sometime not obvious Closed-form pros: one step (no loop) cons: finding it often requires significant effort

48 Outline 1 2 3

49 Recursion: function defined using other values of function Issues can waste computation can lead to infinite loops (bad design) Use when closed/loop form too complicated chains not too long memory table feasible

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