Dynamic Programming

Mar 23, 2018Download as PPTX, PDF0 likes99 views

Sang Woon Park

AlCall 2017 Summer Semminar

Ch 09. 동적 계획법 테크닉
Dynamic Programming
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반복적 동적 계획법
9.21
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반복적 동적 계획법
Iterative Dynamic Programming
반복문(보통 for)을 쓰는 DP
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반복적인

반복적 동적 계획법
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의존성 파악이 쉬워야 하고
= 부분구조가 단순해야 한다

예제 : 삼각형 위의 최대 경로
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@algospot.com

예제 : 삼각형 위의 최대 경로
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@algospot.com

예제 : 삼각형 위의 최대 경로
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Path2(y,x) = triangle[y,x] + max
𝑝𝑎𝑡ℎ2(𝑦 + 1, 𝑥)
𝑝𝑎𝑡ℎ2(𝑦 + 1, 𝑥 + 1)
메모이제이션 : Path2(y,x) 함수를 호출 -> Memo[y, x]의 유무 확인 -> 결과값 반환 (Top-down)
반복적 DP : C[y,x]에 바로 값을 대입하여 사용
( = Path2(y,x)를 계산하기 위한 값들(Path2(y+1,x+1), Path2(y+1,x))이 이미 계산 되어있음. )
(Bottom-up)

예제 : 삼각형 위의 최대 경로
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슬라이딩 윈도우
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6
1 2
3 7 4
9 4 1 7
2 7 5 9 4
16 11 10 16
19 18 20
20 22
28
계산순서가 일정하므로 가능

피보나치 마지막 5자리 DP
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피보나치 마지막 5자리 DP
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𝐶𝑖 =
𝑓𝑖𝑏(𝑖 − 1)
𝑓𝑖𝑏(𝑖)
𝐶𝑖+1 =
𝑓𝑖𝑏(𝑖)
𝑓𝑖𝑏(𝑖 + 1)
=
𝑓𝑖𝑏(𝑖)
𝑓𝑖𝑏 𝑖 − 1 + 𝑓𝑖𝑏(𝑖)
=
𝐶𝑖,1
𝐶𝑖,0 + 𝐶𝑖,1
=
0 1
1 1
𝐶𝑖
𝐶 𝑛 =
0 1
1 1
𝑛−1
𝐶1

피보나치 마지막 5자리 DP
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𝑊 =
0 1
1 1
𝑊 𝑛
= 𝑊 𝑛/2
× 𝑊 𝑛/2
O(log n)

일반화
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• 𝑛 × 𝑚 행렬 A의 선형 결합으로 정의되는 DP 알고리즘
의 시간복잡도는O(𝑛𝑚2)
• 행렬의 빠른 거듭제곱을 사용하면 O(𝑚3 log 𝑛)
• M의 크기에 따라서 취사선택

재귀함수 vs 반복문
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• 재귀함수의 장단점
장점 : 직관적인 코드. 의존관계나 계산순서에 무관.
단점 : 공간낭비가 심함. Call Stack Overflow 발생.
• 반복문의 장단점
장점 : 짧은 코드. Call Stack 부담 X. 공간 절약(슬라이딩 윈
도)
단점 : 구현이 비직관적이고 계산순서에 영향을 많이 받음.

문제 : 회전초밥
9.22
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문제 : 회전초밥 풀이
9.22
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Knapsack
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• 배낭 채우기 문제와 유사
정해진 무게를 넘지 않으면서 최대 가치를 찾기
VS
정해진 예산를 넘지 않으면서 최대 선호도를 찾기

차이점
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• 중복된 초밥을 가져갈 수 있음
• M < 10^9 인데 필요한 DP 테이블 크기 : N x M
-> 슬라이딩 윈도우로 배열크기를 줄이는게 관건

슬라이딩 윈도우
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• T[i][j] = i번 초밥만큼 경우가 있을 때 j원으로 먹을 수 있는 최대 금액
테이블 크기 20 * 10^9 (20000MB)
• i:0->N
Prev[j] : i-1 번 초밥까지 먹었을 때 최대 선호도
T[j] : i번초밥까지 먹었을 때 최대 선호도
T[j] = max(T[j],T[j – price[i]]+ price[i]])
• 가격의 단위가 100이므로 100으로 나눠준 값을 index로 사용

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Dynamic Programming

1. Ch 09. 동적 계획법 테크닉 Dynamic Programming PNU ALCALL 2017 SUMMER 1

2. 반복적 동적 계획법 9.21 PNU ALCALL 2017 SUMMER 2

3. 반복적 동적 계획법 Iterative Dynamic Programming 반복문(보통 for)을 쓰는 DP PNU ALCALL 2017 SUMMER 3 반복적인

