Recursion. February 02, 2018 Cinda Heeren / Geoffrey Tien 1

Transcription

1 Recursion February 02, 2018 Cinda Heeren / Geoffrey Tien 1

2 Function calls in daily life How do you handle interruptions in daily life? You're at home, working on PA1 You stop to look up something in the textbook Your roommate/spouse/partner/parent/etc. asks you for help moving some stuff outside Your neighbour tells you a story The doorbell rings You stop what you're doing, you memorize where you were in your task, you handle the interruption, and then you go back to what you were doing February 02, 2018 Cinda Heeren / Geoffrey Tien 2

3 Call stack in daily life I am reading about inserting at the front of a list in Carrano p. 140 I am working on line 177 of my chain.cpp file February 02, 2018 Cinda Heeren / Geoffrey Tien 3

4 Call stack in daily life I have moved 2 boxes of old papers to the back alley I am reading about inserting at the front of a list in Carrano p. 140 I am working on line 177 of my chain.cpp file February 02, 2018 Cinda Heeren / Geoffrey Tien 4

5 Call stack in daily life I am listening to my neighbour tell me about some wild party he went to last night I have moved 4 boxes of old papers to the back alley I am reading about inserting at the front of a list in Carrano p. 140 I am working on line 177 of my chain.cpp file February 02, 2018 Cinda Heeren / Geoffrey Tien 5

6 Call stack in daily life I am signing for a FedEx delivery My neighbour is just about to get to the part where he pukes... I have moved 4 boxes of old papers to the back alley I am reading about inserting at the front of a list in Carrano p. 140 I am working on line 177 of my chain.cpp file February 02, 2018 Cinda Heeren / Geoffrey Tien 6

7 Call stack in daily life My neighbour has finally finished his story. "Good for you!" I say sarcastically I have moved 4 boxes of old papers to the back alley I am reading about inserting at the front of a list in Carrano p. 140 I am working on line 177 of my chain.cpp file February 02, 2018 Cinda Heeren / Geoffrey Tien 7

8 Call stack in daily life I have moved 86 boxes of old papers to the back alley I am reading about inserting at the front of a list in Carrano p. 140 I am working on line 177 of my chain.cpp file February 02, 2018 Cinda Heeren / Geoffrey Tien 8

9 Call stack in daily life I am reading about the list copy constructor in Carrano p. 146 I am working on line 177 of my chain.cpp file February 02, 2018 Cinda Heeren / Geoffrey Tien 9

10 Call stack in daily life I finished my chain.cpp file! February 02, 2018 Cinda Heeren / Geoffrey Tien 10

11 Rabbits! What happens when you put a pair of rabbits in a field? More rabbits! Let s model the rabbit population, with a few assumptions: Newly-born rabbits take one month to reach maturity and mate Each pair of rabbits produces another pair of rabbits one month after mating Rabbits never die...and recursion February 02, 2018 Cinda Heeren / Geoffrey Tien 11

12 More rabbits... How many rabbit pairs are there after 5 months? Month 1: start 1 pair Month 2: first pair are now mature and mate 1 pair Month 3: first pair give birth to a pair of babies original pair + baby pair = 2 pairs Month 4: first pair give birth to another pair of babies, pair born in month 3 are now mature 3 pairs Month Month 5: the 3 pairs from month 4, and two new pairs 5 pairs Month 6: the 5 pairs from month 5, and three new pairs 8 pairs And so on... February 02, 2018 Cinda Heeren / Geoffrey Tien 12

13 Fibonacci series The n th number in the Fibonacci series, fib(n), is: 0 if n = 0, and 1 if n = 1 fib(n 1) + fib(n 2) for any n > 1 e.g. what is fib(23) Easy if we only knew fib(22) and fib(21) The answer is fib(22) + fib(21) What happens if we actually write a function to calculate Fibonacci numbers like this? February 02, 2018 Cinda Heeren / Geoffrey Tien 13

14 Calculating the Fibonacci series Let s write a function just like the formula fib(n) = 0 if n = 0, 1 if n = 1, otherwise fib(n) = fib(n 1) + fib(n 2) int fib(int n) { if (n <= 1) return max(0, n); else return fib(n-1) + fib(n-2); } The function calls itself February 02, 2018 Cinda Heeren / Geoffrey Tien 14

