Discrete Mathematics: Logic Discrete Mathematics: Lecture 14. Recursive algorithm recursive algorithms an = a·a·a·a·a·a···a = a·an-1 power (a, n) = a ·power (a, n-1) basis step: if n = 0, power(a, 0) = 1, which is correct since a0 = 1 inductive step: inductive hypothesis: power(a, k) = ak for all a ≠ 0 power(a, k+1) = a·power(a, k) = a·ak = ak+1 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 an} 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! = 4·3·2·1! = 4·3·2·1·0! = 4·3·2·1 procedure factorial (n: nonnegtive integer) if n = 0 then return 1 else return n·factorial(n-1) {output is n!} recursive algorithms procedure factorial (n: nonnegtive integer) if n = 0 then return 1 else return n·factorial(n-1) {output is n!} recursive modular exponentiation bn mod m, where b, n, and m are integers with m ≥ 2, n ≥ 0, and 1≤ b < m b0 mod m = 1 bn mod m = ( b·bn-1) mod m = (b·(bn-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 recursive modular exponentiation bn mod m, where b, n, and m are integers with m ≥ 2, n ≥ 0, and 1≤ b < m bn mod m = (bn/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 Fibonacci number 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . 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} 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)} 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)} 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, . recursive Euclidean algorithm a = bq + r, a, b, q, r are integer gcd (a, b) = gcd (b, r) , a > b gcd(8, 5) ? gcd(8, 5) 8 = 5 ·1 + 3 = gcd(5, 3) 5 = 3 ·1 + 2 = gcd(3,2) 3 = 2 ·1 + 1 =1 2 = 1 ·2 procedure gcd (a, b: positive integer, a > b) if b = 0 then return a else return gcd (b, a mod b) linear search procedure linear search(x: integer, a1, a 2, . , a n: 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} recursive linear search search (i, j, x): searches for the first occurrence of x in the sequence ai, a i+1, . a j. 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, a 2, . a n if it appears; otherwise it is 0} binary search procedure binary search(x: integer, a1, a 2, . , a n: 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} 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, a 2, . a n if it appears; otherwise it is 0} merge sort procedure mergesort(L=a1, . a n) if n > 1 then m := ⎣n/2⎦ L1 := a1, a 2, . , a m L2 := am+1, a m+2, . ,a n L := merge(mergesort(L1), mergesort(L2)) {L is now sorted into elements in nondecreasing order} procedure merge(L1, L 2) 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 merge sort 27 13 26 1 15 2 24 38 27 13 26 1 15 2 24 38 divide conquer 1 13 26 27 2 15 24 38 1 2 13 15 24 26 27 38 merge sort 27 13 26 1 15 2 24 38 1 11 27 13 26 1 15 2 24 38 divide 2 6 27 13 26 1 15 2 24 38 3 4 7 8 27 13 26 1 15 2 24 38 5 9 13 27 1 26 2 15 24 38 10 conquer 1 13 26 27 2 15 24 38 21 1 2 13 15 24 26 27 38.