Discrete Mathematics: Lecture 14. Recursive Algorithm Recursive Algorithms

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

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27 13 26 1 15 2 24 38 divide

conquer

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1 2 13 15 24 26 27 38 merge sort

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27 13 26 1 15 2 24 38 3 4 7 8

27 13 26 1 15 2 24 38 5 9

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10 conquer

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