이산수학 (Discrete Mathematics)

3.5 재귀 알고리즘 (Recursive Algorithms)

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문양세문양세강원대학교 컴퓨터과학과강원대학교 컴퓨터과학과

Page 2Discrete Mathematicsby Yang-Sae Moon

IntroductionIntroduction3.5 Recursive Algorithms

Recursive definitions can be used to describe algorithms as

well as functions and sets.( 재귀적 정의를 수행한 경우 , 손쉽게 알고리즘의 함수 / 집합으로 기술할 수 있다 .)

예제 1: A procedure to compute an.

procedure power(a≠0: real, nN)if n = 0 then return 1else return a · power(a, n 1)−

Page 3Discrete Mathematicsby Yang-Sae Moon

Efficiency of Recursive AlgorithmsEfficiency of Recursive Algorithms

The time complexity of a recursive algorithm may depend crit

ically on the number of recursive calls it makes.( 재귀 호출에서의 시간 복잡도는 재귀 호출 횟수에 크게 의존적이다 .)

예제 2: Modular exponentiation to a power n can take log(n)

time if done right, but linear time if done slightly differently. ( 잘하면 O(log(n)) 이나 , 조금만 잘못하면 O(n) 이 된다 .)

• Task: Compute bn mod m, where

m≥2, n≥0, and 1≤b<m.

3.5 Recursive Algorithms

Page 4Discrete Mathematicsby Yang-Sae Moon

Modular Exponentiation Algorithm Modular Exponentiation Algorithm #1#1

Uses the fact that

• bn = b·bn 1− and that

• x·y mod m = x·(y mod m) mod m.

3.5 Recursive Algorithms

Note this algorithm takes (n) steps!

procedure mpower(b≥1,n≥0,m>b N)

{Returns bn mod m.}if n=0 then return 1 elsereturn (b·mpower(b,n 1,− m)) mod m

Page 5Discrete Mathematicsby Yang-Sae Moon

Modular Exponentiation Algorithm Modular Exponentiation Algorithm #2#2

Uses the fact that b2k = bk·2 = (bk)2.

3.5 Recursive Algorithms

What is its time complexity?

procedure mpower(b,n,m) {same signature}

if n=0 then return 1else if 2|n then

return mpower(b,n/2,m)2 mod melse return (mpower(b,n 1,− m)·b) mod m

( 첫 번째 mpower() 는 한번 call 되고 , 그 값을 제곱하는

것임 )

(log n) steps

Page 6Discrete Mathematicsby Yang-Sae Moon

A Slight Variation of Algorithm #2A Slight Variation of Algorithm #2

Nearly identical but takes (n) time instead!

3.5 Recursive Algorithms

The number of recursive calls made is critical.

procedure mpower(b,n,m) {same signature}

if n=0 then return 1

else if 2|n then

return (mpower(b,n/2,m)·mpower(b,n/2,m)) mod m

else return (mpower(b,n 1,− m)·b) mod m

Page 7Discrete Mathematicsby Yang-Sae Moon

Recursive Euclid’s Algorithm (Recursive Euclid’s Algorithm ( 예제 예제 4)4)

Note recursive algorithms are often simpler to code

than iterative ones… (Recursion 이 코드를 보다 간단하게 하지만… )

However, they can consume more stack space, if

your compiler is not smart enough.( 일반적으로 , recursion 은 보다 많은 stack space 를 차지한다 .)

3.5 Recursive Algorithms

procedure gcd(a,bN)

if a = 0 then return b

else return gcd(b mod a, a)

Page 8Discrete Mathematicsby Yang-Sae Moon

Recursion vs. Iteration (1/3)Recursion vs. Iteration (1/3)3.5 Recursive Algorithms

Factorial – Recursion

procedure factorial(nN)if n = 1 then return nelse return n factorial(n – 1)

Factorial – Iteration

procedure factorial(nN)x := 1for i := 1 to n

x := i xreturn x

Page 9Discrete Mathematicsby Yang-Sae Moon

Recursion vs. Iteration (2/3)Recursion vs. Iteration (2/3)3.5 Recursive Algorithms

Fibonacci – Recursion

procedure fibonacci(n: nonnegative integer)if (n = 0 or n = 1) then return nelse return (factorial(n–1) + factorial(n–2))

Fibonacci – Iteration

procedure fibonacci(n: nonnegative integer)if n = 0 then return 0else

x := 0, y := 1for i := 1 to n – 1

z := x + y, x := y, y := zreturn z

Page 10Discrete Mathematicsby Yang-Sae Moon

3.5 Recursive Algorithms

재귀 (recursion) 는 프로그램을 간단히 하고 , 이해하기가

쉬운 장점이 있는 반면에 ,

각 호출이 스택을 사용하므로 , Depth 가 너무 깊어지지

않도록 조심스럽게 프로그래밍해야 함

컴퓨터에게 보다 적합한 방법은 반복 (iteration) 을 사용한

프로그래밍이나 , 적당한 범위에서 재귀를 사용하는 것이

바람직함

Recursion vs. Iteration (3/3)Recursion vs. Iteration (3/3)

Page 11Discrete Mathematicsby Yang-Sae Moon

Merge Sort (Merge Sort ( 예제 예제 8) (1/2)8) (1/2)

The merge takes The merge takes ((nn) steps, and merge-sort takes) steps, and merge-sort takes

((nn log log nn).).

3.5 Recursive Algorithms

procedure sort(L = 1,…, n)

if n>1 then

m := n/2 {this is rough ½-way point}

L := merge(sort(1,…, m), sort(m+1,…, n))

return L

Page 12Discrete Mathematicsby Yang-Sae Moon

The Merge RoutineThe Merge Routine

3.5 Recursive Algorithms

procedure merge(A, B: sorted lists)L = empty listi:=0, j:=0, k:=0while i<|A| j<|B| {|A| is length of A} if i=|A| then Lk := Bj; j := j + 1

else if j=|B| then Lk := Ai; i := i + 1

else if Ai < Bj then Lk := Ai; i := i + 1

else Lk := Bj; j := j + 1

k := k+1return L

Merge Sort (Merge Sort ( 예제 예제 8) (2/2)8) (2/2)

Page 13Discrete Mathematicsby Yang-Sae Moon

Homework #5Homework #5

$3.1 의 연습문제 : 1, 3, 32

• A hint for 32: prove it by cases (n = 2k and n = 2k + 1)

$3.2 의 연습문제 : 2(c,d), 9(d,f), 18(b,d)

$3.3 의 연습문제 : 3, 8, 15

$3.4 의 연습문제 : 2(a,c), 8(b)

$3.5 의 연습문제 : 2, 4

Due Date:

3.5 Recursive Algorithms