Recursion CS GMU

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Thomasina Bruce
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1 Recursion CS GMU

2 Recursion 2

3 Recursion recursion: something defined in terms of itself. function recursion: when a function calls itself. Sometimes this happens directly, sometimes indirectly. direct: function call to itself is right there, in the function body. indirect: example: foo calls bar, bar calls foo.

4 Recursion recursion: something defined in terms of itself. function call recursion: when a function calls itself. Example of Recursion: evenness def is_even(n): if n==0: return True elif n==1: return False else: equivalent = is_even (n-2) return equivalent

5 Indirect Recursion indirect recursion: a function call indirectly causes itself to be called again. Indirect Recursion: evenness and oddness def even(n): if n==0: return True return odd(n-1) def odd(n): if n==0: return False return even(n-1)

6 Recursion Basics Think of your task in terms of "base cases" and "recursive cases". base case: inputs where the answer is directly calculated with no recursive call; the parameters' values are sufficient information to find the answer. even(0) is known to be True. recursive case: inputs where the answer is calculated by recursively calling the function again on a "smaller" problem. "smaller" means we made progress towards a base case. example: even(5) calculates even(3). We've made progress towards calling to even(1), a base case.

7 Recursion Recipe Task To use recursion, you might want to follow this pattern: Thoughts 1. Iden;fy the base cases can be calculated without need for recursive call. 2. Iden;fy the recursive cases: can define solu;on in terms of other func;on calls 3. reach code for the base cases first 4. Write code for the recursive cases a8er the bases cases Is the recursive step using the method on a "smaller" problem? (needs to be yes!) (allows us to 'escape' before commikng to the recursive calls) (only use recursion when base cases aren't sufficient yet) handle any error condi.ons like base cases. Example: factorial shouldn't be called on nega;ve numbers; choose how to exit (or what to return) meaningfully. No recursive call needed/wanted.

8 Example: Factorial Mathematical definition: fact(n)= 1 if n<=1 # base case n*(fact(n-1)) if n>1 # recursive case factorial def fact(n): if n<=1: return 1 else: return n * fact(n-1) draw out memory (in the visualizer) for fact(5).

9 Separate Calls Each recursive call is distinct: separate frame on the execution stack separate local variables it's as if they are separate functions with the same implementation inside. 9

10 Factorial: Running It Suppose we call fact(3). fact(3) uses recursion, calls fact(2) (and is paused) fact(2) uses recursion, calls fact(1) (and is paused) fact(1) uses base case, directly returns value 1. fact(2) unpauses, receives the 1, returns 2*1 == 2. fact(3) unpauses, receives the 2, returns 3*2 == 6.

11 Calculation via recursion fib(n) = 1, if n=0 or n=1 = fib(n-1)+fib(n-2), otherwise fibonacci def fib (n): if n<=1: return 1 return fib(n-1)+fib(n-2) draw out memory/visualize for fib(4). n fib(n) = = = = = 13

12 Practice Problems Thought experiments: How can you use recursion to find the log of a number? (return an int) solve a maze? solve a sudoku?

Practice Problem Write a function, gcd(a,b), that calculates the greatest common

write it with a loop first (iteratively) then try again with recursion.

a's last value is the gcd of the original a and b values.

13 Practice Problem Write a function, gcd(a,b), that calculates the greatest common denominator of a and b. write it with a loop first (iteratively) then try again with recursion. Euclid's Algorithm: Assume a b. Repeatedly calculate (a,b)=(b,a%b) until b==0. a's last value is the gcd of the original a and b values. Sample calcula>ons: gcd(150,60): 150%60 == 30 60%30 == 0 gcd(150,60)==30 gcd(78,18): 78%18 == 6 18% 6 == 0 gcd(78,18)==6 gcd(2730,858): 2730 % 858 == % 156 == % 78 == 0 gcd(2730,858) == 78

14 Practice Problem Write a function, sum_list(xs), which sums all the integers in the list. The catch is that there is an arbitrary number of dimensions at each spot in the list! (something like [1,[2],[3,[4]],[[[5]]] ] could be supplied). use the type() function and recursion to solve this. even better: use isinstance(val,ty). Examples: isinstance( 5, int ) True isinstance( [2,3,4], list) True isinstance( 5, list) False

15 when should I use recursion? Recursion is sometimes much slower than an iterative solution (but not always). Pros: each call has its own local scope/local variables might look quite like the mathematical def'n easy to handle irregular shapes (as did sum_list) Cons: uses more stack space (limits recursion depth!) performing function call takes a bit more time than starting next loop iteration

16 when should I use recursion? Use recursion if the solution is straightforward and easy to verify your calculation is correct. If it runs 'fast enough', congrats! You're done. If you test realistically largest inputs and the stack doesn't overflow, congrats! You'll get away with using recursion.

17 when should I use recursion? If the number of recursive calls isn't expected to be excessively high, it could be a good approach. Given a list of n elements: either need <n recursive calls, or n known to be small always. fact(n) had n recursive calls. Only acceptable on smallish values of n (n>990 or so crashes my machine) fib(n) had far more than n recursive calls! Horrible time to use recursion. gcd(a,b) had far less than a or b calls if this solution is easier for you to write, it's an okay time to use recursion. sum_list(xs) had as many recursive calls as the number of sub-lists. Given the difficulty of writing an imperative solution, this is a good tradeoff.

CS 310 Advanced Data Structures and Algorithms

CS 310 Advanced Data Structures and Algorithms Recursion June 27, 2017 Tong Wang UMass Boston CS 310 June 27, 2017 1 / 20 Recursion Recursion means defining something, such as a function, in terms of itself