Recursion

Transcript

1. Recursion Thierry Sans
2. Programming basics • Variables and scope • Control structures (conditionals, loops) • Functions (with return statement)
3. A classical example - factorial n! {1 1 × 2 × … × (n-1) × n if n = 0 if n > 0 Intuitive definition
4. A classical example - factorial n! {1 1 × 2 × … × (n-1) × n if n = 0 if n > 0 Intuitive definition (n-1)!
5. A classical example - factorial n! {1 n × (n-1)! if n = 0 if n > 0 Recursive definition n! {1 1 × 2 × … × (n-1) × n if n = 0 if n > 0 Intuitive definition (n-1)!
6. Unrolling the function calls fact(4)
7. Unrolling the function calls fact(4) ⤷ return 4*fact(3)
8. Unrolling the function calls fact(4) ⤷ return 4*fact(3) ⤷ return 3*fact(2)
9. Unrolling the function calls fact(4) ⤷ return 4*fact(3) ⤷ return 3*fact(2) ⤷ return 2*fact(1)
10. Unrolling the function calls fact(4) ⤷ return 4*fact(3) ⤷ return 3*fact(2) ⤷ return 2*fact(1) ⤷ return 1*fact(0)
11. Unrolling the function calls fact(4) ⤷ return 4*fact(3) ⤷ return 3*fact(2) ⤷ return 2*fact(1) ⤷ return 1*fact(0) ⤷ return 1
12. Unrolling the function calls fact(4) ⤷ return 4*fact(3) ⤷ return 3*fact(2) ⤷ return 2*fact(1) ⤷ return 1*fact(0) ⤷ return 1 㽉 1
13. Unrolling the function calls fact(4) ⤷ return 4*fact(3) ⤷ return 3*fact(2) ⤷ return 2*fact(1) ⤷ return 1*fact(0) ⤷ return 1 1 㽉 㽉 1
14. Unrolling the function calls fact(4) ⤷ return 4*fact(3) ⤷ return 3*fact(2) ⤷ return 2*fact(1) ⤷ return 1*fact(0) ⤷ return 1 1 㽉 1 㽉 㽉 1
15. Unrolling the function calls fact(4) ⤷ return 4*fact(3) ⤷ return 3*fact(2) ⤷ return 2*fact(1) ⤷ return 1*fact(0) ⤷ return 1 1 㽉 1 㽉 㽉 2 㽉 1
16. Unrolling the function calls fact(4) ⤷ return 4*fact(3) ⤷ return 3*fact(2) ⤷ return 2*fact(1) ⤷ return 1*fact(0) ⤷ return 1 1 㽉 1 㽉 㽉 2 6 㽉 㽉 1
17. Unrolling the function calls fact(4) ⤷ return 4*fact(3) ⤷ return 3*fact(2) ⤷ return 2*fact(1) ⤷ return 1*fact(0) ⤷ return 1 1 㽉 1 㽉 㽉 2 24 6 㽉 㽉 1
18. Definition of a recursive function Base case the result is the simplest version of the problem Recursive case the result is computed based on a reduced version of the same problem
19. Definition of a recursive function Base case the result is the simplest version of the problem Recursive case the result is computed based on a reduced version of the same problem n! {1 n × (n-1)! if n = 0 if n > 0
20. Definition of a recursive function Base case the result is the simplest version of the problem Recursive case the result is computed based on a reduced version of the same problem n! {1 n × (n-1)! if n = 0 if n > 0 Base case Recursive case
21. Cumulative return An investment returns 2.5% every year. If you invest $1000 at the beginning only, how much money do you get after n years of cumulating the interests? agg(n) { Base case Recursive case
22. Cumulative return An investment returns 2.5% every year. If you invest $1000 at the beginning only, how much money do you get after n years of cumulating the interests? agg(n) {1000 agg(n-1) × 1.025 if n = 0 if n > 0 Base case Recursive case
23. Another classical example - Fibonacci (rabbits) fib(n) {1 1 fib(n-1) + fib(n-2) if n = 0 if n = 1 if n > 1 Base case Base case Recursive case
24. Another classical example - Fibonacci (rabbits)
25. Another classical example - Fibonacci (rabbits) Rabbits Newly born Total month 0 0 1 1 month 1 1 0 1 month 2 1 1 2 month 3 2 1 3 month 4 3 2 5 month 5 5 3 8 month 6 8 5 13
26. Another classical example - Fibonacci (rabbits) fib(n) {1 1 fib(n-1) + fib(n-2) if n = 0 if n = 1 if n > 1 Base case Base case Recursive case Rabbits Newly born Total month 0 0 1 1 month 1 1 0 1 month 2 1 1 2 month 3 2 1 3 month 4 3 2 5 month 5 5 3 8 month 6 8 5 13
27. Palindrome A palindrome is a sequence of characters which reads the same backward or forward. “kayak” “emma”
28. Palindrome A palindrome is a sequence of characters which reads the same backward or forward. pal(s) ➡ True if the length of s is 0 or 1 ➡ False if the first and last characters do not match
 (assuming the length is greater than 1) ➡ pal(s’) otherwise, considering s’ the substring of s
 without the first and last character Base case Recursive case Base case “kayak” “emma”
29. Recursion is about problem solving Recursion is a way to solve problems and design solutions ➡ Programming paradigm : functional programming
30. Going further • Recursive data structures list, tree, graph, … • Puzzles tower of Hanoi, Sudoku, … • Art Fractales image sources wikimedia and Deviant Art