Discrete Mathematics: Logic. Discrete Mathematics: Lecture 14. Recursive algorithm

Transcription

1 Discrete Mathematics: Logic Discrete Mathematics: Lecture 14. Recursive algorithm

2 recursive algorithms a n = a a a a a a a = a a n-1 power (a, n) = a power (a, n-1) basis step: if n = 0, power(a, 0) = 1, which is correct since a 0 = 1 inductive step: inductive hypothesis: power(a, k) = a k for all a 0 power(a, k+1) = a power(a, k) = a a k = a k+1

3 recursive algorithms power (a, n) = a power (a, n-1) procedure power (a: nonzero real number, n: nonnegative integer) if n = 0 then return 1 else return a power(a, n-1) {output is a n }

4 recursive algorithms an algorithm is recursive if it solves a problem by reducing it to an instance of the same problem with smaller input 0! = 1 n! = n (n-1)! n is positive integer 4! = 4 3! = 4 3 2! = ! = ! = procedure factorial (n: nonnegtive integer) if n = 0 then return 1 else return n factorial(n-1) {output is n!}

5 recursive algorithms procedure factorial (n: nonnegtive integer) if n = 0 then return 1 else return n factorial(n-1) {output is n!}

6 recursive modular exponentiation b n mod m, where b, n, and m are integers with m 2, n 0, and 1 b < m b 0 mod m = 1 b n mod m = ( b b n-1 ) mod m = (b (b n-1 mod m)) mod m, n>0 procedure mpower (b, n, m) if n = 0 then return 1 else return b mpower(b, n-1, m)mod m

7 recursive modular exponentiation b n mod m, where b, n, and m are integers with m 2, n 0, and 1 b < m b n mod m = (b n/2 mod m) 2 mod m, n is even = ((b n/2 mod m) 2 mod m) (b mod m) mod m n is odd procedure mpower (b, n, m) if n = 0 then return 1 else if n is even then return mpower(b, n/2, m) 2 mod m else return ((mpower(b, n/2, m) 2 mod m) (b mod m) ) mod m

8 Fibonacci number 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...

9 Fibonacci number 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,... procedure iterative fibonacci(n: nonnegative integer) if n=0 then return 0 else x := 0 y := 1 for i :=1 to n-1 z := x + y x := y y := z return y { output is the nth Fibonacci number}

10 Fibonacci number fibonacci(0) = 0 fibonacci(1) = 1 fibonacci(n) = fibonacci(n-1) + fibonacci(n-2) procedure fibonacci(n: nonnegative integer) if n=0 then return 0 else if n=1 then return 1 else return fibonacci(n-1) + fibonacci(n-2) {output is fibonacci(n)}

11 Fibonacci number fibonacci(0) = 0 fibonacci(1) = 1 fibonacci(n) = fibonacci(n-1) + fibonacci(n-2) procedure fibonacci(n: nonnegative integer) if n=0 then return 0 else if n=1 then return 1 else return fibonacci(n-1) + fibonacci(n-2) {output is fibonacci(n)}

12 Fibonacci number procedure fastfibonacci(n: nonnegative integer) if n=0 return 0 return findfib(0,1,1,n) procedure findfib(a, b, m, n) if m=n return b return findfib(b, a+b, m+1, n) findfib(0, 1, 1, 4) return findfib(1,1, 2, 4) return findfib(1,2, 3, 4) return findfib(2, 3, 4, 4) return 3 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...

13 recursive Euclidean algorithm a = bq + r, a, b, q, r are integer gcd (a, b) = gcd (b, r), a > b gcd(8, 5)? 8 = = = = 1 2 gcd(8, 5) = gcd(5, 3) = gcd(3,2) =1 procedure gcd (a, b: positive integer, a > b) if b = 0 then return a else return gcd (b, a mod b)

14 linear search procedure linear search(x: integer, a1, a2,..., an: integers) i := 1 while (i n and x ai) i := i + 1 if i n then location := i else location := 0 return location {location is the subscript of the term that equals x, or 0 if x is not found}

15 recursive linear search search (i, j, x): searches for the first occurrence of x in the sequence ai, ai+1,... aj. procudure search(i, j, x: integers, 1 i j n) if ai = x then return i else if i = j then return 0 else return search(i+1, j, x) {output is the location of x in a1, a2,... an if it appears; otherwise it is 0}

16 binary search procedure binary search(x: integer, a1, a2,..., an: integers in increasing order) i := 1 {i is left endpoint of search interval} j := n {j is right endpoint of search interval} while (i < j) m := (i + j)/2 if x > am then i := m + 1 else j := m if x = ai then location := i else location := 0 return location {location is the subscript i of the term ai equal to x, or 0 if x is not found}

17 recursive binary search procedure binary search(i, j, x: integers, 1 i j n) m := (i+j)/2 if x = am then return m else if (x < am and i < m) then return binary search(i, m-1, x) else if (x > am and j > m) then return binary search(m+1, j, x) else return 0 {i is the start output is the location of x in a1, a2,... an if it appears; otherwise it is 0}

18 merge sort procedure mergesort(l=a1,... an) if n > 1 then m := n/2 L1 := a1, a2,..., am L2 := am+1, am+2,...,an L := merge(mergesort(l1), mergesort(l2)) {L is now sorted into elements in nondecreasing order} procedure merge(l1, L2) L := empty list while L1 and L2 are nonempty remove smaller of first elements of L1 and L2 from its list put it at the right end of L if this removal makes one list empty then remove all elements from the other list and append them to L return L

19 merge sort divide conquer

20 merge sort divide conquer