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2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세

이산수학 (Discrete Mathematics)  알고리즘의 복잡도 (Algorithm Complexity). 2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세. Our focus. What is Algorithm Complexity?. Algorithm Complexity. The algorithmic complexity of a computation is some measure of how difficult it is to perform the computation.

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2014 년 봄학기 강원대학교 컴퓨터과학전공 문양세

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  1. 이산수학(Discrete Mathematics) 알고리즘의 복잡도(Algorithm Complexity) 2014년 봄학기 강원대학교 컴퓨터과학전공 문양세

  2. Our focus What is Algorithm Complexity? Algorithm Complexity • The algorithmic complexity of a computation is some measure of how difficult it is to perform the computation. (문제 계산(computation)이 얼마나 어려운가를 나타내는 측정치이다.) • Measures some aspect of cost of computation (in a general sense of cost). (계산을 위한 비용에 대한 측정치이다.) • Common complexity measures: • Time complexity: # of operations or steps required • Space complexity: # of memory bits required

  3. Complexity Depends on Input Algorithm Complexity Most algorithms have different complexities for inputs of different sizes. (E.g. searching a long list takes more time than searching a short one.) (대부분의 알고리즘은 입력의 크기에 따라 복잡도가 달라진다. 당연!) Therefore, complexity is usually expressed as a function of input length. (따라서, 복잡도는 입력의 크기/길이에 대한 함수로 표현한다.) This function usually gives the complexity for the worst-case input of any given length. (복잡도를 나타내는 함수는 통상 입력 크기가 최악인 경우를 고려한다.)

  4. Example: Max Algorithm Algorithm Complexity proceduremax(a1, a2, …, an: integers) v := a1 {largest element so far} fori := 2 ton {go thru rest of elems} ifai > vthen v := ai{found bigger?} {at this point v’s value is the same as the largest integer in the list} returnv Problem: Find the exact order of growth () of the worst-case time complexity of the max algorithm. Assume that each line of code takes some constant time every time it is executed.

  5. Complexity Analysis of Max Algorithm (1/2) Algorithm Complexity proceduremax(a1, a2, …, an: integers) v := a1 t1 fori := 2 ton t2 ifai > vthen v := ai t3 returnv t4 Times for each execution of each line. (각 line을 하나의 수행으로 볼 때의 시간) What’s an expression for the exact total worst-case time? (Not its order of growth.)(최악의 경우, 정확한 수행 시간을 어떻게 될까?)

  6. Complexity Analysis of Max Algorithm (2/2) Algorithm Complexity proceduremax(a1, a2, …, an: integers) v := a1 t1 fori := 2 ton t2 ifai > vthen v := ai t3 returnv t4 Times for each execution of each line. (각 line을 하나의 수행으로 볼 때의 시간) Worst case execution time:

  7. Example: Linear Search Algorithm Complexity procedurelinear search x: integer, a1, a2, …, an: distinct integers)i := 1 t1while (i n  x  ai) t2i:=i + 1 t3ifi n then location:=i t4elselocation:= 0 t5return location t6 Worst case: Best case: Average case (if item is present):

  8. (1) (1) (1) Example: Binary Search Algorithm Complexity procedurebinary search (x:integer, a1, a2, …, an: distinct integers) i := 1 j := nwhile i<j beginm := (i+j)/2 ifx>amtheni := m+1 else j := mend ifx = aithenlocation:=ielselocation:= 0returnlocation Key Question: How Many Loop Iterations?

  9. Binary Search Analysis Algorithm Complexity Suppose n=2k. Original range from i=1 to j=n contains n elements. Each iteration: Size ji+1 of range is cut in half. (매번 검색 범위(j-i+1)의 절반(½)씩 줄여 나간다.) Loop terminates when size of range is 1=20 (i=j). (검색 범위의 변화: 2k 2k-1  2k-2  …  21  20, 결국 반복 횟수 = k) Therefore, number of iterations is k = log2n = (log2n)= (log n) (반복 횟수 = k이고, log2n = log22k이므로…) Even for n2k (not an integral power of 2), time complexity is still (log2n) = (log n).

