Algorithms. Jordi Planes. Escola Politècnica Superior Universitat de Lleida

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1 Algorithms Jordi Planes Escola Politècnica Superior Universitat de Lleida 2016

2 Vote in Sakai.

3 Syllabus What s been done Formal specification Cost Transformation recursion iteration

4 Syllabus What we ll do today Formal specification Cost Transformation recursion iteration

5 Recursion

6 RecursiveTree by Brentsmith101

8 Recursion Recursion (Wikipedia) Recursion is the process of repeating items in a self-similar way.

9 Recursion Recursion (Wikipedia) Recursion is the process of repeating items in a self-similar way. Example (Fibonacci) fibonacci(x) = x fibonacci(x 1)

10 Recursion (John D. Cook) Be sure of three things: 1. The problem gets smaller each time. 2. You include a solution for the base case. 3. Each case is handled correctly.

11 General scheme function r e c u r s i v e ( x ) i s base c a s e return t r i v i a l ( x ) r e c u r s i v e c a s e y = r e c u r s i v e ( r e d u c e ( x ) ) return compute ( x, y )

12 Example: factorial function f a c t o r i a l ( x ) i s x = 0 return 1 x > 0 y f a c t o r i a l ( x 1 ) return x y

13 Example: product Design the product by additions

14 Example: product function product ( x, y ) i s x = 0 return 0 x > 0 p product ( x 1, y ) return y + p

15 Exercises Digit counter for an integer Addition of all the elements in a list Check if a list is sorted Power by products Division by substrations Integer Division Maximum in a list Lineal search

16 Transformations

17 Transformations Any recursive computation has each equivalent iterative computation, and viceversa

18 Transformations Any recursive computation has each equivalent iterative computation, and viceversa The easiest transformation: tail recursion

19 Transformations Any recursive computation has each equivalent iterative computation, and viceversa The easiest transformation: tail recursion General Scheme: Recursive Tail recursive Iterative

20 Transformations Tail call A tail call is a subroutine call performed as the final action of a procedure. function c a l l e r ( x ) i s y = some o p e r a t i o n s return c a l l e e ( y )

21 Transformations Tail call A tail call is a subroutine call performed as the final action of a procedure. function c a l l e r ( x ) i s y = some o p e r a t i o n s return c a l l e e ( y ) Tail recursion A tail recursion is a tail call which calls the caller. function c a l l e r ( x ) i s y = some o p e r a t i o n s return c a l l e r ( y )

22 General scheme function r e c u r s i v e ( x ) i s base c a s e return t r i v i a l ( x ) r e c u r s i v e c a s e y = r e c u r s i v e ( r e d u c e ( x ) ) return compute ( x, y ) function r e c u r s i v e ( x, a ) i s base c a s e a r e c u r s i v e c a s e a = compute ( x, a ) return r e c u r s i v e ( r e d u c e ( x ), a )

23 General scheme function r e c u r s i v e ( x ) i s base c a s e return t r i v i a l ( x ) r e c u r s i v e c a s e y = r e c u r s i v e ( r e d u c e ( x ) ) return compute ( x, y ) function r e c u r s i v e ( x, a ) i s base c a s e a r e c u r s i v e c a s e a = compute ( x, a ) return r e c u r s i v e ( r e d u c e ( x ), a ) r e c u r s i v e ( x, t r i v i a l ( x ) )

24 Example: factorial function f a c t o r i a l ( x ) i s x = 0 return 1 x > 0 y f a c t o r i a l ( x 1 ) return x y

25 Example: factorial function f a c t o r i a l ( x ) i s x = 0 return 1 x > 0 y f a c t o r i a l ( x 1 ) return x y function f a c t o r i a l ( x, a ) i s x = 0 return a x > 0 a x a return f a c t o r i a l ( x 1, a ) f a c t o r i a l ( x, 1 )

26 Exercises Digit counter for an integer Addition of all the elements in a list Check if a list is sorted Power by products Division by substrations Integer Division Maximum in a list Lineal search

27 General scheme function r e c u r s i v e ( x, a ) i s base c a s e a r e c u r s i v e c a s e return r e c u r s i v e ( r e d u c e ( x ), compute ( x, a r e c u r s i v e ( x, t r i v i a l ( x ) ) i s

28 General scheme function r e c u r s i v e ( x, a ) i s base c a s e a r e c u r s i v e c a s e return r e c u r s i v e ( r e d u c e ( x ), compute ( x, a r e c u r s i v e ( x, t r i v i a l ( x ) ) i s function r e c u r s i v e ( x ) i s a t r i v i a l ( x ) while not base c a s e ( x ) do a compute ( x, a ) x r e d u c e ( x ) return a

29 Example: factorial function f a c t o r i a l ( x, a ) i s x = 0 return a x > 0 a x a return f a c t o r i a l ( x 1, a ) f a c t o r i a l ( x, 1 )

30 Example: factorial function f a c t o r i a l ( x, a ) i s x = 0 return a x > 0 a x a return f a c t o r i a l ( x 1, a ) f a c t o r i a l ( x, 1 ) function f a c t o r i a l ( x ) i s a 1 while not x = 0 do a a x x x 1 return a

31 Exercises Digit counter for an integer Addition of all the elements in a list Check if a list is sorted Power by products Division by substrations Integer Division Maximum in a list Lineal search

32 Compute the costs Compute the costs for the previous exercises.

Algorithms. Jordi Planes. Escola Politècnica Superior Universitat de Lleida

Algorithms. Jordi Planes. Escola Politècnica Superior Universitat de Lleida Algorithms Jordi Planes Escola Politècnica Superior Universitat de Lleida 2016 Syllabus What s been done Formal specification Computational Cost Transformation recursion iteration Divide and conquer Sorting