4. 반복적 동적 계획법 PNU ALCALL 2017 SUMMER 4 의존성 파악이 쉬워야 하고 = 부분구조가 단순해야 한다

5. 예제 : 삼각형 위의 최대 경로 PNU ALCALL 2017 SUMMER 5 @algospot.com

6. 예제 : 삼각형 위의 최대 경로 PNU ALCALL 2017 SUMMER 6 @algospot.com

7. 예제 : 삼각형 위의 최대 경로 PNU ALCALL 2017 SUMMER 7 Path2(y,x) = triangle[y,x] + max 𝑝𝑎𝑡ℎ2(𝑦 + 1, 𝑥) 𝑝𝑎𝑡ℎ2(𝑦 + 1, 𝑥 + 1) 메모이제이션 : Path2(y,x) 함수를 호출 -> Memo[y, x]의 유무 확인 -> 결과값 반환 (Top-down) 반복적 DP : C[y,x]에 바로 값을 대입하여 사용 ( = Path2(y,x)를 계산하기 위한 값들(Path2(y+1,x+1), Path2(y+1,x))이 이미 계산 되어있음. ) (Bottom-up)

8. 예제 : 삼각형 위의 최대 경로 PNU ALCALL 2017 SUMMER 8

9. 슬라이딩 윈도우 PNU ALCALL 2017 SUMMER 9 6 1 2 3 7 4 9 4 1 7 2 7 5 9 4 16 11 10 16 19 18 20 20 22 28 계산순서가 일정하므로 가능

10. 슬라이딩 윈도우 PNU ALCALL 2017 SUMMER 10

11. 피보나치 마지막 5자리 DP PNU ALCALL 2017 SUMMER 11

12. 피보나치 마지막 5자리 DP PNU ALCALL 2017 SUMMER 12 𝐶𝑖 = 𝑓𝑖𝑏(𝑖 − 1) 𝑓𝑖𝑏(𝑖) 𝐶𝑖+1 = 𝑓𝑖𝑏(𝑖) 𝑓𝑖𝑏(𝑖 + 1) = 𝑓𝑖𝑏(𝑖) 𝑓𝑖𝑏 𝑖 − 1 + 𝑓𝑖𝑏(𝑖) = 𝐶𝑖,1 𝐶𝑖,0 + 𝐶𝑖,1 = 0 1 1 1 𝐶𝑖 𝐶 𝑛 = 0 1 1 1 𝑛−1 𝐶1

13. 피보나치 마지막 5자리 DP PNU ALCALL 2017 SUMMER 13 𝑊 = 0 1 1 1 𝑊 𝑛 = 𝑊 𝑛/2 × 𝑊 𝑛/2 O(log n)

14. 일반화 PNU ALCALL 2017 SUMMER 14 • 𝑛 × 𝑚 행렬 A의 선형 결합으로 정의되는 DP 알고리즘 의 시간복잡도는O(𝑛𝑚2) • 행렬의 빠른 거듭제곱을 사용하면 O(𝑚3 log 𝑛) • M의 크기에 따라서 취사선택

15. 재귀함수 vs 반복문 PNU ALCALL 2017 SUMMER 15 • 재귀함수의 장단점 장점 : 직관적인 코드. 의존관계나 계산순서에 무관. 단점 : 공간낭비가 심함. Call Stack Overflow 발생. • 반복문의 장단점 장점 : 짧은 코드. Call Stack 부담 X. 공간 절약(슬라이딩 윈 도) 단점 : 구현이 비직관적이고 계산순서에 영향을 많이 받음.

16. 문제 : 회전초밥 9.22 PNU ALCALL 2017 SUMMER 16

17. 문제 설명 PNU ALCALL 2017 SUMMER 17

18. 문제 설명 PNU ALCALL 2017 SUMMER 18

19. 문제 : 회전초밥 풀이 9.22 PNU ALCALL 2017 SUMMER 19

20. Knapsack PNU ALCALL 2017 SUMMER 20 • 배낭 채우기 문제와 유사 정해진 무게를 넘지 않으면서 최대 가치를 찾기 VS 정해진 예산를 넘지 않으면서 최대 선호도를 찾기

21. 차이점 PNU ALCALL 2017 SUMMER 21 • 중복된 초밥을 가져갈 수 있음 • M < 10^9 인데 필요한 DP 테이블 크기 : N x M -> 슬라이딩 윈도우로 배열크기를 줄이는게 관건

22. 슬라이딩 윈도우 PNU ALCALL 2017 SUMMER 22 • T[i][j] = i번 초밥만큼 경우가 있을 때 j원으로 먹을 수 있는 최대 금액 테이블 크기 20 * 10^9 (20000MB) • i:0->N Prev[j] : i-1 번 초밥까지 먹었을 때 최대 선호도 T[j] : i번초밥까지 먹었을 때 최대 선호도 T[j] = max(T[j],T[j – price[i]]+ price[i]]) • 가격의 단위가 100이므로 100으로 나눠준 값을 index로 사용

23. 소스코드 PNU ALCALL 2017 SUMMER 23

24. PNU ALCALL 2017 SUMMER 24 Q&A FIN

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