  10. Example: Bubble Sort Algorithm Complexity 1st pass 2nd pass 3 2 4 1 5 2 3 4 1 5 2 3 4 1 5 2 3 1 4 5 2 3 1 4 5 2 3 1 4 5 2 1 3 4 5 3rd pass 4th pass 2 1 3 4 5 1 2 3 4 5 1 2 3 4 5 Consider # of compare operations only! (n-1) + (n-2) + … + 2 + 1 = ((n-1)n)/2 = (n2)

  11. Example: Insertion Sort Algorithm Complexity 3 2 4 1 5 2 3 4 1 5 2 3 4 1 5 1 2 3 4 5 2 3 4 1 5 2 3 4 1 5 1 2 3 4 5 1 2 3 4 5 Also, consider # of compare operations only! 1 + 2 + … + (n-2) + (n-1) = ((n-1)n)/2 = (n2) Then, are all sorting algorithm’s complexities (n2)? NO! …, merge sort, heap sort, quick sort, …

  12. Names for Some Orders of Growth Algorithm Complexity (1) Constant (logn) Logarithmic (n) Linear (nlogn) Polylogarithmic (nc) Polynomial (cn), c>1 Exponential (n!) Factorial

  13. Tractable versus Intractable Algorithm Complexity • A problem or algorithm with at most polynomial time complexity is considered tractable (or feasible). P is the set of all tractable problems.(Polynomial 복잡도를 가지면, 풀만한(풀기 쉬운) 문제라 여기며, 이러한 문제들의 집합을 P라 나타낸다.) • A problem or algorithm that has more than polynomial complexity is considered intractable (or infeasible).(Polynomial 복잡도 이상이면 풀기 어려운 문제라 여긴다.) • But, note that • n1,000,000 is technically tractable, but really impossible. • clog log lognis technically intractable, but easy.

  14. Unsolvable Problems Algorithm Complexity • Alan Turing discovered in the 1930’s that there are problems unsolvable by any algorithm. (튜링은 “어떠한 알고리즘으로도 풀 수 없는 문제가 있음”을 밝혔다.) • Example: the halting problem. • Given an arbitrary algorithm and its input, will that algorithm eventually halt, or will it continue forever in an “infinite loop?” (주어진 알고리즘이 결국 정지하는지의 여부를 판단하는 문제는 결국 풀 수가 없다.)

  15. P versus NP (1/3) Algorithm Complexity • P is the set of all tractable problems. (i.e., the problems have at most polynomial time complexity.) (통상적으로 polynomial complexity를 가지면 P이다.) • NP is the set of problems for which there exists a tractable algorithm for checking solutions to see if they are correct. • In other words, NP is the set of decision problems for which a given proposed solution for a given input can be checked quickly (in polynomial time) to see if it really is a solution. • 주어진 입력에 대해 제시된 솔루션이 바른 해인지의 여부를 빠르게(poly-nomial time) 판단할 수 있는 알고리즘이 존재하는 문제들의 집합이 NP이다. • 통상적으로 exponential/factorial complexity를 가지면 NP이다.

  16. P versus NP (2/3) - skip Algorithm Complexity • NP-complete is the term used to describe decision problems that are the hardest ones in NP in the sense that, if there were a polynomial-bounded algorithm for an NP complete problem, then there would be a polynomial-bounded algorithm for each problem in NP. • NP중에서도 어려운 문제들이란 용어를 의미하며, 하나의 NP-complete 문제가 풀리면(P가 되면), 관련된 수많은 모든 NP-complete 문제가 동시에 풀리게 된다. • 많은 학자들은 어려운 많은 문제들에 대해서 NP-complete임을 증명(특정 문제가 다른 NP-complete 문제로 해석(변경)될 수 있음을 보임)하였다.

  17. P versus NP (3/3) - skip Algorithm Complexity We know PNP, but the most famous unproven conjecture in computer science is that this inclusion is proper (i.e., that PNP rather than P=NP). Whoever first proves it will be famous!

  18. Computer Time Examples Algorithm Complexity Assume time = 1 ns (109 second) per operation You should carefully design algorithms and write programs!

  19. Homework #3 Algorithm Complexity

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