15 Recursive functions The Fibonacci function is recursive A recursive function calls itself Each call to a recursive method results in a separate call to the method, with its own input Recursive functions are just like other functions The invocation (e.g. parameters, etc.) is pushed onto the call stack And removed from the call stack when the end of a method or a return statement is reached Execution returns to the previous method call February 02, 2018 Cinda Heeren / Geoffrey Tien 15

16 Recursive function anatomy Recursive functions do not use loops to repeat instructions But use recursive calls, in if statements Recursive functions consist of two or more cases, there must be at least one Base case, and Recursive case February 02, 2018 Cinda Heeren / Geoffrey Tien 16

17 Recursion cases The base case is a smaller problem with a known solution This problem s solution must not be recursive Otherwise the function may never terminate There can be more than one base case And base cases may be implicit The recursive case is the same problem with smaller input The recursive case must include a recursive function call There can be more than one recursive case February 02, 2018 Cinda Heeren / Geoffrey Tien 17

18 Analysis of fib(5) int fib(int n) { if (n <= 1) return max(0, n); else return fib(n-1) + fib(n-2); } 3 5 fib(5) 2 fib(4) fib(3) fib(3) fib(2) fib(2) fib(1) fib(2) fib(1) fib(1) fib(0) fib(1) fib(0) 1 fib(1) 0 fib(0) Later in the course we will explain how this is an extremely inefficient way to compute the Fibonacci series February 02, 2018 Cinda Heeren / Geoffrey Tien 18

19 Example Base cases Target is found, or the end of the array is reached Recursive case Recursive linear search Target not found, search subarray starting from next element // Recursive linear search int reclinsearch(int arr[], int next, int sz, int x) { if (next >= sz) // end of array reached return -1; else if (x == arr[next]) // target found return next; else // not found, search from different starting index return reclinsearch(arr, next + 1, sz, x); } February 02, 2018 Cinda Heeren / Geoffrey Tien 19

20 Stack overflow It's not just a useful website By default, program stack space is extremely limited If many function invocations are placed on the stack without returning, a stack overflow can result int summation(int n) { if (n <= 0) return 0; else return (n + summation(n-1)); } int factorial(int n) { if (n <= 0) return 1; else return (n * factorial(n-1)); } summation( ); But a call to factorial( ); *might* not produce a stack overflow. Why? February 02, 2018 Cinda Heeren / Geoffrey Tien 20

21 An infinite recursion This function's recursive case does not converge to a base case. What happens when we call it? void endlesslygreet() { cout << "Hello world!\n"; endlesslygreet(); } A. This will result in a stack overflow B. This will magically not result in a stack overflow C. It depends D. None of the above February 02, 2018 Cinda Heeren / Geoffrey Tien 21

22 Tail recursion A function is tail-recursive if the recursive call is the absolute last thing the function needs to do before returning No need to wait for a return from a deeper recursive call to compute a result Why bother pushing a new stack frame? There is nothing to remember Just use the old stack frame Most compilers will do this February 02, 2018 Cinda Heeren / Geoffrey Tien 22

23 Ordinary vs tail recursion How are these functions similar/different? int factorial(int n) { if (n <= 0) return 1; else return (n * factorial(n-1)); } void countinggreet(int n) { if (n <= 1) return; cout << "Hello!\n"; countinggreet(n-1); } Think about the program flow of a call e.g. factorial(4) vs countinggreet(4) February 02, 2018 Cinda Heeren / Geoffrey Tien 23

24 Tail recursive factorial Use an additional parameter (and a recursive helper function) to keep track of the computed factorial so far int facttail(int n) { int result = facttailrec(n, 1); return result; } int facttailrec(int n, int acc) { if (n == 0) return acc; else return facttailrec(n-1, n*acc); } February 02, 2018 Cinda Heeren / Geoffrey Tien 24

25 Tail recursive Fibonacci int fibtail(int n) { return fibtailrec(n, 1, 1); } int fibtailrec(int n, int next, int result) { if (n == 1) return result; else return fibtailrec(n-1, result + next, next); } This runs almost as quickly as an iterative implementation, and uses about the same amount of stack space as well February 02, 2018 Cinda Heeren / Geoffrey Tien 25

26 Readings for this lesson Carrano & Henry Chapter 2 (Recursion) Next class Carrano & Henry: Chapter 15.1 (Tree terminology) February 02, 2018 Cinda Heeren / Geoffrey Tien 26